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 [

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

In recent years, more and more attention has been drawn to the stability of the retarded differential inclusions; see [

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 [

The outline of this paper is as follows. In Section

For

A function

It is remarked that, under the condition that the set-valued map

In order to investigate the stability of (

A function

If

In this section, we suppose

The solution

Let

Now we give the main result in this paper.

Suppose that

the solution

if

if

For any

In order to prove the assertion (ii), we need only to show that there is a

It is clear that if

Take

The conclusion (i) of Theorem

Comparing with previous stability results in [

Theorem

Theorem

To investigate the asymptotical stability, Theorem

In this section, an application of the main result obtained in Section

Consider the retarded neural network which is described by the following differential equation:

Following [

It is clear that if

It is evident that

Let

Dynamical behavior of (

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).