diff --git "a/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt" "b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/ANE5T4oBgHgl3EQfSg9u/content/tmp_files/load_file.txt" @@ -0,0 +1,1083 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf,len=1082 +page_content='UNIFORM GLOBAL STABILITY OF SWITCHED NONLINEAR SYSTEMS IN THE KOOPMAN OPERATOR FRAMEWORK∗ CHRISTIAN MUGISHO ZAGABE† AND ALEXANDRE MAUROY ‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we provide a novel solution to an open problem on the global uniform stability of switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results are based on the Koopman operator approach and, to our knowledge, this is the first theoretical contribution to an open problem within that framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk, we define equivalent linear (but infinite-dimensional) switched systems and we construct a common Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by the linearized vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A common Lyapunov function for the original switched nonlinear systems is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Lyapunov function is shown to converge in a bounded region of the state space, which proves global uniform stability of specific switched nonlinear systems on bounded invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator, Hardy space on the polydisk, Switched systems, Uniform stability, Common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 47B32, 47B33, 47D06, 70K20, 93C10, 93D05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Switched systems are hybrid-type models encountered in ap- plications where the dynamics abruptly jump from one behavior to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They are typically described by a family of subsystems that alternate according to a given commutation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Stability properties of switched systems have been the focus of intense research effort (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [32] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this context, a natural question is whether a switched system with an equilibrium point is uniformly stable, that is, stable for any commutation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It turned out that the uniform stability problem is counter-intuitive and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the linear case, it is well-known that stable subsystems may induce an unstable switched system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, uniform stability is guaranteed if the matrices associated with the subsystems are stable and commute pairwise [24], a result which is extended in [15] to subsystems described by stable matrices generating a solvable Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This latter result can be explained by the well-known equivalence between solvable Lie algebra of matrices and the existence of a common invariant flag for those matrices, which allows to construct a common Lyapunov function for the subsystems [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of switched nonlinear systems, an open problem was posed in [13] on the relevance of Lie-algebraic conditions of vector fields for global uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Partial solutions have been proposed in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It was proven in [17] that uniform stability holds if the vector fields are individually stable and commute, in which case a common Lyapunov function can be constructed [30, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Uniform stability was also shown for a pair of vector fields generating a third-order nilpotent Lie algebra [29] and for particular r-order nilpotent Lie algebras [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, no result has been obtained, which solely relies on the more general solvability property of Lie algebras of the subsystems vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we provide a partial solution to the problem introduced in [13] by ∗Submitted to the editors †Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- versity of Namur (christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='mugisho@unamur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='be), ‡Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni- versity of Namur (alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='mauroy@unamur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='be) 1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='05529v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='DS] 13 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY proving global uniform stability results for switched nonlinear systems under a gen- eral solvability property of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To do so, we rely on the Koopman operator framework [3, 21]: we depart from the classical pointwise description of dynami- cal systems and consider instead the evolution of observable functions (here in the Hardy space of holomorphic functions defined on the complex polydisk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Through this approach, equivalent infinite-dimensional dynamics are generated by linear Koopman generators, so that nonlinear systems are represented by Koopman linear systems that are amenable to global stability analysis [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, building on preliminary results obtained in [34], we construct a common Lyapunov functional for switched Koopman linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A key point is to focus on the adjoint of the Koopman generators and notice that these operators have a common invariant maximal flag if the linear parts of the subsystems generate a solvable Lie algebra, a condition that is milder than the original assumption proposed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, we derive a common Lyapunov function for the original switched nonlinear system and prove its conver- gence under specific algebraic conditions on the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This allows us to obtain a bounded invariant region where the switched nonlinear system is globally uniformly asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To our knowledge, this is the first time that a novel solution to an open theoretical problem is obtained within the Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In Section 2, we present some preliminary notions on uniform stability of switched nonlinear systems and give a general introduction to the Koopman operator framework, as well as some specific properties in the Hardy space on the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In Section 3, we state and prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We recast the open problem given in [13] in terms of the existence of an invariant maximal flag and we provide a constructive proof for the existence of a common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Additional corollaries are also given, which focus on specific classes of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our main results are illustrated with two examples in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, concluding remarks and perspectives are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will use the following notation throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For multi-index notations α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', αn) ∈ Nn, we define |α| = α1 + · · · + αn and zα = zα1 1 · · · zαn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The complex conjugate and real part of a complex number a are denoted by ¯a and ℜ(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The transpose-conjugate of a matrix (or vector) A is denoted by A†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Jacobian matrix of the vector field F at x is given by JF(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The complex polydisk centered at 0 and of radius ρ is defined by Dn(0, ρ) = {z ∈ Cn : |z1| < ρ, · · · , |zn| < ρ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, Dn denotes the unit polydisk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' with ρ = 1) and ∂Dn is its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, the floor of a real number is denoted by ⌊x⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this section, we introduce preliminary notions and results on the stability theory for switched systems and on the Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Stability of switched systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We focus on the uniform asymptotic sta- bility property of switched systems and on the existence of a common Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Some existing results that connect these two main concepts are presented in both linear and nonlinear cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 (Switched system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A switched system ˙x = F (σ)(x) is a (finite) set of subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) � ˙x = F (i)(x), x ∈ X ⊂ Rn�m i=1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 3 associated with a commutation law σ : R+ → {1, · · · , m} indicating which subsystem is activated at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we make the following standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The commutation law σ is a piecewise constant function with a finite number of discontinuities on every bounded time interval (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to [16], stability analysis of switched sys- tems revolves around three important