Dual delay-partitioning approach to stability analysis of generalized neural networks with interval time-varying delay

Abstract

This paper is concerned with improved delay-dependent stability criteria for generalized neural networks (GNNs) with interval time-varying delay. A dual delay-partitioning approach is introduced to partition the delay intervals $[0,{\tau }_a]$ and $[{\tau }_a,{\tau }_b]$ into different multi-segments separately. A newly augmented Lyapunov–Krasovskii functional (LKF) with triple integral terms is constructed by dual–partitioning the delay in integral terms, in which the relationships between the augmented state vectors are fully taken into account. The Wirtinger-based integral inequality and Peng-Park's integral inequality are employed to effectively handle the cross-product terms occurred in derivative of the LKF. Therefore, less conservative results can be achieved in terms of e_s and LMIs. Finally, two numerical examples are included to show that the deduced criteria are less conservative than existing ones.

Introduction

It is well known that the back-propagation neural networks and optimization type neural networks can be modeled as static neural networks (SNNs), whereas Hopfield neural networks, bidirectional associative memory neural networks and cellular neural networks are classified as local field neural networks (LFNNs) [1]. In [2], the generalized neural networks (GNNs) model containing the SNNs and LFNNs as special cases was first introduced. Thus, it is enough to study the stability of GNNs instead of both LFNNs and SNNs. In recent years, much effort has been made in stability analysis of GNNs model [1], [2], [5], [6], [7], [17]. During the implementation of artificial neural networks (NNs), time delays are inevitably introduced due to the finite switching speed of amplifiers and the inherent communication time between the neurons [18], [19], which might cause oscillation, divergence, and even instability [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Therefore, the stability of the delayed neural networks (DNNs) has received considerable attention [1], [2], [5], [6], [7], [12], [13], [14], [15], [16], [17], [30], [31]. The stability criteria of DNNs are generally classified into two categories, delay-independent ones [12], [13], [14], [15], [16] and delay-dependent ones [1], [2], [5], [6], [7], [17], [30], [31]. The delay-dependent stability conditions are usually less conservative than delay-independent ones, especially when the time delay is relatively small or it varies within an interval [17].

The main goal of delay-dependent stability criteria is to obtain the maximum admissible upper bounds (MAUBs) of time-delays such that the designed networks are asymptotically stable. As is well known, the reduction of conservatism in delay-dependent stability criteria can be achieved from a priori construction of LKF as well as the application of tighter bounding techniques to estimate the derivative of LKF. Various types of LKFs have been constructed for stability analysis of DNNs, such as discretized LKF [32], triple integral form LKF [33], [30], augmented LKF [6], [7], [17], delay-partitioning-dependent LKF [32], [34], [31], [35]. On the other hand, numerous bounding techniques have been developed to estimate the derivative of the LKF, such as Jensen's inequality [32], reciprocally convex combination (RCC) technique [36], Wirtinger-based integral inequality [37], free-matrix-based integral inequality [38], auxiliary function-based integral inequalities [39] and Bessel–Legendre integral inequality [40].

In stability analysis of delayed (generalized) neural networks with (interval) time-varying delay, the delay-dependent stability criteria have become a hot research topic in recent years [1], [5], [6], [7], [10], [11], [17], [30], [41], [42], [43], [44], [45]. By augmented LKF and RCC technique, [44] has proposed a new approach, which divides the bounding of activation function into two subinterval, to analysis stability of DNNs with interval time-varying delays. In [30], the delay interval is divided into multiple equidistant subintervals and different LKFs on these subintervals are constructed. Then less conservative delay-dependent stability criterion for DNNs with interval time-varying delay was derived via the RCC technique. By augmented LKF and modified Wirtinger-based integral inequality, sufficient conditions for guaranteeing the asymptotic stability of the GNNs with time-varying delay were derived in [6]. For SNNs with interval time-varying delay, the improved stability/dissipativity criteria were derived via employing Wirtinger-based inequality to estimate the derivative of the augmented LKF [42]. By establishing a newly augmented LKF that contains four triple integral terms, an improved stability criterion for recurrent NNs with interval time-varying delays was derived in [43]. In [7], less conservative stability criteria for GNNs with time-varying delay were achieved by means of applying the free-matrix-based integral inequality to bound the derivative of the augmented LKF. On the basis of proposing an improved integral inequality to handle the cross-product terms occurred in derivative of the modified LKF, less conservative stability criteria for GNNs with interval time-varying delays were derived in [17] by combining delay bi-partitioning method with RCC technique. However, when revisiting the aforementioned stability analysis of NNs/GNNs with interval time-varying delay, we find that these works still leave plenty of room for improvement because (a) In [17], [30], [43], [44], an item like ${\dot{x}}^{\mathrm{T}}\left(t\right)\left({\tau }_a^{2}Z\right)\dot{x}\left(t\right)(Z>0)$ has occurred in $\dot{V}(t,x_t)$, which is adverse to the ultimate goal $\dot{V}(t,x_t)<0$ when τ_a is big enough. Therefore, the LKF should be ameliorated; (b) In [1], [17], only delay bi-partitioning method was adopted. If the delay interval is divided into multiple segments and an appropriate LKF is constructed, then improved results may be expected; (c) the treatment of lower bound τ_a is not adequate, such as only single delay-partitioning approach (i.e., only the delay interval $[{\tau }_a,{\tau }_b]$ is divided into bi-/multiple-segments with a non-partitioned $[0,{\tau }_a]$) was adopted in the literature [17], [30]. However, as pointed out by [46], dividing $[0,{\tau }_a]$ into multi-subintervals can yield less conservative results especially when the lower delay bound τ_a is big enough.

