Novel rerults on the global exponential stability of cellular neural networks with variable coefficients and unbounded delays

Abstract. In this article, we study cellular neural networks (CNNs) with time-varying coefficients, bounded and unbounded delays. By introducing a new Liapunov function to approach unbounded delays and using technique of Young inequality, we obtain some sufficient conditions which ensure global exponential stability of CNNs with unbounded delays and without assumption on boundness of active functions. The main results in this paper are new and complement previously known results

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JOURNAL OF SCIENCE OF HNUE Natural Sci., 2008, Vol. 53, N ◦ . 5, pp. 9-19 NOVEL RERULTS ON THE GLOBAL EXPONENTIAL STABILITY OF CELLULAR NEURAL NETWORKS WITH VARIABLE COEFFICIENTS AND UNBOUNDED DELAYS Tran Thi Loan and Duong Anh Tuan Hanoi 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 Liapunov function to approach unbounded delays and using technique of Young inequality, we obtain some sufficient conditions which ensure global exponential stability of CNNs with unbounded delays and without assump- tion on boundness of active functions. The main results in this paper are new and complement previously known results. 1. Introduction It is well-known that CNNs proposed by L.O. Chua and L. Yang in 1988 (see [4]) have been extensively studied in both theory and applications. They have been successfully applied in signal process, pattern recognition, associative memories and especially in static image treatments. Such applications rely on the qualitative properties of the neural networks. In hardware implementation, time delays occur due to finite switching speeds of the amplifiers and communication time. Time delays may lead to an oscillation and furthermore, to an instability of networks [1]. On the other hand, it has also been known that the process of moving images requires the introduction of delays in the signal transmission among the networks [3]. Therefore, the study of stability of neural networks with delay is practically required. We know that fixed time delays in model of delayed feedback systems serve as a good approximation of a simple circuit having a small number of cells. The neural network usually has a spatial nature due to the presence of various parallel pathways, thus it is desirable to model them by introducing unbounded delays. In this paper we consider general neural networks with variable and unbounded time delays of the form dxi(t) dt =− di(t)xi(t) + n∑ j=1 