problems: decide whether an equilibrium is stable under the action of the switched system for any commutation law σ, in which case the equilibrium is said to be uniformly stable, identify the commutation laws for which the equilibrium is stable, and construct the commutation law for which the equilibrium is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper we focus on the first problem related to uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 (Uniform stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that F (i)(xe) = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The equilibrium xe is uniformly asymptotically stable (UAS) if ∀ϵ > 0, ∃δ > 0 such that ∥x(0) − xe∥ ≤ δ ⇒ ∥x(t) − xe∥ ≤ ϵ, ∀t > 0, ∀σ and ∥x(0) − xe∥ ≤ δ ⇒ lim t→∞ x(t) = xe, ∀σ, globally uniformly asymptotically stable (GUAS) on D ⊆ Rn if it is UAS and x(0) ∈ D ⇒ lim t→∞ x(t) = xe, ∀σ, globally uniformly exponentially stable (GUES) on D ⊆ Rn if ∃β, λ > 0 such that x(0) ∈ D ⇒ ∥x(t) − xe∥ ≤ β∥x(0) − xe∥e−λt, ∀t > 0, ∀σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This definition implies that the subsystems share a common equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' More- over, a necessary condition is that this equilibrium is asymptotically stable with re- spect to the dynamics of all individual subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, this condition is not sufficient, since the switched system might be unstable for a specific switching law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A sufficient condition for uniform asymptotic stability is the existence of a common Lyapunov function (CLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3 (Common Lyapunov function [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A positive C1- function V : D ⊆ Rn → R is a common Lyapunov function on D ⊆ Rn for the family of subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) if ∇V · F (i)(x) < 0 ∀x ∈ D \\ {xe}, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For switched systems with a finite number of subsystems, a converse Lyapunov result also holds ([12], [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that D ⊆ Rn is compact and forward-invariant with respect to the flow induced by the subsystems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The switched system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS on D if and only if all subsystems share a CLF on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY A corollary of this result provides a necessary condition for GUAS, which is based on convex combinations of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the equilibrium of the switched system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS, then it is a globally asymptotically stable equilibrium for the dynamics ˙x = αF (i)(x) + (1 − α)F (j)(x), for all i, j ∈ {1, · · · , m} and for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lie-algebraic conditions in the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of switched linear systems { ˙x = A(i)x, A(i) ∈ Cn×n}m i=1, several results related to uniform stability have been proved (see [32] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We focus here on specific results based on Lie-algebraic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let g = span � A(i)� Lie denote the Lie algebra generated by the matrices A(i), with i = 1, · · · , m, and equipped with the Lie bracket [A(i), A(j)] = A(i)A(j)−A(j)A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 (Solvable Lie algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A Lie algebra g equipped with the Lie bracket [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='] is said to be solvable if there exists k ∈ N such that gk = 0, where {gj}j∈N∗ is a descendant sequence of ideals defined by � g1 := g gj+1 := � gj, gj� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A general Lie-algebraic criterion for uniform exponential (asymptotic) stability of switched linear systems is given in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If all matrices A(i), i = 1, · · · , m, are stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' with eigenvalues λ(i) j such that ℜ � λ(i) j � < 0) and if the Lie algebra g is solvable, then the switched linear system { ˙x = A(i)x}m i=1 is GUES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' As shown in [23, 31], this result follows from the simultaneous triangularization of the matrices A(i), which is a well-known property of solvable Lie algebras (see Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This property is in fact equivalent to the existence of a common invariant flag for complex matrices [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 (Invariant flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' An invariant maximal flag of the set of matrices {A(i)}m i=1 is a set of subspaces {Sj}n j=1 ⊆ Cn such that (i) A(i)Sj ⊂ Sj for all i, j, (ii) dim(Sj) = j for all j, and (iii) Sj ⊂ Sj+1 for all j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The subspaces Sj can be described through an orthonormal basis (v1, · · · , vn), so that Sj = span {v1, · · · , vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that the vector v1 is a common eigenvector of the matrices A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This basis can be used to construct a CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 ([34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) � ˙x = A(i) x, A(i) ∈ Cn×n, x ∈ Cn�m i=1 be a switched linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that all matrices A(i) are stable and admit a common invariant maximal flag {0} ⊂ S1 ⊂ · · · ⊂ Sn = Cn, Sj = span{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 5 Then there exist ϵj > 0, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , n, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) V (x) = n � j=1 ϵj|v† jx|2 is a CLF for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The values ϵj must satisfy the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) ϵj > max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',m} k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1} ϵk (n − 1)2 4 ���v† kA(i)vj ��� 2 ���ℜ � λ(i) j ���� ���ℜ � λ(i) k ���� where λ(i) j are the eigenvalues of A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They can be obtained iteratively from an arbitrary value ϵ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The geometric approach followed in [34] provides a constructive way to obtain a CLF, a result that we will leverage in an infinite-dimensional setting for switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lie-algebraic condition in the nonlinear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the context of switched nonlinear systems, one has to consider the Lie algebra of vector fields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) gF = span � F (i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m � Lie equipped with the Lie bracket (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) [F (i), F (j)](x) = JF (j)(x) F (i)(x) − JF (i)(x) F (j)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It has been conjectured in [13] that Lie-algebraic conditions on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) could be used to characterize uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This problem has been solved partially in [29] for third-order nilpotent Lie algebras and in [18] for particular r-order nilpotent Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Another step toward more general Lie-algebraic conditions based on solvability has been made in [34], a preliminary result that relies on the so-called Koopman operator framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, the results obtained in [34] are restricted to specific switched nonlinear systems that can be represented as finite-dimensional linear ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this paper, we build on this preliminary work, further exploiting the Koopman operator framework to obtain general conditions that characterize the GUAS property of switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator approach to dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this section, we present the Koopman operator framework, which is key to extend the result of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 to switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We introduce the Koopman semigroup along with its Koopman generator, cast the framework in the context of Lie groups, and describe the finite-dimensional approximation of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider a continuous-time dynamical system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) ˙x = F(x), x ∈ X ⊂ Rn, F ∈ C1 which generates a flow ϕt : X → X, with t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Koopman operator is defined on a (Banach) space F and acts on observables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' functions f : X → R, f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10 (Koopman semigroup [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The semigroup of Koopman opera- tors (in short, Koopman semigroup) is the family of linear operators (Ut)t≥0 defined by Ut : F → F, Utf = f ◦ ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY We can also define the associated Koopman generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11 (Koopman generator [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Koopman generator associated with the vector field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is the linear operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) LF : D(LF ) → F, LF f := F · ∇f with the domain D(LF ) = {f ∈ F : F · ∇f ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' As shown below (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13), the Koopman semigroup and the Koopman gen- erators are directly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When the Koopman semigroup is strongly continuous [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' lim t→0+ ∥Utf − f∥F = 0, the Koopman generator is the infinitesimal generator LF f := lim t→0+(Utf −f)/t of the Koopman semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the Koopman operator Ut and the generator LF are both linear, we can describe the dynamics of an observable f on F through the linear abstract ordinary differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) ˙f = LF f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We can also briefly discuss the spectral properties of the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12 (Koopman eigenfunction and eigenvalue [3, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' An eigenfunc- tion of the Koopman operator is an observable φλ ∈ F \\ {0} such that LF φλ = λφλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The value λ ∈ C is the associated Koopman eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Under the strong continuity property, the Koopman eigenfunction also satisfies Utφλ = eλtφλ, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For a linear system ˙x = Ax, with x ∈ Rn, we denote an eigenvalue of A by ˜λj and its associated left eigenvector by wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then ˜λj is a Koopman eigenvalue and the associated Koopman eigenfunction is given by φ˜λj(x) = w† jx [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For a nonlinear system of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) which admits a stable equilibrium xe, the eigenvalues of JF(xe) are typically Koopman eigenvalues and the associated eigenfunctions are the so-called principal Koopman eigenfunctions (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman operator in the Hardy space H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' From this point on, we define the Koopman operator in the Hardy space on the polydisk (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' [25, 26, 28] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This choice is well-suited to the case of analytic vector fields that admit a stable hyperbolic equilibrium, where it allows to exploit convenient spectral properties of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let D be the open unit disk in C, ∂D its boundary, and Dn the unit polydisk in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Hardy space of holomorphic functions on Dn is the space H2(Dn) = � f : Dn → C, holomorphic : ∥f∥2 = lim r→1− � (∂D)n |f (rω) |2dmn(ω) < ∞ � , where mn is the normalized Lebesgue measure on (∂D)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is equipped with an inner product defined by ⟨f, g⟩ = � (∂D)n f (ω) ¯g (ω) dmn(ω), This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 7 so that the set {zα : α ∈ Nn} is the standard orthonormal basis of monomials on H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The monomials will be denoted by ek(z) = zα(k), where the map α : N → Nn, k �→ α(k) refers to the lexicographic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ek1 < ek2 if |α(k1)| < |α(k2)|, or if |α(k1)| = |α(k2)| and αj(k1) > αj(k2) for the smallest j such that αj(k1) ̸= αj(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For f and g in H2(Dn), with f = � k∈N fkek and g = � k∈N gkek, the isomorphism � k∈N fkek �→ (fk)k≥0 between H2(Dn) and the l2-space allows to rewrite the norm and the inner product as ∥f∥2 = � k∈N |fk|2 and ⟨f, g⟩ = � k∈N fk ¯gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We also note that H2(Dn) is a reproducing kernel Hilbert space (RKHS) with the Cauchy kernel ([25, Chapter 1]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) k (z, ξ) = n � i=1 1 1 − ¯ξizi , z, ξ ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that one can define the evaluation functional f(z) = ⟨f, kz⟩ with kz(ω) = k (z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the vector field F is analytic, we can consider its analytic continuation on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, if it generates a holomorphic flow that is invariant in Dn, we can define the Koopman semigroup on H2(Dn), which is also known as the composition operator with symbol ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The required assumptions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The components Fl, l = 1, · · · , n, of the vector field F belong to the Hardy space H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, F generates a flow which is holomorphic and maps Dn to Dn (forward invariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is shown in [4] that the flow ϕt is holomorphic on Dn if and only if the vector field components have a specific form (see Proposition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) in the Appendix, and the works [4, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note also that this property holds if the dynamics possess a globally stable hyperbolic equilibrium (Assumption 3 below) in the case of non-resonant eigenvalues (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Now, we recall some important properties that we will use to prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider a function f ∈ H2(Dn) and an evaluation functional kz, with z ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' LF zα ∈ H2(Dn) and the domain D (LF ) is dense in H2(Dn), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' U ∗ t kz = kϕt(z), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' d dt ⟨U ∗ t kz, f⟩ = ⟨L∗ F U ∗ t kz, f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all z ∈ Dn, we have LF zα = F(z) · ∇zk = n � l=1 Fl(z) αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ∥fzα∥ = ∥f∥ for all f ∈ H2(Dn) and for all α ∈ Nn, it follows from Assumption 2 that ∥LF eα∥ = ����� n � l=1 αlFl ����� ≤ n � l=1 |αl| ∥Fl∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover D (LF ) is dense in H2(Dn) since the monomials zα form a complete basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all f ∈ H2(Dn), we have ⟨U ∗ t kz, f⟩ = ⟨kz, Utf⟩ = (Utf) (z) and � kϕt(z), f � = f (ϕt(z)) = (Utf) (z), so that U ∗ t kz = kϕt(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all z ∈ Dn and all f ∈ D(LF ), d dt ⟨U ∗ t kz, f⟩ = d dt � kϕt(z), f � = d dtf ◦ ϕt(z) = F (ϕt(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='∇f (ϕt(z)) = � kϕt(z), LF f � = ⟨L∗ F U ∗ t kz, f⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows for all f since D(LF ) is dense in H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the previous lemma, the second property is a well-known property of the composi- tion operator on a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The third property is also known in the context of strongly continuous semigroup theory (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, we make the following additional standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The vector field F admits on Dn a unique hyperbolic stable equi- librium at 0 (without loss of generality), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' F(0) = 0 and the eigenvalues ˜λj of the Jacobian matrix JF(0) satisfy ℜ{˜λj} < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14 (Holomorphic flow and spectral properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If Assumption 3 holds and if the eigenvalues ˜λj are non-resonant1, then the Poincar´e linearization theorem [2] implies that the flow ϕt is topologically conjugated to the linear flow ˜ϕt(z) = eJF (0)tz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' there exists a bi-holomorphic map h such that ϕt = h−1 ◦ ˜ϕt ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, the flow ϕt is clearly holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the components of h are associated with holomorphic Koopman eigenfunctions φ˜λj ∈ H2(Dn) associated with the eigenvalues ˜λj [9, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' These eigenfunctions are called principal eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, it can easily be shown that, for all α ∈ Nn, n � j=1 αj˜λj is a Koopman eigenvalue associated with the eigenfunction φα1 ˜λ1 · · · φαn ˜λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Koopman infinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since H2(Dn) is isomorphic to l2, the Koop- man generator can be represented by the Koopman infinite matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) ¯LF = � � � � � � � � � ⟨LF e0, e0⟩ ⟨LF e0, e1⟩ ⟨LF e0, e2⟩ ⟨LF e0, e3⟩ · · ⟨LF e1, e0⟩ ⟨LF e1, e1⟩ ⟨LF e1, e2⟩ ⟨LF e1, e3⟩ · · ⟨LF e2, e0⟩ ⟨LF e2, e1⟩ ⟨LF e2, e2⟩ ⟨LF e2, e3⟩ · · ⟨LF e3, e0⟩ ⟨LF e3, e1⟩ ⟨LF e3, e2⟩ ⟨LF e3, e3⟩ · · ⟨LF e4, e0⟩ ⟨LF e4, e1⟩ ⟨LF e4, e2⟩ ⟨LF e4, e3⟩ · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' · · � � � � � � � � � , 1The eigenvalues ˜λj are non-resonant if n � j=1 αj ˜λj = 0 with α ∈ Zn implies that α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 9 where the kth row contains the components of LF ek in the basis of monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For f = � k∈N fkek, we also have that ⟨LF f, ej⟩ = � k∈N fk⟨LF ek, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that, since LF e0 = 0, the first row and column of ¯LF contains only zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By removing the first row and column, one obtains the representation of the restriction of the Koopman generator to the subspace of functions f that satisfy f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This subspace