Abstract

We establish a new LaSalle's invariance principle and discuss the asymptotic behavior of a class of first-order evolution variational inequalities.

1. Introduction

Nonsmooth systems, roughly speaking, are those systems whose trajectories may not be differentiable everywhere. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution variational inequalities, and so on [1]. Since they play important roles in numerous fields, there appeared an increasing interest in the study of their dynamics in recent years.

In this paper, we consider a class of typical nonsmooth dynamical systems given by the following first-order evolution variational inequalities: It is known that many important mechanical systems arising from applications can be transformed into an variational inequality as above. In case is a proper convex and lower semicontinuous function from to and is a continuous operator with being monotone for some , Adly and Goeleven [2] made a systematic study on the asymptotic behavior of the system (1.1). The existence and uniqueness of solutions were established, and the asymptotic behavior was discussed.

In this present work, we are basically interested in the case where is only continuous. On the other hand, to avoid some technical difficulties, we will always assume is a proper convex and lower semicontinuous function from to . Note that, in our case, (1.1) may fail to have uniqueness. The main purpose is to establish a LaSalle’s invariance principle and discuss asymptotic stability of the equilibria of the system.

LaSalle’s invariance principle plays a key role in stability analysis and control. In the past decades, there appeared many important extensions. Results closely related to ours can be found in [2–6] and so forth.

This paper is organized as follows. In Section 2, we provide some basic definitions and auxiliary results. In Section 3, we develop a LaSalle’s invariance theorem and discuss the strong stability and strong asymptotic stability of the system.

2. Preliminaries

This section is concerned with some preliminary works. For convenience, we will denote by the usual inner product in with the corresponding norm .

2.1. Subdifferential

Let be a function from to . For , define is said to be the derivative of at in the direction .

If exists for all directions , we say that is differentiable at .

Definition 2.1. The following closed convex subset (possibly empty) is called the subdifferential of at , and we say that the elements of are the subgradients of at .

It is known that if is differentiable at in the classical sense, then

Proposition 2.2 (see [7]). If is a convex function from to , then, for each fixed , the mapping is convex and positively homogeneous with the following inequalities hold: Furthermore,

Proposition 2.3 (see [7]). Let one assume that is convex and lower semicontinuous. Then,(a)for each , is a nonempty and bounded set,(b)the mapping is upper semicontinuous (and thus is upper semicontinuous as well),(c)the following regularity property holds:

2.2. Some Basic Facts on the Evolution Variational Inequality

Let be a convex and lower semicontinuous function, and let be a continuous vector field. Consider the following evolution variational inequality.(VP) For any given , find a with , such that

Thanks to Proposition 2.2, one can easily rewrite (2.7) as the initial value problem of a differential inclusion: By Definition 2.1 and Proposition 2.3, we see that, for each , is a nonempty compact and convex subset of ; moreover, the multifunction is upper semicontinuous in . This guarantees by the basic theory on differential inclusions (see, e.g., [7–9], etc.) the local existence of solutions for (2.7).

Let be a solution to differential inclusion (2.7) defined on . Then, the -limit set is defined as

We infer from [10] that the following basic facts on -limit sets hold.

Proposition 2.4. If a solution of (2.7) is bounded on , then is a nonempty compact weakly invariant set, namely, for each , there is a complete solution on which is contained in with . Moreover,

3. LaSalle’s Invariance Principle

We are now ready to establish a LaSalle’s invariance principle for (2.7). For convenience, we will denote by the set In case , we simply write as .

3.1. Invariance Principle

In this subsection, we provide a LaSalle’s invariance principle for the system (2.7) involving a mapping that is only assumed continuous. The approach followed by Adly and Goeleven [2] has been proved with being continuous and monotone.

The main result is contained in the following theorem. The weak invariance of -limit set plays an important role in the proof of the theorem.

