Global exponential stability for nonautonomous cellular neural networks with unbounded delays

In this article, we study cellular neural networks (CNNs) with timevarying coefficients, bounded and unbounded delays. By introducing a new Liapunov functional to approach unbounded delays and using the continuation theorem of coincidence degree, we obtain some sufficient conditions to ensure the existence periodic solutions and global exponential stability of CNNs.
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Global exponential stability for nonautonomous cellular neural networks with unbounded delays Journal of Science of Hanoi National University of Education Natural sciences, Volume 52, Number 4, 2007, pp. 38- 46 GLOBAL EXPONENTIAL STABILITY FOR NONAUTONOMOUS CELLULAR NEURAL NETWORKS WITH UNBOUNDED DELAYS Tran Thi Loan and Duong Anh Tuan Department of Mathematics, Ha Noi National University of Education Abstract. In this article, we study cellular neural networks (CNNs) with time- varying coefficients, bounded and unbounded delays. By introducing a new Lia- punov functional to approach unbounded delays and using the continuation theorem of coincidence degree, we obtain some sufficient conditions to ensure the existence periodic solutions and global exponential stability of CNNs. Many of the existing results in previous literature are extended and improved in this paper.1 IntroductionIt is well known that cellular neural networks (CNNs) proposed by L.O.Chu and L.Yangin 1988 have been extensively studied both in theory and applications, such as [1], [2], [3],[4] in refs. They have been successfully applied in signal processing, pattern recognitionand associative memories and especially in static image treatment. Such applications relyon the qualitative properties of the neural networks. Usually, the Liapunov functional method is used to study qualitative properties of CNNs.Such a method is performed in three steps. In step 1, we construct a Liapunov functionsV (t). In step 2, we use suitable technique to estimate V (t). In step 3, we put some condi-tions on CNNs such that the function V (t) satisfies necessary properties. Thus, we obtainsufficient criteria to check the qualitative properties of CNNs. In our knowledge, results about the neural networks with variable, unbounded delaysand time varying coefficients have not been widely studied. Moreover, we easy see that,other authors have not used scale Liapunov functions in their studies (refs [3], [4]). In thispaper, we use scale Liapunov functions and the continuation theorem of coincidence degreeto establish conditions.2 Definitions and assumptionsIn this paper we consider the general neural networks with variable and unbounded timedelays38 Global exponential stability for nonautonomous cellular neural networks with unbounded delay Xn Xn dxi (t) = − di (t)xi (t) + aij (t)fij (xj (t)) + bij (t)gij (xj (t − τij (t))) dt j=1 j=1 Xn Z t + cij (t) kij (t − s)hij (xj (s))ds + Ii (t), (i = 1, n), (1) j=1 −∞where xi is the state of neuron i (i = 1..n), n is the number of neuron; A(t) = (aij (t))n×n ,B(t) = (bij (t))n×n , C(t) = (cij (t))n×n are connection matrices; I(t) = (I1 (t), ..., In (t))T isthe input vector; fij , gij , hij are the activation functions of the neurons. D(t) = diag(d1 (t),..., dn (t)), di (t) represents the rate with which the ith unit will reset its potential to theresting state in isolation when disconnected from the network. kij (t) (i, j = 1..n) are thekernel functions; τij (t)(i, j = 1..n) are the delays. We consider System (1) under some following assumptions (H1 ) Functions di (t), aij (t), bij (t), cij (t) and Ii (t)(i, j = 1...n) are bounded and continu- +ously defined on R . Functions τij (t)(i, j = 1..n) are nonnegative, bounded by the constantτ and continuously differentiable defined on R+ , inf (1 − τ˙ij (t)) > 0, where τ˙ij (t) is the t∈R+derivative of τij (t) with respect to t. (H2 ) Functions R∞ kij : [0, ∞) → [0, ∞)(i, j = 1..n) are piecewise continuous on [0, ∞)and satisfy 0 es kij (s)ds = pij (), where pij () are continuous functions in [0, δ), δ > 0,pij (0) = 1. (H3 )There are positive constants Hij , Kij , Lij (i, j = 1, ..., n) such that 0 ≤ |fij (u) −fij (u∗ )| ≤ Hij |u − u∗ |; |gij (u) − gij (u∗ )| ≤ Kij |u − u∗ |; |hij (u) − hij (u∗ )| ≤ Lij |u − u∗ | for ∗all u, u ∈ R and i, j = 1, ..., n. (H4 ) There are positive constant a and the functions αi (t), i = 1, ..., n such thatinf t≥0 αi (t) > 0, n X n X −1 −1 |bji (ψji (t))| αi (t)di (t) − α˙i (t) − αj (t)|aji (t)|Hji − Kji αj (ψji (t)) −1 j=1 j=1 1 − τ˙ji (ψji (t)) n X Z ∞ − Lji kji (s)αj (t + s)|cji (t + s)|ds ≥ a for all t≥0 j=1 0 −1where ψij (t) ψij (t) = t − τij (t). is inverse function of We denote by BC the Banach space of bounded continuous functions φ : (−∞, 0] → Rn ...