Triple-integral method for the stability analysis of delayed neural networks

Abstract

This paper addresses the stability problem of a delayed neural networks. Combined with the property of convex function, by generalizing the famous Jensen integral inequality, a new triple-integral Lyapunov function is constructed, and a new improved delay-dependent stability criterion is derived. Two numerical examples are presented to illustrate the less conservatism and the effectiveness of the main results.

Introduction

Recently, recurrent neural networks (RNNs) have attracted considerable attention due to their successful applications in various areas including optimization solvers, model identification, signal processing, and other engineering areas. However, because of the existence of time delays, stochastic disturbances, parameter uncertainties and so on, the convergence of a neural network may often be destroyed. This makes the design or performance for the corresponding closed-loop systems become difficult. Therefore, stability analysis of delayed uncertain neural network has received much attention. Up to now, various stability conditions have been obtained, and many excellent papers and monographs have been available (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]). Generally speaking, these so-far obtained stability results for delayed RNNs can be mainly classified into two types: that is, delay-independent and delay-dependent. Since sufficiently considered the information of time delays, delay-dependent criteria may be less conservative than delay-independent ones when the size of time delay is small. For delay-dependent type, the size of the allowable upper bound of delay is always regarded as an important criterion to discriminate the quality between different criteria. This motivates researchers to attempt all kinds of different approaches to reduce the criteria's conservatism. Among these methods, free-weighting matrix technique, delay decomposition technique, and weighting-matrix decomposition method are the most effective ones.

In [11], the authors derived some less conservative stability criteria by considering some useful terms and using free-weighting matrix technique. By considering the relationship between the time-varying delay and its lower and upper bound, the results obtained in [11] were improved in [12]. By constructing a new Lyapunov functional and using free-weighting matrix method, some more less conservative criteria than those obtained in [12] were proposed in [13]. Recently, by introducing triple-integral terms and convex optimization approach, the results obtained in the above literature were improved further in [14], [15], [16], [17], [18], respectively.

On the other hand, it can be see that the Jensen inequality used in these references only focuses on the relationship between ${\int }_{t-\tau }^t{x}^T(s)\mathit{Qx}(s)\mathit{ds}$ and ${({\int }_{t-\tau }^{t}x(s)\mathit{ds})}^{T}Q({\int }_{t-\tau }^{t}x(s)\mathit{ds})$ or between ${\int }_{-\tau }^0{\int }_{t+\theta }^t{x}^T(s)\mathit{Qx}(s)\mathit{ds}d\theta$ and ${({\int }_{-\tau }^0{\int }_{t+\theta }^{t}x(s)\mathit{ds}d\theta )}^T$ $Q({\int }_{-\tau }^0{\int }_{t+\theta }^{t}x(s)\mathit{ds}d\theta )$. One natural question is whether there exists a relationship between ${\int }_{t-\tau }^t{x}^T(s)\mathit{Qx}(s)\mathit{ds}$ and ${({\int }_{-\tau }^0{\int }_{t+\theta }^{t}x(s)\mathit{ds}d\theta )}^{T}Q({\int }_{-\tau }^0{\int }_{t+\theta }^t$ $x(s)\mathit{ds}d\theta )$. This idea motivates this study. By using the property of convex function, we establish a new integral inequality between ${\int }_{t-\tau }^t$$x^T(s)\mathit{Qx}(s)\mathit{ds}$ and ${({\int }_{-\tau }^0{\int }_{t+\theta }^{t}x(s)\mathit{ds}d\theta )}^{T}Q({\int }_{-\tau }^0{\int }_{t+\theta }^{t}x(s)$ $\mathit{ds}d\theta )$. On the basis of this new established inequality, a class of new Lyapunov functional including triple-integral is proposed, and some less conservative delay-dependent stability criteria are derived. Finally, two numerical examples are presented to illustrate the validity of the main results.

Notation: The notations are used in our paper except where otherwise specified. $\parallel ·\parallel$ denotes a vector or a matrix norm; $\mathcal{R},{\mathcal{R}}^n$ are real and n-dimension real number sets, respectively; Real matrix $P>0(<0)$ denotes P is a positive-definite (negative-definite) matrix.