aij(t)fj(xj(t)) + n∑ j=1 bij(t)gj(xj(t− τij(t))) + n∑ j=1 cij(t) ∫ t −∞ kij(t− s)hj(xj(s))ds+ Ii(t), (i = 1, n), (1.1) 9 Tran Thi Loan and Duong Anh Tuan where xi is the state of ith neuron (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 is the input vector; fi, gi, hi are the active functions of the neurons; D(t) = diag(d1(t), ..., dn(t)), di(t) represents the rate in which the ith unit will reset its potentiality to the resting state in isolation when disconnected from the network; kij(t) (i, j = 1, n) are the kernel functions; τij(t)(i, j = 1, n) are the delays. Results about the stability of this model are still few. Moreover, most of those results have been derived in models with constant coefficients [5,7]. On the other hand, the authors have assumed that the active function f is bounded (see [7,8]) or f(0)=0 (see [8]). In this paper, we use the Young inequality and construct a suitable Liapunov function to give some new sufficient conditions for the global exponential stability of the system (1.1). We do not require that the active function f is bounded, f(0) = 0 and the system (1.1) must have equilibrium point. Moreover, the main results in [2,6] are special cases of the main results in this paper in some aspects. The rest of the paper is organized as follows. Section 2 presents some defini- tions and assumptions. In Section 3, the global exponential stability is obtained. An example is given in Section 4 to illustrate our results. 2. Definitions and assumptions In this section, we give some assumptions that are used in next section. We consider system (1.1) under some following assumptions (H1) Functions di(t), aij(t), bij(t), cij(t) and Ii(t)(i, j = 1, n) are defined, bounded and continuous on R+. Functions τij(t)(i, j = 1, n) are defined nonnegative, bounded by the constant τ and continuously differentiable on R+, inf t∈R+ (1− τ˙ij(t)) > 0, where τ˙ij(t) is the derivative of τij(t) with respect to t. (H2) Functions kij : [0,∞) → [0,∞)(i, j = 1, n) are piecewise continuous on [0,∞) and satisfy ∫∞ 0 eskij(s)ds = pij(), where pij() are continuous functions on [0, δ), δ > 0, pij(0) = 1. (H3) There are positive constants Hi, Ki, Li (i = 1, n) such that 0 ≤ |fi(u)− fi(u ∗)| ≤ Hi|u− u ∗|; |gi(u)− gi(u ∗)| ≤ Ki|u− u ∗|; |hi(u)− hi(u ∗)| ≤ Li|u− u ∗| for all u, u∗ ∈ R and i = 1, n. (H4) There is a positive constant a, bounded functions hij(t), lij(t), pij(t), qij(t), mij(t), γij(t), βij(t), ωi(t), inf t≥0 ωi(t) > 0, i, j = 1, n and r ≥ 1 such that − ω˙i(t) + rωi(t)di(t)− (r − 1) n∑ j=1 ωi(t)|aij(t)| r−hij(t) r−1 H r−qij(t) r−1 j − n∑ j=1 ωj(t)|aji(t)| hji(t)H qji(t) i − (r − 1) n∑ j=1 ωi(t)|bij(t)| r−lij(t) r−1 K r−pij(t) r−1 j 10 Novel results on the global exponential stability of cellular neural networks... − n∑ j=1 ωj(ψ −1 ji (t)) |bji(ψ −1 ji (t))| lji(ψ −1 ji (t)) 1− τ˙ji(ψ −1 ji (t)) K pji(ψ −1 ji (t)) i − (r − 1) n∑ j=1 ωi(t) ∫ ∞ 0 |kij(s)| r−γij(t) r−1 |cij(t)| r−mij(t) r−1 L r−βij (t) r−1 j ds − n∑ j=1 ∫ ∞ 0 |kji(s)| γji(t+s)ωj(t+ s)|cji(t+ s)| mji(t+s)L βji(t+s) i ds ≥ a for all t ≥ 0. Case r = 1, (H4) will transform into the following case. (H∗4) There is a positive constant a and bounded functions ωi(t), inf t≥0 ωi(t) > 0 such that − ω˙i(t) + ωi(t)di(t)−Hi n∑ j=1 ωj(t)|aji(t)| −Ki n∑ j=1 ωj(ψ −1 ji (t)) |bji(ψ −1 ji (t))| 1− τ˙ji(ψ −1 ji (t)) − Li n∑ j=1 ∫ ∞ 0 kji(s)ωj(t+ s)|cji(t+ s)|ds ≥ a for all t ≥ 0, where ψ−1ij (t) is an inverse function of ψij(t) = t− τij(t). We denote by BC the Banach space of bounded continuous functions φ : (−∞, 0]→ Rn with norm ‖φ‖ = ( n∑ i=1 sup s≤0 |φi(s)| r ) 1 r . The initial condition associated with (1.1) has the following form x(θ) = φ(θ), θ ∈ (−∞, 0], where φ ∈ BC. (2.1) We know that if hypotheses (H1), (H2) are satisfied, then the system (1.1) has a unique solution x(t) = (x1(t), ..., xn(t)) T satisfying the initial condition (2.1) (see [11]). Definition 2.1. The system (1.1) is said to be globally exponentially stable, if there are constants ε > 0 and M ≥ 1 such that for any two solutions x(t), y(t) of the system (1.1) with the initial functions φ, ψ, respectively, one has ‖x(t)− y(t)‖ = ( n∑ i=1 |xi(t)− yi(t)| r ) 1 r ≤M‖ψ − φ‖e−εt for all t ∈ R+. 3. Global exponential stability In this section, by constructing a suitable Liapunov function and using the technique of Young inequality, we derive some sufficient conditions for the global exponential stability of the system (1.1). 11 Tran Thi Loan and Duong Anh Tuan Theorem 3.1. If the hypotheses (H1), (H2), (H3) and (H4) are satisfied then the system (1.1) is globally exponentially stable. Proof. Let x(t), y(t) be two arbitrary solutions of the system (1.1) with initial value ψ, φ, respectively. Setting zi(t) = xi(t)− yi(t), we have dzi(t) dt = −di(t)zi(t) + n∑ j=1 aij(t)[fj(xj(t))− fj(yj(t))] + n∑ j=1 bij(t)[gj(xj(t− τij(t)))− gj(yj(t− τij(t)))] + n∑ j=1 cij(t) ∫ t −∞ kij(t− s)[hj(xj(s))− hj(yj(s))]ds. We consider two cases: r > 1 and r = 1. When r > 1, define a Liapunov function as follows V (t, zt) = V1(t, zt) + V2(t, zt) + V3(t, zt), where V1(t, zt) = n∑ i=1 ωi(t)|zi(t)| rerεt, V2(t, zt) = n∑ i=1 n∑ j=1 ∫ t t−τij(t) ωi(ψ −1 ij (s)) |bij(ψ −1 ij (s))| lij(ψ −1 ij (s)) 1− τ˙ij(ψ −1 ij (s)) K pij(ψ −1 ij (s)) j |zj(s)| r× × erε(s+τij(ψ −1 ij (s))ds, V3(t, zt) = n∑ i=1 n∑ j=1 ∫ ∞ 0 ∫ t t−s |kij(s)| γij(u+s)ωi(u+ s)|cij(u+ s)| mij(u+s)L βij(u+s) j × × erε(u+s)|zj(u)| rduds, ε will be determined later on. Calculating the Dini derivative of V (t, zt), we get D+V1(t, zt) ≤ e rεt|zi(t)| r−1 n∑ i=1 { (ω˙i(t) + ωi(t)rε)|zi(t)| + r n∑ i=1 ωi(t) [ − di(t)|zi(t)|+ n∑ j=1 |aij(t)||fj(xj(t))− fj(yj(t))| + n∑ j=1 |bij(t)||gj(xj(t− τij(t)))− gj(yj(t− τij(t)))| + n∑ j=1 |cij(t)| ∫ t −∞ kij(t− s)|hj(xj(s))− hj(yj(s))|ds ]} 12 Novel results on the global exponential stability of cellular neural networks... ≤ erεt n∑ i=1 {( ω˙i(t) + ωi(t)rε− rdi(t)ωi(t) ) |zi(t)| + r n∑ i=1 ωi(t) [ n∑ j=1 |aij(t)|Hj|zj(t)|+ n∑ j=1 |bij(t)|Kj |zj(t− τij(t))| + n∑ j=1 |cij(t)| ∫ t −∞ kij(t− s)Lj |zj(s)|ds ]} |zi(t)| r−1. D+V2(t, zt) = e rεt n∑ i=1 n∑ j=1 ( ωi(ψ −1 ij (t)) |bij(ψ −1 ij (t))| lij(ψ −1 ij (t)) 1− τ˙ij(ψ −1 ij (t)) K pij(ψ −1 ij (t)) j e rετij(ψ −1 ij (t))× × |zj(t)| r − ωi(t)|bij(t)| lij(t)K pij(t) j |zj(t− τij(t))| r ) . D+V3(t, zt) = e rεt n∑ i=1 n∑ j=1 (∫ ∞ 0 |kij(s)| γij(t+s)ωi(t+ s)|cij(t+ s)| mij(t+s)L βij(t+s) j × × erεs|zj(t)| rds− ∫ ∞ 0 |kij(s)| γij(t)ωi(t)|cij(t)| mij(t)L βij(t) j |zj(t− s)| rds ) . By using Young inequality ab ≤ ap/p+ bq/q with a > 0, b > 0, p > 1, 1/p+ 1/q = 1, we have n∑ j=1 r|zi(t)| r−1|aij(t)|Hj|zj(t)| = r n∑ j=1 ( |aij(t)| r−hij(t) r−1 H r−qij (t) r−1 j |zi(t)| r ) r−1 r × ( |aij(t)| hij(t)H qij(t) j |zj(t)| r ) 1 r ≤ (r − 1) n∑ j=1 |aij(t)| r−hij(t) r−1 H r−qij (t) r−1 j |zi(t)| r + n∑ j=1 |aij(t)| hij(t)H qij(t) j |zj(t)| r and n∑ j=1 r|zi(t)| r−1|bij(t)|Kj|zj(t− τij(t))| ≤ (r − 1) n∑ j=1 |bij(t)| r−lij(t) r−1 K r−pij(t) r−1 j |zi(t)| r + ∑ j=1 |bij(t)| lij(t)K pij(t) j |zj(t− τij(t))| r, n∑ j=1 r ∫ t −∞ kij(t− s)|cij(t)||zi(t)| r−1Lj |zj(s)|ds = n∑ j=1 r ∫ ∞ 0 kij(s)|cij(t)||zi(t)| r−1Lj |zj(t− s)|ds 13 Tran Thi Loan and Duong Anh Tuan ≤ (r − 1) n∑ j=1 ∫ ∞ 0 |kij(s)| r−γij(t) r−1 |cij(t)| r−mij (t) r−1 L r−βij (t) r−1 j |zi(t)| rds + n∑ j=1 ∫ ∞ 0 |kij(s)| γij(t)|cij(t)| mij(t)L βij(t) j |zj(t− s)| rds. From above inequalities and (H4) we can choose ε > 0 such that the following estimate holds D+V (t, zt) ≤ e rεt n∑ i=1 ( ω˙i(t) + ωi(t)[−rdi(t) + rε] + (r − 1) n∑ j=1 ωi(t)|aij(t)| r−hij(t) r−1 H r−qij (t) r−1 j + n∑ j=1 ωj(t)|aji(t)| hji(t)H qji(t) i + (r − 1) n∑ j=1 ωi(t)|bij(t)| r−lij(t) r−1 K r−pij(t) r−1 j + n∑ j=1 ωj(ψ −1 ji (t)) |bji(ψ −1 ji (t))| lji(ψ −1 ji (t)) 1− τ˙ji(ψ −1 ji (t)) K pji(ψ −1 ji (t)) i e rετ + (r − 1) n∑ j=1 ωi(t) ∫ ∞ 0 |kij(s)| r−γij(t) r−1 |cij(t)| r−mij(t) r−1 