Global Asymptotic Stability of Switched Neural Networks with Delays

This paper investigates the global asymptotic stability of a class of switched neural networks with delays. Several new criteria ensuring global asymptotic stability in terms of linear matrix inequalities (LMIs) are obtained via Lyapunov-Krasovskii functional. And here, we adopt the quadratic convex approach, which is different from the linear and reciprocal convex combinations that are extensively used in recent literature. In addition, the proposed results here are very easy to be verified and complemented. Finally, a numerical example is provided to illustrate the effectiveness of the results.


Introduction
In the past thirty years, neural networks have found extensive applications in associative memory, pattern recognition, and image processing [1][2][3].It is true that most applications of neural networks are heavily dependent on the dynamic behaviors of neural networks, especially on global asymptotic stability of neural networks.On the other hand, time delays are inevitably encountered in the hardware implementation due to the finite switching speed of amplifier, which may destroy the system performance and become a source of oscillation or instability in neural networks.Therefore, stability of neural networks with delays has attracted increasing attention and lots of stability criteria have been reported in the literature [4,5].
As a special class of hybrid systems, switched systems are organized by a switching rule that orchestrates the switching.In reality, neural networks sometimes have finite modes that switch from one to another at different times according to a switching law.In [6,7], the authors studied the stability problem of different kinds of switched neural networks with time delays.Different from the model in these works, in this paper, we consider a class of neural networks with state-dependent switchings.Our switched neural networks model is general and it generalizes the conventional neural networks.
Recently, convex analysis has been significantly employed in the stability analysis of time-delay systems [8][9][10][11][12][13][14][15][16][17].According to the feature of different convex functions, different convex combination approaches are adopted in the literature, such as the linear convex combination [8][9][10], reciprocal convex combination [11][12][13], and quadratic convex combination [15][16][17].In [8,9,12,14,16,17], convex combination technology was successfully used to derive some stability criteria for neural networks with time delays.It should be pointed out that the lower bound of the time delay in [8,9,16] is zero, which means the information on the lower bound of the time delay cannot be sufficiently used.Namely, the conditions obtained in [9,14,16] fail to take effect on the stability of neural networks when the lower bound of the time delay is strictly greater than zero.
In this paper, some delay-dependent stability criteria in terms of LMIs are derived.The advantages are as follows.Firstly, differential inclusions and set-valued maps are Mathematical Problems in Engineering employed to deal with the switched neural networks with discontinuous right-hand sides.Secondly, our results employ the quadratic convex approach, which is different from the linear and reciprocal convex combinations that are extensively used in recent literature on stability.Thirdly, the lower bound  1 of the time-varying delays is not zero and its information is adequately used to construct the Lyapunov-Krasovskii functional.Fourthly, we resort to neither Jensen's inequality with delay-dividing approach nor the free-weighting matrix method compared with previous results.
The organization of this paper is as follows.Some preliminaries are introduced in Section 2. In Section 3, based on the quadratic convex approach, delay-dependent stability criteria in terms of LMIs are established for switched neural networks with time-varying delays.Then, an example is given to demonstrate the effectiveness of the obtained results in Section 4. Finally, conclusions are given in Section 5.
Notations.Throughout this paper,   denotes the -dimensional Euclidean space.  and  −1 denote the transpose and the inverse of the matrix , respectively. > 0 ( ≥ 0) means that the matrix  is symmetric and positive definite (semipositive definite).* represents the elements below the main diagonal of a symmetric matrix.The identity and zero matrices of appropriate dimensions are denoted by  and 0, respectively.SYM() is defined as SYM() =  +   .diag{⋅ ⋅ ⋅ } denotes a block-diagonal matrix.

System Description and Preliminaries
In this paper, we consider a class of switched neural networks with delays as follows: where The following assumptions are given for system (1): (H1) For  ∈ 1, 2, . . ., ,   is bounded and there exist constants ℎ −  , ℎ +  such that (H2) The transmission delay   () is a differential function and there exist constants 0 ≤  1 <  2 ,  such that for all  ≥ 0,  = 1, 2, . . ., .
Obviously, system (1) is a discontinuous system; then its solution is different from the classic solution and cannot be defined in the conventional sense.In order to obtain the solution of system (1), some definitions and lemmas are given.Definition 1.For a system with discontinuous right-hand sides, d d =  (, ) , (0) =  0 ,  ∈   ,  ≥ 0.
By applying the theories of set-valued maps and differential inclusions [18][19][20], system (1) can be rewritten as the following differential inclusion: where The other parameters are the same as in system (1).
It is easy to find that the origin (0, 0, . . ., 0)  is an equilibrium point of system (1).
Before giving our main results, we present the following important lemmas that will be used in the proof to derive the stability conditions of the switched neural networks.
On the other hand, the augmented vector () includes the distributed delay terms.
Remark 10.We use three inequalities in Lemma 7 combined with the quadratic convex combination implied by Lemma 6, rather than Jensen's inequality and the linear convex combination.In addition, our theoretical proof is not concerned with free-weighting matrix method.
Remark 11.To use the quadratic convex approach, we construct the Lyapunov-Krasovskii functional with the following term: ∫ In the case  1 = 0, we have the following result.
Remark 16.Because the parameters of system (1) are discontinuous, the results obtained in [6] about neural networks with continuous right-hand sides cannot be used here.In addition, the lower bounds of the delays of system (36) are not zero, so the results obtained in [8,9,16] cannot be used here.

Conclusions
In this paper, the delay-dependent stability for a class of switched neural networks with time-varying delays has been studied by using the quadratic convex combination.Some delay-dependent criteria in terms of LMIs have been obtained.The lower bound  1 of the time-varying delays is considered to be nonzero so that the information of  1 can be used adequately.It is worth noting that we resort to neither Jensen's inequality with delay-dividing approach nor the freeweighting matrix method compared with previous results.