Asymptotic stability of uniformly bounded nonlinear switched systems

Abstract

We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function.We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets.The results, which extend those of [5], are illustrated by many examples.

Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

“…Since sup k≥0 w k ≤ U , then y k ≥ α(U ) implies that y k ≥ α(w k ). By (33), then for each k for which y k ≥ α(U ), we have y k − y k+1 ≥ y k − h 2 (y k , w k ) ≥ y k − h 2 (y k , U ), where the last inequality follows because h 2 (a, ·) is nondecreasing. Let l = inf{k ∈ N 0 : y k < α(U )}.…”

“…We then have h 3 (a, b) < a/2 for all a ≥ α(b), a > 0, which implies that h 2 (a, b) < a for all a ≥ α(b), a > 0. We have now established (33).…”

“…If a ≥ α(b), by (33) and the fact that a ≤ (a, b), then h 2 (a, b) < (a, b). As a consequence, if a ≥ α(b), then h 2 (a, b) is the unique value of c ∈ [0, (a, b)] satisfying h 1 (a, b, c) = c. Consider the set C = {(a, b) ∈ R 2 ≥0 : a ≥ α(b), b > 0} and a sequence {(a k , b k )} in C that converges to (a, b) ∈ C. We have 0 < h 2 (a k , b k ) < a k for all k, and there exists ≤ (a, b).…”

“…Thus, h 2 (V k , u k ) ≤ h 2 (V k , U ). Then, h 2 (V k , U ) < V k follows from V k ≥ α(U ) and (33).…”

“…In many of these cases, however, it is indeed possible to find a common weak Lyapunov function (see the interesting discussion in Example 2 in Section VI of [21]). This fact motivated the development of several stability results for switched timeinvariant systems: extensions of LaSalle's invariance principle [22]- [29] and other approaches [30]- [33]. To the best of our knowledge, results based on weak Lyapunov functions for switched time-varying nonlinear systems can only be found in [34]- [36], where the concept of persistence of excitation plays a fundamental role.…”