L r−βij(t) r−1 j ds + n∑ j=1 ∫ ∞ 0 |kji(s)| γji(t+s)ωj(t+ s)|cji(t+ s)| mji(t+s)L βji(t+s) i e rεsds ) |zi(t)| r ≤ − a 2 n∑ i=1 |zi(t)| rerεt ≤ 0 for all t ≥ 0. Therefore setting ω = min i=1,n {inf t≥0 |ωi(t)|} then ω n∑ i=1 |zi(t)| rerεt ≤ V (t, zt) ≤ V (0, z0), ∀t > 0. Since V (0, z0) = V1(0, z0) + V2(0, z0) + V3(0, z0) = n∑ i=1 ωi(0)|zi(0)| r + n∑ i=1 n∑ j=1 ∫ 0 −τij(0) ωi(ψ −1 ij (s)) |bij(ψ −1 ij (s))| lij(ψ −1 ij (s)) 1− τ˙ij(ψ −1 ij (s)) K pij(ψ −1 ij (s)) j |zj(s)| rerε(s+τij(ψ −1 ij (s))ds + n∑ i=1 n∑ j=1 ∫ ∞ 0 ∫ 0 −s |kij(s)| γij(u+s)ωi(u+ s)|cij(u+ s)| mij(u+s)L βij(u+s) j e rε(u+s)|zj(u)| rduds, we obtain V (0, z0) ≤ P n∑ i=0 sup t∈(−∞,0] |zi(t)| r, 14 Novel results on the global exponential stability of cellular neural networks... where P does not depend on the solutions of the system (1.1). When r = 1. Define a Liapunov function as follows V (t, zt) = n∑ i=1 ωi(t)|zi(t)|e εt + n∑ i=1 n∑ j=1 Kj ∫ t t−τij (t) ωi(ψ −1 ij (s)) |bij(ψ −1 ij (s))| 1− τ˙ij(ψ −1 ij (s)) |zj(s)|e ε(s+τij(ψ −1 ij (s))ds + n∑ i=1 n∑ j=1 Lj ∫ ∞ 0 kij(s) ∫ t t−s ωi(u+ s)|cij(u+ s)|e ε(u+s)|zj(u)|duds, (3.1) By (H∗4), we can choose a positive constant ε such that ωi(t)(di(t)− ε)− ω˙i(t)− n∑ j=1 ωj(t)|aji(t)|Hi − n∑ j=1 Kie ετωj(ψ −1 ji (t)) |bji(ψ −1 ji (t))| 1− τ˙ji(ψ −1 ji (t)) − n∑ j=1 Li ∫ ∞ 0 kji(s)ωj(t+ s)|cji(t+ s)|e εsds ≥ a 2 for all t ≥ 0. Calculating the Dini derivative of V (t, zt), we get D+V (t, zt) ≤ e εt { n∑ i=1 [ω˙i(t) + ωi(t)(ε− di(t))]|zi(t)| + n∑ i=1 [ ωi(t) n∑ j=1 |aij(t)||fj(xj(t))− fj(yj(t))| + n∑ j=1 ωi(t)|bij(t)||gj(xj(t− τij(t)))− gj(yj(t− τij(t)))| + n∑ j=1 ωi(t)|cij(t)| ∫ t −∞ kij(t− s)|hj(xj(s))− hj(yj(s))|ds + n∑ j=1 Kjωi(ψ −1 ij (t)) |bij(ψ −1 ij (t))| 1− τ˙ij(ψ −1 ij (t)) eετij(ψ −1 ij (t))|zj(t)| − n∑ j=1 ωi(t)Kj |bij(t)||zj(t− τij(t))| + n∑ j=1 Lj ∫ ∞ 0 kij(s)ωi(t+ s)|cij(t+ s)|e εs|zj(t)|ds − n∑ j=1 Lj ∫ ∞ 0 kij(s)ωi(t)|cij(t)||zj(t− s)|ds ]} ≤ eεt { n∑ i=1 [ω˙i(t) + ωi(t)(ε− di(t))]|zi(t)|+ n∑ i=1 [ n∑ j=1 ωi(t)|aij(t)|Hj|zj(t)| 15 Tran Thi Loan and Duong Anh Tuan + n∑ j=1 ωi(t)Kj |bij(t)||zj(t− τij(t))|+ n∑ j=1 ωi(t)Lj |cij(t)| ∫ t −∞ kij(t− s)|zj(s)|ds + n∑ j=1 Kjωi(ψ −1 ij (t)) |bij(ψ −1 ij (t))| 1− τ˙ij(ψ −1 ij (t)) eετij(ψ −1 ij (t))|zj(t)| − n∑ j=1 ωi(t)Kj|bij(t)||zj(t− τij(t))| + n∑ j=1 Lj ∫ ∞ 0 kij(s)ωi(t+ s)|cij(t+ s)|e εs|zj(t)|ds − n∑ j=1 Lj ∫ ∞ 0 kij(s)ωi(t)|cij(t)||zj(t− s)|ds ]} ≤ −eεt n∑ i=1 { ωi(t)(di(t)− ε)− ω˙i(t)− n∑ j=1 ωj(t)|aji(t)|Hi − n∑ j=1 Kiωj(ψ −1 ji (t)) |bji(ψ −1 ji (t))| 1− τ˙ji(ψ −1 ji (t)) eετ − n∑ j=1 Li ∫ ∞ 0 kji(s)ωj(t+ s)|cji(t+ s)|e εsds } |zi(t)|. Thus, D+V (t, zt) ≤ − a 2 n∑ i=1 