Theorem 3.1. Let be closed. Assume that there exists such that Let be the largest weakly invariant set of .
Then, for each and each bounded solution of (2.7) in , we have

Proof. For each , by Proposition 2.2, we find that Taking in (3.4), one gets Equation (3.2) then implies Now, by (3.5) and (3.6), we deduce that
Set Denote by the largest weakly invariant set of . In the following, we will check that, for each and each bounded solution of (2.7) in , we have
By Proposition 2.4, we know that is a nonempty compact weakly invariant set, and To prove (3.9), it suffices to check that
Note that is closed, we have . Since for every , by the definition of -limit set, there is such that . Here, is a bounded solution of (2.7) in , and is closed, hence .
In what follows we first show that Indeed, by (2.8) and (3.7), we see that It then follows from the proof of Lemma  2 in [2] that is nonincreasing on . Moreover, is bounded from below on since and is continuous on the closed set . This provides with an existence of the limit of . Hence, exists.
For each , by definition of -limit set, there exists such that Further, by continuity, we deduce that This verifies the validity of (3.12).
Now, we check that . Let . As is weakly invariant, there exists a complete solution starting from with for all . By what we have just proved, it holds that Take a sequence such that is differentiable at each with Then, This implies that Thus, one deduces that . By continuity of , we know that . This proves what we desired, and (3.11) follows directly from the weak invariance of .
Finally, we verify that , which implies and completes the proof of the theorem. Let . Then, by (3.8), we have Invoking (3.5) and (3.6), we find that Thus, .

As a particular case of Theorem 3.1, we have the following.

Theorem 3.2. Assume that there exists such that(1), ,(2) as , . Denote by the largest weakly invariant set of .
Then, for each , every solution of (2.7) is bounded and

Proof. Let be given. We set Then, by assumption (2), we see that is a bounded closed subset of . We infer from the proof of Theorem 3.1 that is decreasing along any solution of (2.7). Thus, is actually positively invariant. Therefore, by Theorem 3.1, one concludes that for each solution , where is the largest weakly invariant subset of . Clearly, , and the conclusion follows.

3.2. Asymptotic Stability of Equilibria

As simple applications of the LaSalle’s invariance principle established above, we make some further discussions on the asymptotic behavior of the system (1.1). For this purpose, we denote by the set of stationary solutions to (1.1), that is, In what follows, we will always assume that so that is the trivial stationary solution of (1.1).

Let us first prove the strong stability of the trivial stationary solution 0. For , we denote by the closed ball of radius ,

Theorem 3.3. Suppose that there exists and such that(1) for all , where satisfies   ,(2), (3), . Then, the stationary solution 0 is strongly stable, that is, for any , there exists a such that for any solution of (1.1) with , one has

Proof. For any , let be the minimum value of on the boundary of . Then, by assumptions (1) and (2), we find that . Let It is clear that is a neighborhood of 0. Since is decreasing along each solution in , one trivially checks that is strongly positive invariant. This implies the desired result.

Propositions 3.4–3.6 below can be proved by the same arguments as the ones in [2]. We omit the details.

Proposition 3.4. Suppose that is a subset of , and there exists such that Then,

Proposition 3.5. Suppose that there exist a and such that(1), ,(2). Then, the stationary solution 0 is isolated in .

Proposition 3.6. Suppose that there exists such that Then, .

Now, we can easily prove the following result.

Theorem 3.7. Suppose that there exists and such that (1)–(3) in Theorem 3.3 hold; moreover, Then, the trivial stationary solution 0 is strongly asymptotically stable.

Proof. The strong stability is readily implied in Theorem 3.3. Define Then, as in the proof of Theorem 3.3, we know that is a strongly positively invariant neighborhood of 0. Applying Theorem 3.1, we deduce that, for , On the other hand, by (3.34), we see that . Therefore, the trivial stationary solution 0 is strongly asymptotically stable.

Theorem 3.8. Suppose that there exists such that(1) for all , where is a continuous strictly increasing function with ,(2), (3), ,(4). Then, the trivial stationary solution to (2.7) is globally strongly asymptotically stable.

Proof. The strong asymptotic stability can be directly deduced from Theorem 3.8. Repeating the same argument as in Theorem 3.2, we can show that each solution of the system approaches .

Acknowledgments

This paper was supported by NNSF of China (11071185) and NSF of Tianjin (09JCYBJC01800).