On Exponential Stabilizability for a Class of Switched Nonlinear Systems with Mixed Time-Varying Delays

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 [1– 5]. 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 [6, 7]. The other is to identify those switching signals for which the switched system is asymptotically stable, that is, stability of switched systems under constrained switching. For this case, themultiple-Lyapunov functions are a powerful and effective tool, and average dwell time (ADT) approaches have been used to investigate the stability and stabilization problems in [8]. Recently, positive switched system receives much attention. In the theory of positive switched systems, the stability problem is investigated extensively by many researchers [9– 13], especially for the stability under arbitrary switching. It is well known that a common linear copositive Lyapunov function (CLCLF) is usually applied to the asymptotic stability of positive switched systems under arbitrary switching. Recently, necessary and sufficient conditions for the existence of CLCLFs were established in [13, 14]. For the case when the positive switched system does not share a CLCLF, a multiple linear copositive Lyapunov functional method was used in [15]. Some other methods to stability of switched nonlinear systems were proposed in [16–21]. In this paper, we study the exponential stabilizability for a class of switched nonlinear systems with mixed timevarying 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.


Introduction
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 [1][2][3][4][5].
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 [6,7].The other is to identify those switching signals for which the switched system is asymptotically stable, that is, stability of switched systems under constrained switching.For this case, the multiple-Lyapunov functions are a powerful and effective tool, and average dwell time (ADT) approaches have been used to investigate the stability and stabilization problems in [8].
Recently, positive switched system receives much attention.In the theory of positive switched systems, the stability problem is investigated extensively by many researchers [9][10][11][12][13], especially for the stability under arbitrary switching.It is well known that a common linear copositive Lyapunov function (CLCLF) is usually applied to the asymptotic stability of positive switched systems under arbitrary switching.Recently, necessary and sufficient conditions for the existence of CLCLFs were established in [13,14].For the case when the positive switched system does not share a CLCLF, a multiple linear copositive Lyapunov functional method was used in [15].Some other methods to stability of switched nonlinear systems were proposed in [16][17][18][19][20][21].
In this paper, we study the exponential stabilizability for a class of switched nonlinear systems with mixed timevarying 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.
Notation.Throughout this paper, ⟨⟩ is the set of integers {1, 2, . . ., } for any positive integer .Say a real vector  ≻ 0 (≺ 0) if all entries of  are positive (negative).The norm of the vector  ∈   is defined to be ‖‖ = max =1,2,..., {|  |}.Say a square matrix is Metzler if its off-diagonal entries are nonnegative.Say a matrix is nonnegative if all its entries are nonnegative.
Let ([−, 0],   ) be the Banach space of all continuous functions on [−, 0] with values in   normed by the maximum norm For a switching signal () and any  2 >  1 ≥ 0, let   ( 1 ,  2 ) denote the number of discontinuities of () in the open interval ( 1 ,  2 ).We say that () has an ADT   > 0 if   satisfies Throughout this paper, system (1) is said to be exponentially stabilizable via ADT switching, if for any initial function  ∈ ([−,0],  ) there exist positive constants  > 0,  > 0, and   > 0 (which are usually relative to the given initial function ) such that the corresponding solution () of system (1) under any switching with ADT   satisfies ‖()‖ ≤  − for  ≥ 0.

Main Results
In the sequel, we assume that there exist vectors  () ≻ 0,  ∈ , such that Then, we have the following global exponential stability criterion for system (1).
If there exists a common vector  ≻ 0 such that we have that  = 1.Then, Theorem 1 yields the following corollary.
Corollary 2. If there exists a common vector  ≻ 0 such that (25) holds, then system (1) is globally exponentially stable under any switching.

Conclusion
This paper has investigated the exponential stabilizability for a class of switched nonlinear systems with mixed timevarying 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.