is spanned by the basis (ek)k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that kz − k0 belongs to this subspace, since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) implies that kz(0) − k0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For Fl(z) = � |β|≥1 al,βzβ, the action of the Koopman operator on a basis element is given by LF zα = n � l=1 Fl(z) αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn) = n � l=1 � |β|≥1 al,βzβ αl z(α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αl−1,αl−1,αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',αn) = n � l=1 αl � � � (β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn)∈Nn al,(β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn)z(β1+α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βl+αl−1,βl+1+αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',βn+αn) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By setting γ1 = β1 + α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , γl = βl + αl − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , γn = βn + αn we obtain LF zα = n � l=1 αl � |γ|≥|α| al,(γ1−α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γl−αl+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γn−αn)z(γ1,γ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',γn) = n � l=1 αl � |γ|≥|α| al,(γ−α)lzγ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) where we denote (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) (γ − α)l = (γ1 − α1, · · · , γl − αl + 1, · · · , γn − αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that the entries of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) ⟨LF ek, ej⟩ = � � � � � n � l=1 αl(k) al,(α(j)−α(k))l if |α(j)| ≥ |α(k)| 0 if |α(j)| < |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the linear part of the vector field F, where |α(j)| = 1, j = 1, · · · , n, it is clear that α(j) is the canonical basis vector of Cn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' αi(j) = δij, and we have that a(l) α(j) = [JF(0)]lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, if |α(j)| = |α(k)|, we have that (α(j)−α(k))l = α(r) for some r ≤ n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' |α(r)| = 1), with αr(j) = αr(k) + 1, αl(j) = αl(k) − 1, and This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY αi(j) = αi(k) for all i /∈ {l, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) ⟨LF ek, ej⟩ = � � � � � � � � � n � l=1 αl(j) [JF(0)]ll if j = k αl(k) [JF(0)]lr if α(j) = (α1(k), · · · , αl(k) − 1, · · · , αr(k) + 1, · · · , αn(k)), 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Switched Koopman systems and Lie-algebraic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case of a switched nonlinear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1), the Koopman operator description yields a switched linear infinite-dimensional system (in short, switched Koopman system) of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) � ˙f = LF (i)f, f ∈ D �m i=1 with D = ∩m i=1D(LF (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Similarly, the Lie algebra gF spanned by F (i) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5)) is replaced by gL = span {LF (i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m}Lie, equipped with the Lie bracket [LF (i), LF (j)] = LF (i)LF (j) − LF (j)LF (i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, we have the well-known relationship (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) [LF (i), LF (j)] = L[F (i),F (j)] so that the two algebras gF and gL are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that Lie-algebraic conditions in gF can be recast into Lie-algebraic criteria in gL, a framework where we can expect to obtain new results on switched systems that are reminiscent to the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In particular, since the solvability property of gF is equivalent to the solvability property of gL, we will investigate whether this latter condition implies the existence of a common Lyapunov functional for the switched Koopman system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This section presents our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We first use an illus- trative example to show that Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 cannot be used for nonlinear vector fields, in contrast to the linear case (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We then relax the algebraic conditions suggested in [13] in order to obtain a triangular form in the Koopman matrix representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11), a property which is equivalent to the existence of an in- variant flag for the adjoint operator L∗ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We finally prove uniform stability of switched nonlinear systems under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A first remark on the existence of the common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following example shows that Lie’s theorem does not hold for infinite-dimensional switched Koopman systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the two vector fields F (1)(x1, x2) = (−αx1, −αx2) and F (2)(x1, x2) = (−βx1+γ � x2 1 − x2 2 � , −βx2+2γx1x2), where α, β and γ are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' These two vector fields generate the Lie alge- bra g = span � F (1), F (2), F (3)� Lie with F (3)(x1, x2) = (αγ(x2 1 − x2 2), 2αγx1x2) since [F (1), F (2)] = F (3), [F (1), F (3)] = αF (3) and [F (2), F (3)] = βF (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, one has g1 = [g, g] = span � F (3)� Lie and g2 = [g1, g1] = 0, which implies that g is a This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 11 solvable Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, the Koopman generators LF (1) and LF (2) associated with the two vector fields do not share a common eigenfunction, and therefore can- not have a common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, the principal eigenfunctions of LF (1) are φ˜λ(1) 1 (x1, x2) = x1 and φ˜λ(1) 2 (x1, x2) = x2, while those of LF (2) are given by φ˜λ(2) 1 (x1, x2) = βγ � βx1 − γ � x2 1 − x2 2 �� (β − γx1)2 + γ2x2 2 and φ˜λ(2) 2 (x1, x2) = β2γx2 (β − γx1)2 + γ2x2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We conclude that Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 does not hold setting for the above example, so that we cannot directly extend Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The two Koopman generators are not simultaneous triangularizable and do not have a common invariant flag (see [10] for more details about simultaneous triangularization of operators and its connection to the existence of an invariant infinite maximal flag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, it can be easily seen that the Koopman infinite matrices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) related to the vector fields F (1) and F (2) are both upper triangular, and therefore admit a common infinite invariant maximal flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In fact, this implies that the adjoint operators L∗ F (i) have a common invariant flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For this reason, we will depart from the solvability condition on vector fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' on Koopman generators), and we will deal with simultaneous triangularization of adjoints of Koopman generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following result provides a sufficient condition on the vector fields for the simultaneous triangularization of ad- joints of Koopman generators, which appears to be less restrictive than the solvability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let F be an analytic vector field on Dn such that the Jacobian matrix JF(0) is upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then the Koopman matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) is upper triangular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ⟨LF ek, ej⟩ = 0 for all k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the adjoint L∗ F of the Koopman generator admits an infinite invariant maximal flag generated by the monomials ek, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Sk = span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , ek}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that ⟨LF ek, ej⟩ = 0 if |α(k)| > |α(j)| (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' the Koop- man matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) is always upper triangular by matrix blocks related to monomials of the same total degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case |α(k)| = |α(j)| with k > j, the lexicographic order implies that one can have α(j) = (α1(k), · · · , αl(k)−1, · · · , αr(k)+1, · · · , αn(k)) only with r < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since [JF(0)]lr = 0 for all l > r, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) that ⟨LF ek, ej⟩ = 0 when k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, it is clear that L∗ F ej ∈ span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , ej} since ⟨ek, L∗ F ej⟩ = 0 for all k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When the Jacobian matrix is upper triangular, it is well-known that [JF(0)]jj = ˜λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) that the diagonal entries of the (upper triangular) Koopman matrix are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) ⟨LF ej, ej⟩ = n � l=1 αl(j)˜λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since these values are the Koopman eigenvalues in the case of non-resonant eigenvalues ˜λj (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), we will denote λj = ⟨LF ej, ej⟩ by