Stability Analysis of Retarded Differential Inclusions
Retarded differential inclusions have drawn more and more attention, due to the development of feedback control systems with delays and dynamical systems determined by retarded differential equations with a discontinuous right-hand side. The purpose of this paper is to establish a result on the stability and asymptotical stability for retarded differential inclusions. Comparing with the previous results, the main result obtained in this paper allows Lyapunov functions to be nonsmooth. Moreover, to deal with the asymptotical stability, it is not required that Lyapunov functions should have an infinitesimal upper limit, but this condition is needed in most of the previous results. To demonstrate applicability, we use the main result in the analysis of asymptotical stability of a class of neural networks with discontinuous activations and delays.
It has been known for ages that the future state of a system might depend not only on the present states but also on the past states [1–4]. In particular, in some problems it is meaningless not to have dependence on the past. Consequently, there is much concern about retarded differential equations: where “” represents the derivative, is a vector-valued function, is defined by , and is a given function. To our knowledge, most results for (1) require the function to be continuous. However, in many applications the function may be discontinuous or even multivalued. For example, a lot of phenomena in biology are characterized by strongly localized coupling, that is, by interaction of an almost on-off nature; see [5–7]. Thus it is realistic to consider retarded differential equations with a discontinuous right-hand side . This gives rise to the study of systems described by retarded differential inclusions.
Another great impetus to study retarded differential inclusion comes from the development of control theory. A specific class of systems of retarded differential inclusion arising in technology consists of feedback control systems which can be described by equations of the form where denotes controllers. Let be the set of allowable controllers and ; then many of the qualitative properties of the control system (2) can be deduced from the corresponding qualitative properties of the system of retarded differential inclusions . This approach has played an important role in investigating the absolute stability problem of regulator systems; see [9, 10].
In recent years, more and more attention has been drawn to the stability of the retarded differential inclusions; see [9–14]. In [9, 10], stability and asymptotical stability are investigated for retarded differential inclusions, and the Lyapunov functions are required to be differentiable. To study the asymptotical stability of retarded differential inclusions as well as ordinary differential inclusions, most of results in the literature are under the condition that the Lyapunov functions have an infinitesimal upper limit; see [1, 9, 10, 13], for example. In [11, 15], nonsmooth Lyapunov functions are successfully applied in discussing the stability of ordinary differential inclusions or ordinary differential equations with a discontinuous right-hand side. However, to the best of our knowledge, there is very little literature using nonsmooth Lyapunov functions to deal with the stability for retarded differential inclusions.
Based on these motivations, the objective of this paper is to make use of nonsmooth Lyapunov functions to study the stability of retarded differential inclusions. Dropping the condition that Lyapunov functions have an infinitesimal upper limit, we manage to obtain the asymptotical stability for retarded differential inclusions. Our method is based on the generalized Lyapunov approach introduced by [15, 16].
The outline of this paper is as follows. In Section 2, we give some preliminaries which are needed in this paper. Section 3 is devoted to investigate stability and asymptotical stability for retarded differential inclusions. In Section 4, an application of the main result obtained in Section 3 is given for analysis of stability of neural networks with discontinuous activations and delays.
For , denotes a norm of , where “” is the transpose. Let be a given real number and be the Banach space of continuous functions mapping the interval into . For an element , the norm of is defined by . If and is continuous, then is defined by , for any . In the present paper, we consider the following retarded differential inclusion: where is a set-valued map with nonempty convex compact values.
Definition 1. A function is said to be a solution of differential inclusion (3) on if there are and such that is continuous on and absolutely continuous on any compact subinterval of and satisfies (3) for almost all (a.a.) . For given , , is said to be a solution of differential inclusion (3) with initial value at if is a solution of (3) on and .
A function , which is locally Lipschitz continuous at , is said to be C-regular at if exists and , where is the directional derivative of at in the direction and is the generalized directional derivative of at in the direction . The function is said to be C-regular in , if it is C-regular at any . The generalized gradient of at is defined by
Lemma 2 (see ). If is C-regular and is absolutely continuous on any compact subinterval of then for a.a. , and are differentiable, and
3. Main Result
In this section, we suppose and focus on the stability of the zero solution of the differential inclusion (3), since by a transformation the stability of any solution could be investigated in terms of the zero solution of the corresponding differential inclusion. The definition of stability of the solution can be given as in . Let .
Definition 3. The solution of differential inclusion (3) is said to be stable if for any , , there is a such that implies for . The solution is said to be asymptotically stable if it is stable and there is a such that implies as . The solution is said to be globally asymptotically stable if it is stable and as for any .
Now we give the main result in this paper.
Theorem 4. Suppose that takes into bounded sets of , and are continuous nondecreasing functions, and for . If there is a locally Lipschitz continuous function such that and for any and a.a. then one has the following: (i)the solution of the differential inclusion (3) is stable;(ii)if for , the solution of the differential inclusion (3) is asymptotically stable;(iii)if for and as , the solution of the differential inclusion (3) is globally asymptotically stable.
Proof. For any and , there is a such that implies that from the continuity of the function . Since is locally Lipschitz continuous, we have that is absolutely continuous on any compact subinterval of and
It follows from (9) and (10) that
which means that
and thus for . This proves the stability.
In order to prove the assertion (ii), we need only to show that there is a such that for any . Using the approach of contradiction, assume for any there is such that (14) fails to hold; then there exist and a sequence such that as and Note that, as long as is small enough, there is such that . It follows from (9) and (10) that and thus which means is bounded. Notice that the map takes into bounded sets of , there is such that Take ; then for and hence from (15) and (18) Therefore, by (10) we have for a.a. By taking a subsequence of , if necessary, we can assume that and the intervals do not overlap, and (10) implies that If , then which is a contradiction. This proves the asymptotical stability.
It is clear that if as , then (14) is true for all . Therefore, the solution is globally asymptotically stable.
Take ; then is a continuous nondecreasing function. Thus we have the following remark.
Remark 6. Comparing with previous stability results in [1, 9, 10, 15], the advantages of Theorem 4 are as follows.(i)Theorem 4 can be used to deal with the stability for retarded differential inclusions or retarded differential equations with a discontinuous right-hand side (see Section 4).(ii)Theorem 4 allows us to use nonsmooth Lyapunov function to discuss the stability.(iii)To investigate the asymptotical stability, Theorem 4 drops the condition that the Lyapunov function should have an infinitesimal upper limit; that is, there exists a function such that , for , and
In this section, an application of the main result obtained in Section 3 is given for analysis of stability of retarded neural networks with discontinuous activations.
Consider the retarded neural network which is described by the following differential equation: where and are neuron states, and are constants representing the neuron interconnection coefficients, and is the neuron input-output activation, which is defined by
Following [18, 19], is a solution of the system (26) on with initial value at , if is continuous on and absolutely continuous on any compact subinterval of , and there exists a measurable function such that , , for a.a. and where and is called an output associated with the solution .
It is clear that if is a solution of the system (26), then it is a solution of the following retarded differential inclusion: where
Let be a solution of the system (26) on with initial value at and let the output associated with be . Consider the function where then , is regular, and the generalized gradient of at is It follows from Lemma 2 that is differentiable and for a.a. . Define ; then , , and . Note that ; we have where ; thus it follows from Theorem 4 that the solution is globally asymptotically stable; see Figure 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Grant nos. 11301551 and 11226151) and by Hunan Provincial Natural Science Foundation of China (13JJ4088).
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