Geometry of the Limit Sets of Linear Switched Systems

Abstract

The paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a non strict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call non chaotic inputs, which generalize the different notions of inputs with dwell time.Next we turn our attention to the behaviour for possibly chaotic inputs. To finish we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

Asymptotic stability of uniformly bounded nonlinear switched systems

“…. , p. Therefore the assumptions of Theorem 2 are satisfied as soon as no connected As well as in [5] we deduce from Theorem 3 some geometric sufficient conditions for asymptotic stability. They deal with linear objects and are quite easily checkable.…”

“…Definition 2 (see [5]) An input u is said to be chaotic if there exists a sequence ([t k , t k + τ ]) k∈N of intervals which satisfies the following conditions:…”

“…The paper [5] deals with the linear case but consider quadratic Lyapunov functions only. Actually Theorems 3 and 4 of [5] are direct consequences of the results of the previous sections.…”

Chaotic behavior of discrete-time linear inclusion dynamical systems

“…. , S K , yet if they share a common Lyapunov matrix P such that S T k PS k − P ≤ 0 for 1 ≤ k ≤ K (which implies that S k P ≤ 1 for all 1 ≤ k ≤ K), then it is useful for us to capture the asymptotic stability of System (1.1) governed by Balde-Jouan nonchaotic switching laws σ; for example, see [3] and also Theorem 5.5 below.…”

Convergence rate of nonlinear switched systems

“…, p. Indeed the f i 's being analytic and globally asymptotically stable the Lyapunov function (even only weak) is strictly decreasing along all nonzero solutions of each individual vector field. Notice that these assumptions do not imply the GUAS property for all inputs (see Example 4.1 and [2,5]).…”