a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let � F (i)�m i=1 be a switched nonlinear system on Dn and sup- pose that the Lie algebra of matrices span � JF (i)(0) � Lie is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then there ex- ists a change of variables z �→ �z = P −1z on Cn such that the adjoint operators L∗ � F (i) of the Koopman generators (with �F (i)(�z) = P −1F (i)(P �z)) admit a common in- This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY finite invariant maximal flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, � �F (i)�m i=1 is a switched nonlinear system on Dn � 0, ∥P −1∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since span � JF (i)(0) � Lie is solvable, Lie’s theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 implies that the matrices JF (i)(0) are simultaneously triangularizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' there exists a matrix P such that JF (i)(0) = PT (i)P −1 for all i, where T (i) is upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let set F (i)(z) = JF (i)(0)z + ˜F (i)(z) to separate the linear and the nonlinear parts of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the new coordinates �z = P −1z, we obtain the dynamics �F (i)(�z) = P −1JF (i)(0)P �z + P −1 ˜Fi(P �z) = T (i)�z + �˜F i(�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 that monomials �ek, with �ek(�z) = (�z)α(k), generate a common invariant maximal flag for L∗ � F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In addition, for all z ∈ Dn and all j = 1, · · · , n, we have |�zj| ≤ ∥�z∥∞ = ��P −1z �� ∞ ≤ ��P −1�� ∞ ∥z∥∞ < ∥P −1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear that the change of coordinates z �→ �z = P −1z is defined up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Without loss of generality, we will consider in the sequel that ∥P −1∥∞ = 1, so that � �F (i)�m i=1 is a switched nonlinear system on the unit polydisk Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Instead of a nilpotency or solvability condition on the vector fields F (i), we only require a milder solvability condition on the Jacobian matrices JF (i)(xe) to guarantee the triangular form of the Koopman matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is noticeable that this local condition is much less restrictive than the global solvability condition mentioned in the original open problem [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Also, it was shown in [1] that the triangular form of the vector fields (and therefore of the Jacobian matrices) is not sufficient to guarantee the GUAS property of a switched nonlinear system on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the next section, however, we use the solvability condition on the Jacobian matrices to prove the GUAS property in a bounded invariant region of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This result is consistent with the local stability result derived in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' A common Lyapunov function for switched nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We now aim to show that, for some positive sequence (ϵk)∞ k=1, the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) V(f) = ∞ � k=1 ϵk |⟨f, ek⟩|2 is a Lyapunov functional for the switched Koopman system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Before starting our main result, we need a few lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let ˙z = F(z) be a vector field on the polydisk Dn which generates a flow ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that there exist a sequence of positive numbers (ϵk)k≥1 and ρ ∈]0, 1] such that Dn(0, ρ) is forward invariant with respect to ϕt and such that the series � k≥1 |α(k)|ϵkρ2|α(k)| is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) V(kz − k0) = ∞ � k=1 ϵk |⟨kz, ek⟩|2 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 13 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) ∞ � k=1 ϵk d dt |⟨U ∗ t (kz − k0), ek⟩|2 are absolutely and uniformly convergent on Dn(0, ρ) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the first series, we have ϵk |⟨kz − k0, ek⟩|2 = ϵk |⟨kz, ek⟩|2 = ϵk ���zα(k)��� 2 < ϵkρ2|α(k)| ≤ |α(k)|ϵkρ2|α(k)| for all z ∈ Dn(0, ρ) and all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the second series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' we have ϵk ���� d dt |⟨U ∗ t (kz − k0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ek⟩|2 ���� = ϵk ���� d dt ���(ϕt(z))α(k) − (ϕt(0))α(k)��� 2���� = 2ϵk ����ℜ � d dt (ϕt(z))α(k) · � ϕt(z) �α(k)����� ≤ 2ϵk ���� d dt (ϕt(z))α(k) ���� · ���� � ϕt(z) �α(k)���� < 2ϵkρ|α(k)| ���� d dt (ϕt(z))α(k) ���� ≤ 2ϵkρ|α(k)| n � s=1 αs(k) ���F (s) (ϕt(z)) ��� ��(ϕt(z))s ��αs(k)−1 n � l=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='l̸=s ��(ϕt(z))l ��αl(k) < 2ϵkρ2|α(k)|−1 n � s=1 αs(k) ���F (s) (ϕt(z)) ��� for all z ∈ Dn(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' t > 0 and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By using the maximum modulus principle for bounded domains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 with the holomorphic function F (s) ◦ ϕt, we can denote M = max z∈∂Dn,s=1,··· ,n |(F (s) ◦ ϕt)(z)| and we obtain ϵk ���� d dt |⟨U ∗ t (kz − k0), ek⟩|2 ���� < 2M|α(k)|ϵkρ2|α(k)|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, absolute and uniform convergence of both series follow from the Weierstrass test (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let ˙z = F(z) be a nonlinear system on Dn with an upper triangular Jacobian matrix JF(0) and let ˙f = LF f be its corresponding Koopman system on D (LF ) ⊂ H2(Dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely and uniformly convergent in Dn for all t > 0, then, for all double sequences of positive real numbers (bjk)j≥1,k≥1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5) ∞ � k=1 bjk ≤ 1, This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY one has d dtV (U ∗ t (kz − k0)) ≤2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 � bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) with cj = ⟨U ∗ t (kz − k0), ej⟩ and λj = ⟨LF ej, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Suppose that z ∈ Dn is such that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely and uniformly convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13 (3), we obtain d dtϵk |⟨U ∗ t (kz − k0), ek⟩|2 = 2ϵkℜ �� d dtU ∗ t (kz − k0), ek � ⟨U ∗ t (kz − k0), ek⟩ � = 2ϵkℜ � ⟨L∗ F U ∗ t (kz − k0), ek⟩ ⟨U ∗ t (kz − k0), ek⟩ � = 2ϵk ∞ � j=1 ℜ (cj¯ck ⟨ej, LF ek⟩) where we used the decomposition U ∗ t (kz − k0) = ∞ � j=1 cjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) is absolutely and uniformly convergent, term by term derivation yields d dtV (U ∗ t (kz − k0)) = ∞ � k=1 d dtϵk |⟨U ∗ t (kz − k0), ek⟩|2 = 2 ∞ � k=1 ∞ � j=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 ϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) where we used the triangular form of ¯LF (which follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 since JF(0) is triangular) and λj = ⟨LF ej, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5), we have d dtV (U ∗ t (kz − k0)) ≤ 2 ∞ � j=1 � � j � k=1 bjk + ∞ � k=j+1 bjk � � ϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 bjkϵj |cj|2 ℜ (λj) + 2 ∞ � j=1 ∞ � k=j+1 bjkϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 ϵkℜ (cj¯ck ⟨ej, LF ek⟩) = 2 ∞ � j=1 bjjϵj |cj|2 ℜ (λj) + 2 ∞ � j=2 j−1 � k=1 � bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 15 Under Assumption 3, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) that ℜ{λj} < 0 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, the time derivative (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6) of the Lyapunov functional is negative if negative terms related to the diagonal entries ⟨LF ej, ej⟩ = λj and ⟨LF ek, ek⟩ = λk compensate (possibly positive) cross-terms related to ⟨ej, LF ek⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that a term associated with a diagonal entry will be used to compensate an infinity of cross-terms (associated with entries in the corresponding row and column of the Koopman matrix), and the values bjk play the role of weights in the compensation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We are now in position to state our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn and assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on the polydisk Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that P −1JF (i)(0)P are upper triangular), there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to the flows ϕt i of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider double sequences of positive real numbers � b(i) jk � j≥1,k≥1, with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m, such that b(i) jk b(i) kj > 0 if ⟨L � F (i)�ek, �ej⟩ ̸= 0 (where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z) and such that ∞ � k=1 b(i) jk ≤ 1, and define the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) � � �Q(i) jk def = � � � � � ��� L � F (i)�ek, �ej ���2 4 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� 1 b(i) jk b(i) kj if � L � F (i)�ek, �ej � ̸= 0 0 otherwise � � � j≥2,1≤k≤j−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) +∞ � k=1 |α(k)| ϵk ρ2|α(k)| is convergent with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) ϵj > max i=1,··· ,m k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ϵk Q(i) jk , then the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover the series V (z) = ∞ � k=1 ϵk ��� � P −1z �α(k)��� 2 is a common global Lyapunov function on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='11) � ˙�z = �F (i)(�z) �m i=1 defined on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3, the monomials �ek(�z) = (�z)α(k) generate a common infinite invariant maximal flag for ¯L � F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We first show that the candidate Lyapunov functional �V(f) = ∞ � k=1 ϵk |⟨f, �ek⟩|2 satisfies d dt �V � (�U (i) t )∗ (k�z − k0) � < 0 for all i = 1, · · · , m, where �U (i) t denotes the Koopman semigroup associated with the subsystem ˙�z = �F (i)(�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) implies that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4) are absolutely convergent on Dn (0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then, it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 that d dt �V � (�U (i) t )∗ (k�z − k0) � ≤2 ∞ � j=1 b(i) jj ϵj ���c(i) j ��� 2 ℜ � λ(i) j � + 2 ∞ � j=2 j−1 � k=1 b(i) jk ϵj ���c(i) j ��� 2 ℜ � λ(i) j � + b(i) kj ϵk ���c(i) k ��� 2 ℜ � λ(i) k � + ϵkℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek �� where c(i) j = � (�U (i) t )∗ (k�z − k0) , �ej � and λ(i) j = � L � F (i)�ej, �ej � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ℜ{λ(i) j } < 0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1)), one has to find a sequence of positive numbers (ϵj)j≥1 such that b(i) jk ϵj ���c(i) j ��� 2 ���ℜ � λ(i) j ���� + b(i) kj ϵk ���c(i) k ��� 2 ���ℜ � λ(i) k ���� > ϵk ���ℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek ����� for all i = 1, · · · , m and for all j, k with j > k such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) � �ej, L � F (i)�ek � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' By using the inequality ���ℜ � c(i) j ¯c(i) k � �ej, L � F (i)�ek ����� ≤ ���c(i) j ��� ���c(i) k ��� ��� �ej, L � F (i)�ek ��� , one has to satisfy b(i) jk ϵj ���c(i) j ��� 2 ���ℜ � λ(i) j ���� + b(i) kj ϵk ���c(i) k ��� 2 ���ℜ � λ(i) k ���� > ϵk ���c(i) j ��� ���c(i) k ��� ��� L � F (i)�ek, �ej ��� or equivalently (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) ϵj > ϵk � �− b(i) kj b(i) jk ���ℜ � λ(i) k ���� ���ℜ � λ(i) j ���� ����� c(i) k c(i) j ����� 2 + ��� L � F (i)�ek, �ej ��� b(i) jk ���ℜ � λ(i) j ���� ����� c(i) k c(i) j ����� � � def = ϵk h ������ c(i) k c(i) j ����� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is easy to see that the real quadratic function h has the maximal value Q(i) jk = ��� L � F (i)�ek, �ej ���2 4 ���ℜ � λ(i) j ���� ����� � λ(i) k ���� 1 b(i) jk b(i) kj This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 17 so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13) is satisfied if we choose iteratively ϵj according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that we have d dt �V � (�U (i) t )∗ (k�z − k0) � < 2 ∞ � j=1 b(i) jj ϵj ���c(i) j ��� 2 ℜ � λ(i) j � < − min j � b(i) jj ���ℜ � λ(i) j ���� � ∞ � j=1 ϵj ���c(i) j ��� 2 = − min j � b(i) jj ���ℜ � λ(i) j ���� � �V � (�U (i) t )∗ (k�z − k0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' With the evaluation functional k�z, we can define �V : Dn (0, ρ) → R+, �V (�z) = �V (k�z − k0) and, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='13, we verify that �V � �ϕ(i) t (�z) � = �V � k�ϕ(i) t (�z) − k0 � = �V � k�ϕ(i) t (�z) − k�ϕ(i) t (0) � = �V � (�U (i) t )∗ (k�z − k0) � < �V (k�z − k0) = �V (�z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In addition, if we define V = �V ◦ P −1 : Dn (0, ρ) → R+, we have V � ϕ(i) t (z) � = �V � P −1ϕ(i) t (P �z) � = �V � �ϕ(i) t (�z � < �V (�z) = V (P �z) = V (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, we have the CLF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) V (z) = ∞ � k=1 ϵk |⟨k�z − k0, �ek⟩|2 = ∞ � k=1 ϵk |⟨k�z, �ek⟩|2 = ∞ � k=1 ϵk ��� � P −1z �α(k)��� 2 for the switched nonlinear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, since Dn (0, ρ) is forward invariant with respect to ϕ(i) t , the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that, if the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 are satisfied but the polydisk Dn(0, ρ) is not forward invariant with respect to the flow generated by the subsys- tems, then the switched system is GUAS in the largest sublevel set of the Lyapunov function that is contained in Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition on the boundedness of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) could be inter- preted as the dominance of diagonal entries of the matrix ¯L � F (i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=', the Koopman eigenvalues (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1)) with respect to the other entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the number of non- zero cross-terms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='12) to be compensated affects the way we define the sequence of weights b(i) jk and therefore the sequence ϵj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) has an upper bound Q < 1, one can set ϵk = Q for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' However, such case rarely appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Instead, if Q > 1, one might have ϵk = O(Qk) and it is clear that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) diverges for all ρ since k − 2|α(k)| → ∞ as k → ∞ (except in the case n = 1 where |α(k)| = k, see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the following, we will consider specific vector fields such that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) converges for a proper choice of sequence b(i) jk , so that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For polynomial vector fields of the form F (i) l (z) = r � k=1 a(i) l,kzα(k), we denote by K(i) the number of nonzero terms (without counting the monomial zl in F (i) l ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) K(i) = m � l=1 # � k ̸= l : a(i) l,k ̸= 0 � This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where # is the cardinal of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In this case, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn, where F (i) are polynomial vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that PJF (i)(0)P −1 are upper triangular), the unit polydisk Dn is forward invariant with respect to the flows ϕ(i) t gener- ated by F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ( �K(i))2 ��� L � F (i)�ek, �ej ���2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� < 1, where �K(i) is the number of nonzero terms of �F (i)(�z) = P −1F (i)(P �z) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15)) and where �ej(�z) = �zα(j) are the monomials in the new coordinates �z = P −1z, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) is GUAS on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 with the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='18) � � � � � � � b(i) jj = (1 − ξ) b(i) jk = ξ 2K(i) if j ̸= k with � L � F (i)�ek, �ej � ̸= 0 or � L � F (i)�ej, �ek � ̸= 0 b(i) jk = 0, if j ̸= k with � L � F (i)�ek, �ej � = 0 or � L � F (i)�ej, �ek � = 0, with ξ ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that, for a fixed j and for all k ∈ N \\ {j}, there are at most K(i) nonzero values ⟨L � F (i)�ek, �ej⟩ and at most K(i) nonzero values ⟨L � F (i)�ej, �ek⟩, so that the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='18) satisfies ∞ � k=1 b(i) jk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The elements Q(i) jk of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='19) Q(i) jk = ( �K(i))2 ��� L � F (i)�ek, �ej ���2 ξ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) implies that max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 Q(i) jk def = Q < 1 for some ξ ∈]0, 1[, so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) is satisfied with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='20) ϵj ∼ max k∈Kj {ϵk Q} for j ≫ 1, with Kj = {k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , j − 1} : � L � F (i)�ek, �ej � ̸= 0 for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='20) yields ϵj = O(Qj) for j ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) is convergent for any ρ ≤ 1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 implies that the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16) is GUAS on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 19 Another result is obtained when a diagonal dominance property is assumed for the Jacobian matrices JF (i)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) � ˙z = F (i)(z) �m i=1 be a switched nonlinear system on Dn, with F (i) l (z) = +∞ � k=1 a(i) l,kzα(k) and +∞ � k=1 |a(i) l,k| < ∞ for all i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m} and l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Assume that all subsystems of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) have a common hyperbolic equilibrium ze = 0 that is globally asymptotically stable on Dn, the Lie algebra span � JF (i)(0) � Lie is solvable (and therefore there exists a matrix P such that PJF (i)(0)P −1 are upper triangular), there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to the flows ϕ(i) t generated by F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exist ξ ∈]0, 1[ and κ ∈]0, 1[ with ξ + κ < 1 such that, for all q, r ∈ {1, · · · , n} with q < r (when n > 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) ���J �F (i)(0)]qr ��� 2 < � 2ξ n2 − n �2 ���ℜ([J �F (i)(0)]rr) ��� ���ℜ([J �F (i)(0)]qq) ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) ���[J �F (i)(0)]qr ��� < 2ξ n2 − n ���ℜ([J �F (i)(0)]qq) ��� and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 ⟨L � F (i) ��ek,�ej⟩̸=0 �∞ l=1 ��� L � F (i)�el, �ej ��� �∞ l=1 ��� L � F (i)�ek, �el ��� κ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� < 1 ρ2 , where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will denote by D def = (n2 − n)/2 the number of upper off-diagonal entries of the Jacobian matrices J �F (i)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 with the sequence � � � � � � � � � � � � � � � � � � � � � � � � � b(i) jj = (1 − ξ − κ) b(i) jk = ξ 2D if j ̸= k with |α(j)| = |α(k)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' and if � L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej � ̸= 0 or � L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek � ̸= 0 b(i) jk = 0 if |α(j)| = |α(k)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej � = 0 and � L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek � = 0 b(i) jk = κ 2 ��� L � F (i)�ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej ��� �∞ l=1 ��� L � F (i)�el,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ej ��� if |α(k)| < |α(j)| b(i) jk = κ 2 ��� L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �ek ��� �∞ l=1 ��� L � F (i)�ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' �el ��� if |α(k)| > |α(j)| with ξ ∈]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1[ and κ ∈]0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='16 and the fact that the Jacobian matrices J �F (i)(0) are upper triangular that, for a fixed j and all k ̸= j This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY with |α(k)| = |α(j)|, there are at most D nonzero values � L � F (i)�ek, �ej � and at most D nonzero values � L � F (i)�ej, �ek � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Therefore, the sequence b(i) jk satisfies ∞ � k=1 b(i) jk < (1 − ξ − κ) + ξ + κ 2 �j k=1 ��� L � F (i)�ek, �ej ��� �∞ l=1 ��� L � F (i)�el, �ej ��� + κ 2 �∞ k=j+1 ��� L � F (i)�ej, �ek ��� �∞ l=1 ��� L � F (i)�ej, �el ��� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The elements Q(i) jk of the double sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='25) Q(i) jk = � � � � � � � � � � � � � D2 ��� L � F (i)�ek, �ej ���2 ξ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� if |α(j)| = |α(k)| �∞ l=1 ��� L � F (i)�el, �ej ��� �∞ l=1 ��� L � F (i)�ek, �el ��� κ2 ��ℜ �� L � F (i)�ej, �ej ���� ��ℜ �� L � F (i)�ek, �ek ���� if |α(k)| ̸= |α(j)| and � L � F (i)�ek, �ej � ̸= 0 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We note that ∞ � l=1 ��� L � F (i)�el, �ej ��� and ∞ � l=1 ��� L � F (i)�ek, �el ��� are finite according to the as- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Next, we show that the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) imply that Q(i) jk < 1 if |α(j)| = |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='25) that this latter inequality is equivalent to α2 q(k)|[J �F (i)(0)]qr|2 < ξ2 D2 ����� n � l=1 αl(j)ℜ([J �F (i)(0)]ll) ����� ����� n � l=1 αl(k)ℜ([J �F (i)(0)]ll) ����� for all j > k such that α(j) = (α1(k), · · · , αq(k)−1, · · · , αr(k)+1, · · · , αn(k)) for some q < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the diagonal entries of the (upper-triangular) Jacobian matrices J �F (i)(0) are the eigenvalues and therefore have negative real parts, the most restrictive case is obtained with αl(k) = 0 for all l ̸= q, which yields α2 q(k)|[J �F (i)(0)]qr|2 < ξ2 D2 ���(αq(k) − 1)ℜ([J �F (i)(0)]qq) + ℜ([J �F (i)(0)]rr) ��� ���αq(k)ℜ([J �F (i)(0)]qq) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When αq(k) = 1, this inequality is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' When αq(k) > 1, we can rewrite (αq(k) − 1)|[J �F (i)(0)]qr|2 + |[J �F (i)(0)]qr|2 < ξ2 D2 � (αq(k) − 1) ���ℜ([J �F (i)(0)]qq) ��� 2 + ���ℜ([J �F (i)(0)]rr) ��� ���ℜ([J �F (i)(0)]qq) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22), we have that the above inequality is satisfied if (αq(k) − 1)|[J �F (i)(0)]qr|2 < ξ2 D2 (αq(k) − 1) ���ℜ([J �F (i)(0)]qq) ��� 2 , which is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' While Q(i) jk < 1 for |α(j)| = |α(k)|, it is easy to see that Q(i) jk > 1 for |α(j)| > |α(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) therefore implies that max i=1,··· ,m lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 Q(i) jk def = Q < 1/ρ2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10) is satisfied with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='26) ϵj ∼ max k∈Kj {ϵk Q} This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 21 for j ≫ 1, with Kj = {k ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , j − 1 : � L � F (i)�ek, �ej � ̸= 0 for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , m} and |α(k)| < |α(j)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Hence, the sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='26) yields ϵj = O(Q|α(j)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9) is convergent and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 implies that the switched system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='21) is GUAS on Dn(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For the particular case where the Jacobian matrices JF (i)(0) are simultaneously diagonalizable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' they are diagonalizable and they commute), the diagonal dom- inance conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We should mention that the Lie-algebraic property of commutation is only needed for the Jacobian matrices JF (i)(0), an assumption which contrasts with the commutation property imposed on vector fields in [17], [30] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the case n = 1, we recover the trivial GUAS property of switched systems from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, consider the vector fields F (i)(z) = ∞ � k=1 a(i) k zk on D, with ∞ � k=1 |a(i) k | < ∞, and assume that the subsystems have a globally stable equilibrium at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Lie-algebra generated by the scalars JF (i)(0) is trivially solvable and D(0, ρ) is forward invariant for all ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 implies that the switched system is GUAS on D(0, ρ) for ρ ∈]0, 1[ which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14) that ∞ � l=1 |⟨LF (i)el, ej⟩| = j � l=1 l|a(i) j−l+1| = j � l=1 (j − l + 1)|a(i) l |, ∞ � l=1 |⟨LF (i)ek, el⟩| = k ∞ � l=1 |a(i) l |, and |ℜ (⟨ej, LF (i)ej⟩)| = j ���ℜ(a(i) 1 ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' With κ arbitrarily close to 1 (since ξ can be taken arbitrarily small in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23)), condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) is rewritten as max i=1,··· ,m lim sup j∈N �j l=1(j − l + 1)|a(i) l | �∞ l=1 |a(i) l | j ���ℜ(a(i) 1 ) ��� 2 < 1 ρ2 and, using (j − l + 1)|a(i) l | ≤ j|a(i) l | for all l, we obtain ρ < min i=1,··· ,m �∞ l=1 |a(i) l | ���ℜ(a(i) 1 ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This section presents two examples that illustrate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We will focus on specific cases that satisfy the assumptions of Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 and, without loss of generality, we will directly consider Jacobian matrices in triangular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 1: polynomial vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Similarly to Example 1, we con- sider the vector fields on the bidisk D2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) F (1)(z1, z2) = � −az1 −az2 and F (2)(z1, z2) = � � � −az1 + b � z2 1 − z1z2 2 � −az2 + b 2z1z2, This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where b > 0 and a > 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect to the flows of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, for all z ∈ ∂D2(0, ρ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' |zl| = ρ for some l), one has to verify that ℜ � F (i) l (z) ¯zl � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We have |zl| = ρ ⇒ ℜ � F (1) l (z)¯zl � = −aρ2 < 0, |z1| = ρ ⇒ ℜ � F (2) 1 (z)¯z1 � = −aρ2 + bρ2ℜ � z1 − z2 2 � < 0, |z2| = ρ ⇒ ℜ � F (2) 2 (z)¯z2 � = ρ2 � −a + b 2ℜ (z1) � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It is clear that vector field F (1) generates a holomorphic flow on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same property holds for F (2) since the conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 are satisfied with h1 (z′ 1) = h2 (z′ 2) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ℜ{G1(z)} = ℜ{−a + b � z1 − z2 2 � } < 0 and ℜ{G2(z)} = ℜ{−a + b 2z1} < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The unique global stable equilibrium of the subsystems in the bidisk D2 is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given by ⟨LF (1)ek, ej⟩ = � −a |α(j)| if k = j 0 otherwise and ⟨LF (2)ek, ej⟩ = � � � � � � � � � � � � � −a |α(j)| if k = j b � α1(j) + α2(j) − 2 2 � if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) −b α1(j) if α1(k) = α1(j) and α2(k) = α2(j) − 2 ≥ 0 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since ⟨LF (1)ek, ej⟩=0 for all k ̸= j, we have Q(1) jk = 0 for all k, j in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, K(2) = 3 and the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='17) can be rewritten as Q = lim sup j∈N |α(j)|>1 max � � � � � 9b2 a2 � α1(j) + α2(j)−2 2 �2 |α(j)| (|α(j)| − 1) , 9b2 a2 (α1(j))2 |α(j)| (|α(j)| − 2) � � � � � = 9b2 a2 < 1 and is satisfied since a > 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Hence, it follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 that the switched system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1) is GUAS in D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that, in this case, a CLF is given by V (z) = ∞ � k=1 Q−k ���zα(k)��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Example 2: analytic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The following example is taken from [1] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Consider the switched system defined by the vector fields F (1)(x1, x2) = � −x1 + 1 µ sin2(x1)x2 1x2, −x2 � F (2)(x1, x2) = � −x1 + 1 µ cos2(x1)x2 1x2, −x2 � , where µ ≥ 12/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Both subsystems