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This paper is concerned with the boundedness and attractiveness of nonlinear switched delay systems whose subsystems have different equilibria. Some sufficient conditions which can guarantee the system’s boundedness are obtained. In addition, we work out the region where the solution will remain and furthermore the relationship between the initial function and the bounded region. Based on the new concept of attractor with switching laws, we show that the nonlinear switched delay system is attractive and then obtain the attractive region.

A switched system is a collection of finite continuous variable systems (called subsystems) along with a discrete event governing the “switching” among them (called the switching law). In reality, there are many switched systems that occur naturally or by design, such as those in the fields of control, communication, computer, and signal processes. Indeed, these systems are always suited to describe practical dynamical behaviors with hybrid nature in engineering and technology [

Up to now, most of the studies on switched systems require that all subsystems share a common equilibrium. However, in many real world problems, the assumption that all subsystems share a common equilibrium may not hold and thus the assumption may limit the applicability of stability results. Recently, it is pointed out in [

In this paper, we focus on the boundedness and attractiveness problems for a simple yet important class of nonlinear switched delay systems, which includes three main contributions. First, we propose the concept of attractor with the switching law and the notation of a function

Consider the nonlinear switched delay system

there exist

there exists

In this section, we will study the boundedness problem of the switched delay system (

Suppose that

A nonnegative number

Let

Suppose that

Lemma

Let

We use mathematical induction to prove the result. When

(i) The result in Theorem

(ii) Theorem

By the same procedure of Lemma

Assume that

there exist

there exists

Let

Let

In this section, we first give the concept of an attractor of the switched system with switching law, and then we study the existence of the attractor of the nonlinear switched delay system (

A set

According to Definition

For any given

Let

If

Suppose that the dwell time

From Lemma

Similarly, we can study the existence of the attractor of the switched delay system (

For any given

Suppose that the dwell time

This work was partially supported by the NSFC (nos. 11071257, 61290325, and 11171019), the ARC Discovery Projects, HUST Startup Research Fund, HUST Independent Innovation Research Fund (GF and Natural Science 2013), the Natural Science Foundation of Beijing (no. 1122009), and the Grant of Beijing Education Committee Key Project (no. KZ201310028031). General Program of Science and Technology Development Project of Beijing Municipal Education Commission (no. KM200810028001), and Beijing Municipal Commission of Education (KM2014).