Nonlinear Functional Analysis of Boundary Value Problems 2013View this Special Issue
Research Article | Open Access
On Boundedness and Attractiveness of Nonlinear Switched Delay Systems
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 [1, 2]. In the last decades, they have attracted considerable attention among control theorists, computer scientists, and practicing engineers in the study of switched systems and switching control design and hence rich theoretical results have been obtained (see [3–20] and the references therein).
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 [21, 22] that, when subsystems have different equilibria or no equilibrium, a switched system can still exhibit interesting behaviors under appropriate switching laws. Such behaviors are similar to those of a conventional bounded or stable system near an equilibrium point. In this paper, we introduce some boundedness and attractiveness notions to define such behaviors for switched delay systems. Such notions are extensions of the traditional boundedness and attractiveness concepts in [15, 16].
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 in the neighbourhood of a subset which is different from the conventional concept of the distance between two subsets of . Second, we propose sufficient conditions which guarantee that the switched delay system is bounded or attractive. Third, we explicitly construct a switching law and the region in which the solutions remain (boundedness) and explicitly construct the attractor for the switched delay systems.
2. Problem Statement
Consider the nonlinear switched delay system where , , and , . It is assumed that(H1) there exist such that ; (H2)there exists , , such that , ;(H3) has only one solution and for , . Let , where and denote that , , and for , define Let be the solution of the initial value problem If especially, denote that , and let be the solution of (1) with the switching law as where . Since all the subsystems are autonomous, we can take without loss of generality. Let be the switching law of the switched system (1), where and ; that is, under the switching law , the subsystem is active during the time interval . Suppose that holds, which means that a subsystem will be definitely switched to a different subsystem when the switching triggers.
In this section, we will study the boundedness problem of the switched delay system (1).
Definition 1. Suppose that , , and then a neighborhood of in is defined as where the distance between and is defined as
Definition 2. A nonnegative number is called the dwell time of the switching law if holds.
Lemma 3. Let be the dwell time of the switching law , and is a constant. If , then implies that , where
Proof. Suppose that and the switching law is defined as . Then, we will use induction to prove the result. When , , we will prove that In fact, Then, Furthermore, since the right hand side of the inequality is increasing function, we have and then, by applying the Bellman-Gronwall inequality to (14), we have and thus Let ; we get and (11) is proved. Suppose that, when , the result is right; that is, , and then we will prove that ; that is, In fact, when , , and thus and then we have Since the right hand side of the inequality is increasing function, we have Applying the Bellman-Gronwall inequality to (21) yields Let , and then we get and (18) is proved. The proof is completed.
Remark 4. Lemma 3 shows that if the initial function is in the neighborhood of , then the solution with the switching law whose dwell time satisfies has the following property: at the switching point, the function is in the neighborhood of .
Theorem 5. Let be the dwell time of the switching law and is a constant. If , then implies that the solution of (1) satisfies .
Proof. We use mathematical induction to prove the result. When , which implies that Thus, for and , Suppose that, when , (26) holds. Now we prove that, when , In fact, by Lemma 3, we know that and which implies that and thus Thus, for all , we have The proof is completed.
Remark 6. (i) The result in Theorem 5 shows that if the initial function is in the neighborhood of , then the solution with the switching law whose dwell time satisfies is in the neighborhood of the set .
(ii) Theorem 5 not only shows the boundedness of the switched system but also gives the region in which the solution will remain.
Assume that (H1)’ there exist such that ;(H2)’ there exists , , such that , , and ;(H3)’ has only one solution and for , . Let . Similarly, we can obtain the following.
Lemma 7. Let be the dwell time of the switching law , and is a constant. If , then implies that the solution of (32) satisfies .
Theorem 8. Let be the dwell time of the switching law , and is a constant. If , then implies that the solution of (32) satisfies for all .
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 (1). We will establish sufficient conditions which guarantee the existence of the attractor and furthermore we can find out the switching law and the attractor of the switching delay system (1).
Definition 9. A set is called an attractor of the switched system (1) with a switching law , if there exists a such that, for any , there exists a , such that, for any , implies .
Remark 10. According to Definition 9, is an attractor of the switched system (1) with a switching law if and only if there exists such that implies In this section, we will prove that is the attractor of system (1). For convenience, we first show that is equivalent to
Lemma 11. For any given , , there exists such that if , , then the solution of the subsystem of (1) satisfies , where ., If and especially, then one can take such that when implies the solution of the subsystem of (1) satisfies .
Proof. Let be the solution of the subsystem of (1) and , and then
which implies that
So, when , we have
If specially, it follows from (37), that which implies that and thus, when , we have ; that is, . This completes the proof.
Theorem 12. Suppose that the dwell time of the switching law satisfies . If , then the set is an attractor of the switched delay system (1).
Proof. From Lemma 11, we can see that we only need to prove that, for any , , the solution satisfies . In fact, we take and in Lemma 11, and then and implies that if , . On the other hand, for , we have
which implies that
Since the right hand side of the above inequality is an increasing function, we have
since . When , since , we have
which implies that
since . Suppose that, for ,
When , since , (take in Lemma 11), we have
which implies that
Therefore, we have
which means that is the attractor of system (1) with a switching law .
Similarly, we can study the existence of the attractor of the switched delay system (32) with the assumptions (H1)’–(H3)’ and obtain the following results.
Lemma 13. For any given , , there exists such that if and , then the solution of the subsystem of (32) satisfies . If especially, then one can take such that when implies the solution of the subsystem of (32) satisfies .
Theorem 14. Suppose that the dwell time of the switching law satisfies ; if , then the set is an attractor of the switched delay system (32).
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).
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