The global exponential stability issues are considered for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous neuron activations. Some sufficient conditions for the existence, uniqueness, and global exponential stability of almost periodic solution are achieved in terms of certain linear matrix inequalities (LMIs), by applying differential inclusions theory, matrix inequality analysis technique, and generalized Lyapunov functional approach. In addition, the existence and asymptotically almost periodic behavior of the solution of the neural networks are also investigated under the framework of the solution in the sense of Filippov. Two simulation examples are given to illustrate the validity of the theoretical results.

When applying neural networks to solve many practical problems in optimization, control, and signal processing, neural networks are usually designed to be globally asymptotical or exponentially stable to avoid spurious responses or the problems of local minima. Hence exploring the global stability of the neural networks is of primary importance. In recent years, neural networks with discontinuous activations, as a special class of dynamical systems described by differential equations with discontinuous right-hand sides, have been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, switching in electronic circuits, systems oscillating under the effect of an earthquake, and control synthesis of uncertain systems, and therefore have received extensive attention from a lot of scholars so far; see, for example, [

It is well known that any equilibrium point can be regarded as a special case of periodic solution for a neuron system with arbitrary period or zero amplitude. Through the study on periodic solution, more general results can be obtained than those of the study on equilibrium point for a neuron system. Hence to study the global stability of the equilibrium point of the neural networks with discontinuous activation functions, much attention has been paid to deal with the stability of periodic solution for various neural network systems with discontinuous activations; see, for example, [

It should be noted that all the results reported in the literature above are concerned with the issue of stability analysis of equilibrium point or periodic solution and neglect the effects of almost periodicity for the neural networks with discontinuous activation functions. As far as we know, the almost periodicity is one of the basic properties for dynamical neural systems and appears to retrace their paths through phase space but not exactly. Almost periodic functions, with a superior spatial structure, can be regarded as a generalization of periodic functions. As pointed out by [

In this paper, our aim is to study the delay-dependent exponential stability problem for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous activation functions. Under the framework of Filippov differential inclusions, by applying the nonsmooth Lyapunov stability theory and highly efficient LMI approach, new delay-dependent sufficient conditions are presented to ensure the existence and GES of almost periodic solution in terms of LMIs, which can be solved efficiently by using recently developed convex optimization algorithms [

For convenience, some notations are introduced as follows.

Given a set

Let

Let

Let

The set-valued map

The rest of this paper is organized as follows. In Section

In this paper, we consider a general class of the neural networks whose dynamics is described by the system of differential equation

Under the assumption

A function

By the assumption

The function

For any continuous function

A continuous function

The almost periodic solution

To obtain the main results of this paper, the following lemmas will be needed.

If

For any constant matrix

Let scalar

Before proceeding to the main results, we further make the following assumptions.

In this section, the main results concerned with the stability of the almost periodic solution are addressed for a general class of neural networks (

Firstly, we give the proof of the existence of the solution and discuss the asymptotically almost periodic behavior of the solution for the system (

If the assumption

Let

If the assumptions

For any initial value

Consider the following Lyapunov functional candidate:

Suppose that the assumptions

Let

Next, we will discuss the existence, uniqueness, and global exponential stability of the almost periodic solutions for the system (

Suppose that the assumptions

Firstly, we prove the existence of the almost periodic solution for the system (

Under the assumptions of Theorem

By using (

On the other hand, since

for

By Lebesgue's dominated convergence theorem, we can obtain

Notice that

Consider the changes of variables

Similar to

Under the assumptions

then, the conclusion of Theorem

Consider the following Lyapunov functional candidate:

In addition, under the case above, if the assumption

Notice that periodic function can be regarded as special almost periodic function. Hence based on Theorems

Suppose that

the neural network system (

the neural network system (

When

If the assumptions

the neural network system (

the neural network system (

Consider the second-order neural network (

Solving the LMI in (

When the external input of the network

Figures

When the external input of the network

Figure

Time-domain behavior of the state variables

Phase plane behavior of the state variables

Time-domain behavior of the state variables

Consider the third-order neural network (

By solving the LMI in (

When the external input of the network

Figures

When the external input of the network

Figure

Time-domain behavior of the state variables

Phase plane behavior of the state variables

Time-domain behavior of the state variables

In this paper, the almost periodic oscillation issue has been investigated for the neural networks with mixed time-varying delays and discontinuous activation functions. Some sufficient conditions which ensure the existence, uniqueness, and global exponential stability of almost periodic solution have been obtained in terms of LMIs, which is easy to be checked and applied in practice. Two numerical examples have been given to illustrate the effectiveness of the present results.

In [

This work was supported by the National Science and Technology Major Project of China (2011ZX05020-006) and the Natural Science Foundation of Hebei Province of China (A2011203103).