are globally asymptotically stable, but the switched system is not GUAS in R2, as shown in [1] by using the fact that the convex com- bination F = � F (1) + F (2)� /2 of the two subsystems is not globally asymptotically This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 23 stable in R2 (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Yet, our result allows to infer the GUAS property in a specific region of the invariant real square ] − 1, 1[2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' To do so, we complexify the dynamics and define the vector fields on the bidisk D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect to the flows of F (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, we have |z1| = ρ ⇒ ℜ � F (1) 1 (z)¯z1 � = ρ2 � −1 + 1 µℜ � z1z2 sin2(z1) �� < 0 since 1 µ ��ℜ � z1z2 sin2(z1) ��� ≤ ρ2 ��sin2(z1) �� µ < 1 where we used max z1∈D ��sin2(z1) �� < 12/5 ≤ µ, |z2| = ρ ⇒ ℜ � F (1) 2 (z)¯z2 � = −ρ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same result follows for F (2) (with max z1∈D ��cos2(z1) �� < 12/5 ≤ µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The vector field F (1) generates a holomorphic flow on D2 since the conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 are satisfied with h1 (z′ 1) = h2 (z′ 2) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ℜ{G1(z)} = ℜ{−1 + 1 µ sin2(z1)z1z2} < 0 and ℜ{G2(z)} = −1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The same result holds for F (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The Taylor expansion of the vector fields yields F (1)(z) = � −z1 + 1 µ ∞ � p=1 (−1)p+122p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' z2p+2 1 z2, −z2 � F (2)(z) = � −z1 + 1 µz2 1z2 + 1 µ ∞ � p=1 (−1)p22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' z2p+2 1 z2, −z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given by ⟨LF (1)ek, ej⟩ = � � � � � � � � � − |α(k)| if k = j α1(k) µ (−1)p+122p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 0 otherwise and ⟨LF (2)ek, ej⟩ = � � � � � � � � � � � � � � � − |α(k)| if k = j α1(k) µ if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 α1(k) µ (−1)p22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0 0 otherwise This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY where p = 1, · · · , �α1(j) − 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This implies that we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) ∞ � l=1 |⟨LF (1)el, ej⟩| = |α(j)| + 1 µ ⌊ α1(j)−1 2 ⌋ � l=1 (α1(j) − 1 − 2l)22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ∞ � l=1 |⟨LF (2)el, ej⟩| = |α(j)| + α1(j) − 1 µ + 1 µ ⌊ α1(j)−1 2 ⌋ � l=1 (α1(j) − 1 − 2l)22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ∞ � l=1 |⟨LF (1)ek, el⟩| = |α(k)| + α1(k) µ ∞ � p=1 22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' = |α(k)| + α1(k) 2µ (cosh(2) − 1) ∞ � l=1 |⟨LF (2)ek, el⟩| = |α(k)| + α1(k) µ � 1 + ∞ � p=1 22p−1 (2p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � = |α(k)| + α1(k) 2µ (cosh(2) + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Since the Jacobian matrices JF (i)(0) are diagonal, the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='23) are trivially satisfied (with ξ arbitrarily small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Moreover, we observe from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2) that ∞ � l=1 |⟨LF (1)el, ej⟩| ≤ ∞ � l=1 |⟨LF (2)el, ej⟩| and ∞ � l=1 |⟨LF (1)ek, el⟩| ≤ ∞ � l=1 |⟨LF (2)ek, el⟩| for all k, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows that, with κ arbitrarily close to 1, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='24) can be rewritten as lim sup j∈N max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=',j−1 �∞ l=1 |⟨LF (2)el, ej⟩| �∞ l=1 |⟨LF (2)ek, el⟩| |ℜ (⟨LF (2)ej, ej⟩)| |ℜ (⟨LF (2)ek, ek⟩)| < 1 ρ2 , which is verified for ρ = � 1 + 1 2µ (cosh(2) + 1) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Indeed, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2), we have ∞ � l=1 |⟨LF (2)el, ej⟩| < |α(j)| + α1(j) µ � 1 + ∞ � l=1 22l−1 (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' � = |α(j)| + α1(j) 2µ (cosh(2) + 1) ≤ |α(j)| � 1 + 1 2µ (cosh(2) + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' and ∞ � l=1 |⟨LF (2)ek, el⟩| ≤ |α(k)| � 1 + 1 2µ (cosh(2) + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' It follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 that the switched system is GUAS on D2(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' See Figure 1 for the different values of ρ depending on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Note that a CLF is given by V (z) = ∞ � k=1 Q−2|α(k)| ���zα(k)��� 2 where Q = 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Conclusion and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' This paper provides new advances on the uniform stability problem for switched nonlinear systems satisfying Lie-algebraic solv- ability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' First, we have shown that the solvability condition on nonlinear This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' The switched system is shown to be GUAS on a polydisk of radius ρ that depends on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' vector fields does not guarantee the existence of a common invariant flag and, instead, we have imposed the solvability condition only on the linear part of the vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then we have constructed a common Lyapunov functional for an equivalent infinite- dimensional switched linear system obtained with the adjoint of the Koopman genera- tor on the Hardy space of the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally we have derived a common Lyapunov function via evaluation functionals to prove that specific switched nonlinear systems are uniformly globally asymptotically stable on invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results heavily rely on the Koopman operator framework, which appears to be a valid tool to tackle theoretical questions from a novel angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We envision several perspectives for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Our results apply to specific types of switched nonlinear systems within the frame of Lie-algebraic solvability con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' They could be extended to more general dynamics, including dynamics that possess a limit cycle or a general attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' In the same line, the Koopman operator- based techniques developed in this paper could be applied to other types of stability than uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' More importantly, the obtained stability results are limited to bounded invariant sets, mainly due to the convergence properties of the Lyapunov functions and the very definition of the Hardy space on the polydisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We envision that these results could possibly be adapted to infer global stability in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Finally, our results are not restricted to switched systems and have direct implications in the global stability properties of nonlinear dynamical systems, which will be investigated in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' General theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' We recall here some general results that are used in the proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let F : Dn → Cn be holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then F is an infinitesimal generator on Dn if and only if, for all l = 1, · · · , n and for all z ∈ Dn, Fl(z) = Gl(z) (zl − hl (z′ l)) where z′ l = (z1, · · · , zl−1, zl+1, · · · zn), hl : Dn−1 → D is holomorphic, Gl : Dn → C is holomorphic, and ℜ ((1 − hl (z′ l) ¯zl) Gl(z)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='2 (Maximum Modulus Principle for bounded domains [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let Dn ⊂ Cn be a bounded domain and f : Dn → C be a continuous function, whose restriction to Dn is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then |f| attains a maximum on the boundary ∂Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='3 (Abel’s multidimensional lemma [27] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let � α∈Nn aαzα be This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 0 10 20 30 40 50 μ26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' ZAGABE AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' MAUROY a power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exist r ∈ Cn such that sup α∈Nn |aαrα| < ∞, then the series � α∈Nn aαzα is normally convergent for all z ∈ Cn such that |z1| < |r1|, · · · , |zn| < |rn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='4 (Weierstrass’s M-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let +∞ � k=1 fn(z) be a series of functions on a domain Dn of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' If there exists a sequence of real numbers Mk such that Mk > 0 for all k, the numerical series +∞ � k=1 Mk is convergent and ∀k, ∀z ∈ Dn, |fk(z)| ≤ Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then the series +∞ � k=1 fn(z) is absolutely and uniformly convergent on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='5 (Lie’s theorem [8] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Let X be a nonzero n-complex vector space, and g be a solvable Lie subalgebra of the Lie algebra of n × n complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' Then X has a basis (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE5T4oBgHgl3EQfSg9u/content/2301.05529v1.pdf'} +page_content=' , vn) with 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