Motivated by the above-mentioned discussion, the main purpose of this paper is to develop further less conservative delay-dependent stability criteria for GNNs with interval time-varying delay via dual delay-partitioning approach. The main contributions of this paper are summarized as follows:

• The dual delay-partitioning approach is newly introduced, i.e., the delay intervals $[0,{\tau }_a]$ and $[{\tau }_a,{\tau }_b]$ are separately partitioned into different multi-segments.

• A newly augmented LKF is established by dual-partitioning the time-delay in integral terms, in which the relationships among the augmented state vectors ${[x^{\mathrm{T}}(t),x^{\mathrm{T}}(t-{\delta }^{\prime }),\dots ,x^{\mathrm{T}}(t-l{\delta }^{\prime })]}^{\mathrm{T}},{[x^{\mathrm{T}}(t-{\tau }_1),x^{\mathrm{T}}(t-{\tau }_2),\dots ,x^{\mathrm{T}}(t-{\tau }_{m+1})]}^{\mathrm{T}}$ and ${\left[\frac{1}{\delta }{\int }_{t-{\tau }_2}^{t-{\tau }_1}{x}^{\mathrm{T}}\right(\theta )d\theta ,\cdots ,\frac{1}{\delta }{\int }_{t-{\tau }_{m+1}}^{t-{\tau }_m}{x}^{\mathrm{T}}(\theta \left)d\theta \right]}^{\mathrm{T}}$ have been taken a full consideration.

• A delay-partitioning-dependent triple integral term ${\sum }_{j=2}^{m+1}{\int }_{-{\tau }_j}^{-{\tau }_{j-1}}{\int }_{\theta }^{-{\tau }_{j-1}}{\int }_{t+\alpha }^t{\dot{x}}^{\mathrm{T}}\left(s\right)V_j\dot{x}\left(s\right)\mathit{dsd}\alpha d\theta$ is newly introduced in the augmented LKF.

• The Wirtinger-based integral inequality and Peng-Park's integral inequality are employed to effectively handle the cross-product terms occurred in derivative of the augmented LKF;

• Less conservative results than existing ones are obtained in terms of e_s and LMIs.

The rest of this paper is organized as follows. The main problem is formulated in Section 2 and new stability criteria for the GNNs with interval time-varying delay are derived in Section 3. In Section 4, two numerical examples are provided; and a concluding remark is given in Section 5.

Notation. Throughout this paper, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote, respectively, the n-dimensional Euclidean space and the set of all $n\times m$ real matrices; the notation $A>(\ge )B$ means that $A-B$ is positive (semi-positive) definite; I (0) is the identity (zero) matrix with appropriate dimension; $A^{\mathrm{T}}$ denotes the transpose; $\mathrm{He}\left\{A\right\}$ represents the sum of A and $A^{\mathrm{T}}$; $\parallel •\parallel$ denotes the Euclidean norm in ${\mathbb{R}}^n$; “⁎” denotes the elements below the main diagonal of a symmetric block matrix; $C\left(\right[-\tau ,0],{\mathbb{R}}^n)$ is the family of continuous functions ϕ from interval $[-\tau ,0]$ to ${\mathbb{R}}^n$ with the norm ${\parallel \varphi \parallel }_{\tau }={\sup }_{-\tau \le \theta \le 0}\parallel \varphi \left(\theta \right)\parallel$; let $x_t\left(\theta \right)=x(t+\theta ),\theta \in [-\tau ,0]$.