zi(t) ≤ 0 for all t ≥ 0. Therefore, we obtain V (t, zt) ≤ V (0, z0) for all t ≥ 0. From (3.1) we have V (t, zt) ≥ ω n∑ i=1 |zi(t)| for all t ≥ 0, where ω = min i=1,n inf t≥0 ωi(t). It is easy to see that V (0, z0) ≤ P n∑ i=0 sup t∈(−∞,0] |zi(t)|, where P does not depend on the solutions of (1.1). Hence, we get ( n∑ i=1 |xi(t)− yi(t)| r ) 1 r ≤M ( n∑ i=1 sup s∈[−∞,0] |xi(s)− yi(s)| r ) 1 r e−εt for all t ≥ 0, r ≥ 1 that is ‖x(t)− y(t)‖ ≤M‖φ−ψ‖e−εt for all t ≥ 0, where M > 1 is independent of solutions of (1.1). This completes the proof of Theorem 3.1. Remark 3.1. In our knowledge, all conditions in previous literature always demand that parameters are constants. It is difficult to find them. But in this article, we only need that parameters are functions. It supplies us with more choices. 16 Novel results on the global exponential stability of cellular neural networks... Remark 3.2. By this method, we can obtain similar results when we substitute fijl, gijl, hijl for fi, gi, hi. Corollary 3.1. Assume that (H1), (H2), (H3) hold and there exist positive con- stants ωi, hij , lij, pij, qij , mij , γij, βij , i = 1, n and r > 1 such that rωidi(t)− (r − 1) n∑ j=1 ωi|aij(t)| r−hij r−1 H r−qij r−1 j − n∑ j=1 ωj|aji(t)| hjiH qji i − (r − 1) n∑ j=1 ωj|bij(t)| r−lij r−1 K r−pij r−1 j − n∑ j=1 ωj |bji(ψ −1 ji (t))| lji 1− τ˙ji(ψ −1 ji (t)) K pji i − (r − 1) n∑ j=1 ∫ ∞ 0 |kij(s)| r−γij r−1 ωi|cij(t)| r−mij r−1 L r−βij r−1 j ds − n∑ j=1 ∫ ∞ 0 |kji(s)| γjiωj|cji(t+ s)| mjiL βji i ds ≥ a > 0 for all t ≥ 0. Then the system (1.1) is globally exponentially stable. Corollary 3.2. Assume that (H1), (H2), (H3) hold and there exist positive con- stants ωi, i = 1, n such that ωidi(t)− n∑ j=1 ωj |aji(t)|Hi − n∑ j=1 ωj |bji(ψ −1 ji (t))| 1− τ˙ji(ψ −1 ji (t)) Ki − n∑ j=1 ∫ ∞ 0 kji(s)ωj|cji(t+ s)|Lids ≥ a > 0 for all t ≥ 0, then the system (1.1) is globally exponentially stable. Corollary 3.3. Assume that (H1), (H2), (H3) hold and there exist positive con- stants ωi, i = 1, n such that diωi − n∑ j=1 ωjajiHi − n∑ j=1 Kiωj bji inft≥0(1− τ˙ji(t)) − n∑ j=1 Liωjcji ≥ a > 0. Then the system (1.1) is globally exponentially stable, where di = inf t≥0 di(t), aij = sup t≥0 |aij(t)|, bij = sup t≥0 |bij(t)|, cij = sup t≥0 |cij(t)|. We consider the following autonomous neural networks dxi(t) dt =− dixi(t) + n∑ j=1 aijfj(xj(t)) + n∑ j=1 bijgj(xj(t− τij)) + n∑ j=1 cij ∫ t −∞ kij(t− s)hij(xj(s))ds+ Ii, (i = 1, n). (3.2) 17 Tran Thi Loan and Duong Anh Tuan Corollary 3.4. Assume that (H1), (H2), (H3) hold and there exist positive con- stants ωi, hij, lij, pij , qij, mij , γij, βij, i = 1, n and r > 1 such that rωidi − (r − 1) n∑ j=1 ωj|aij | r−hij r−1 H r−qij r−1 j − n∑ j=1 ωj |aji| hjiH qji i − (r − 1) n∑ j=1 ωj|bij | r−lij r−1 K r−pij r−1 j − n∑ j=1 ωj |bji| ljiK pji i − (r − 1) n∑ j=1 ∫ ∞ 0 |kij(s)| r−γij r−1 ωi|cij| r−mij r−1 L r−βij r−1 j − n∑ j=1 ∫ ∞ 0 |kji(s)| γjiωj|cji| mjiL βji i ds ≥ a ≥ 0. Then the system (3.2) is globally exponentially stable. Corollary 3.5. Assume that (H2), (H3) hold and there exist positive constants ωi > 0, i = 1, n such that diωi − n∑ j=1 ωjajiHi − n∑ j=1 Kiωjbji − n∑ j=1 Liωjcji ≥ a > 0, (3.3) then the system (3.2) is globally exponentially stable. 4. An example We consider when n = 2, the system (1.1) becomes dxi(t) dt = −di(t)xi(t) + 2∑ j=1 aij(t)fj(xj(t)) + 2∑ j=1 bij(t)gj(xj(t− τij(t))) + 2∑ j=1 cij(t) ∫ t −∞ kij(t− s)hj(xj(s))ds+ Ii(t), (i = 1, 2), (4.1) where D(t) = ( 11 + 1 t2+1 0 0 11 + e−t ) , A(t) = ( 1− e−t 1 2 (1 + 1 t2+1 ) 1 t2+1 sin t ) , B(t) = ( cos t e−t 1− | sin t| 2 3t2+2 ) , C(t) = ( 1 + sin t 2 sin2 t 2 cos2 t 1− sin t ) , kij(t) = 5e −5t, (i, j = 1, 2; t ≥ 0), fj(u) = gj(u) = hj(u) = 1 2 (u + arctan u), (i, j = 1, 2, u ∈ R), τij(t) = τ(t) = 1 + 1 2 sin t. Hence, we have di = inf t≥0 di(t) = 11, aij = sup t≥0 |aij(t)| = 1, 18 Novel results on the global exponential stability of cellular neural networks... bij = sup t≥0 |bij(t)| = 1, cij = sup t≥0 |cij(t)| = 2, inf t≥0 (1 − τ(t)) ≥ 1 2 , Hj = Kj = Lj = 1, i, j = 1, 2. If we choose ωi = 2, i = 1, 2 then diωi − 2∑ j=1 ωjajiHi − 2∑ j=1 Kiωj bji inft≥0(1− τ˙ji(t)) − 2∑ j=1 Liωjcji ≥ 1. According to Corollary 3.3, we have the system (4.1) is globally exponentially stable. 5. Conclusions In this paper, the general neural networks with variable and unbounded time delays have been studied. By using Young inequality and constructing a suitable Liapunov function, we give some sufficient conditions for the global exponential sta- bility of the system (1.1) without assumption of boundedness on the active function and existence equilibrium point of (1.1). The results obtained in this paper are new and complement previously known results. REFERENCES [1] Cillvallrri PP, Gill LM, Pandolfi L, 1993. On stability of cellular neural networks with delay. IEEE Transactions on Circuits and Systems-I, 40, pp. 157-164. [2] M. Rehim, H. Jiang, Z. Teng, 2004. Boundedness and stability for nonau- tonomous cellular neural networks with delay. Neural networks, Vol. 17, pp. 1017-1025. [3] Roska, Chua LO, 1992. Cellualr neural networks with delay type template ele- ments and non-uniform grids. 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