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We study the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays. By using a new technique developed for positive systems, we design the average dwell time switching under which the switched nonlinear system is exponentially stable for any bounded delays. Finally, numerical examples are worked out to illustrate the main theoretical result.

A switched system which consists of a series of dynamical subsystems and a switching signal is a type of hybrid dynamical systems. Switched systems can be used to model many phenomena which cannot be described by purely continuous or purely discrete processes. Due to its broad applications in traffic control, chemical processing, switching power converters, and network control, the theory of switched systems has historically a position of great importance in systems theory and has been studied extensively in recent years [

Up to now, the stability of switched systems has attracted many researchers’ attention. For stability issues, two main problems have been investigated in the literature. One is to find conditions that guarantee asymptotic stability of the switched system under arbitrary switching. For this case, the common Lyapunov function is required for all subsystems [

Recently, positive switched system receives much attention. In the theory of positive switched systems, the stability problem is investigated extensively by many researchers [

In this paper, we study the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays. Due to the the existence of both discrete and distributed time-varying delays and the assumption that the system is not necessarily positive, a new technique developed for positive systems is employed to the exponential stability under ADT switching for a class of switched nonlinear systems with mixed time-varying delays.

Consider the following switched nonlinear systems with mixed time-varying delays:

For the particular case when

If we do not assume that

However, in many cases, condition (

Let

For a switching signal

Throughout this paper, system (

In the sequel, we assume that there exist vectors

System (

For a given switching sequence

First, we get from (

If there exists a common vector

If there exists a common vector

Consider system (

Consider system (

This paper has investigated the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays by using a new technique developed for positive systems. By using a new method developed for positive systems, we design the appropriate ADT switching under which the system is exponentially stable. The main results generalize some existing results in the literature. Two numerical examples are also worked out to illustrate the effectiveness and sharpness of the given theoretical result. Stability analysis for the more general switched nonlinear systems with mixed time delays will be further investigated in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the reviewers for their valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant no. JQ201119 and the National Natural Science Foundation of China under Grant nos. 61174217, 61374074, and 61473133.