Global exponential stability for nonautonomous cellular neural networks with unbounded delays

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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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Nội dung Text: 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 Introduction It is well known that cellular neural networks (CNNs) proposed by L.O.Chu and L.Yang in 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 recognition and associative memories and especially in static image treatment. Such applications rely on 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 functions V (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 obtain sufficient criteria to check the qualitative properties of CNNs. In our knowledge, results about the neural networks with variable, unbounded delays and 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 this paper, we use scale Liapunov functions and the continuation theorem of coincidence degree to establish conditions. 2 Definitions and assumptions In this paper we consider the general neural networks with variable and unbounded time delays 38
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 is the 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 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. 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 constant τ 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 that inf 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 −1 where 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 n P with norm kφk = sup |φi (s)|. i=1 −∞0 and M ≥1 such that for any two solutions x(t), y(t) of the system (1) 39
TRAN THI LOAN, DUONG ANH TUAN with the initial functions φ, ψ , respectively, one has n X |x(t) − y(t)| = |xi (t) − yi (t)| ≤ M kψ − φke−εt for all t ∈ R+ . i=1 3 Global exponential stability On the global exponential stability of solutions of the system (1), we obtain the following result. Theorem 3.1 If the hypotheses (H1 ), (H2 ), (H3 ) and (H4 ) are satisfied then the system (1) is globally exponentially stable. Proof. Let x(t), y(t) be two arbitrary solutions of the system (1) with initial value ψ, φ, respectively. Setting zi (t) = |xi (t) − yi (t)|. By (H4 ), we can choose a positive constant ε such that n X n X −1 −1 |bji (ψji (t))| αi (t)(di (t) − ε) − α˙i (t) − αj (t)|aji (t)|Hji − Kji eετ αj (ψji (t)) −1 j=1 j=1 1 − τ˙ji (ψji (t)) n X Z ∞ a − Lji kji (s)αj (t + s)|cji (t + s)|eεs ds ≥ for all t≥0 0 2 j=1 Define a Liapunov function as follows n X X n n X Z t −1 |bij (ψij (s))| −1 V (t, zt ) = αi (t)zi (t)eεt + Kij −1 αi (ψij (s)) −1 zj (s)eε(s+τij (ψij (s)) ds i=1 i=1 j=1 t−τij (t) 1− τ˙ij (ψij (s)) Xn Xn Z ∞ Z t + Lij kij (s) αi (u + s)|cij (u + s)|eε(u+s) zj (u)duds (3) i=1 j=1 0 t−s 40
Global exponential stability for nonautonomous cellular neural networks with unbounded delay where αi (t), i = 1, ..., n are given by (H4 ). Caculating the derivative of V (t, zt ), we get dV (t, zt ) n X n X h n X ≤ eεt (α˙ i (t) + αi (t)ε)zi (t) + αi (t) − di (t) + |aij (t)||fij (xj (t)) − fij (yj (t))| dt i=1 i=1 j=1 n X + |bij (t)||gij (xj (t − τij (t))) − gij (yj (t − τij (t)))| j=1 Xn Z t i + |cij (t)| kij (t − s)|hij (xj (s)) − hij (yj (s))|ds j=1 −∞ Xn Xn −1 |bij (ψij (t))| −1 + −1 Kij αi (ψij (t)) −1 eετij (ψij (t)) zj (t) − αi (t)bij (t)zj (t − τij (t)) i=1 j=1 1 − τ˙ij (ψij (t)) Z ! n X X n ∞ + Lij kij (s) αi (t + s)|cij (t + s)|zj (t) − αi (t)cij (t)zj (t − s) eεs ds i=1 j=1 0 n X n X h n X ≤ eεt (α˙ i (t) + αi (t)ε)zi (t) + αi (t) − di (t) + |aij (t)||fij (xj (t)) − fij (yj (t))| i=1 i=1 j=1 n X n X Z t i + Kij |bij (t)||zj (t − τij (t))| + Lij |cij (t)| kij (t − s)|zj (s)|ds j=1 j=1 −∞ n X X n |bij (ψij−1 (t))| ετij (ψ−1 (t)) −1 + Kij αi (ψij (t)) −1 e ij zj (t) − αi (t)|bij (t)|zj (t − τ ij (t)) i=1 j=1 1 − τ˙ij (ψij (t)) Z ∞ ! Xn X n εs + Lij kij (s) αi (t + s)|cij (t + s)|zj (t) − αi (t)|cij (t)|zj (t − s) e ds i=1 j=1 0 n X n X ≤ −eεt zi (t) αi (t)(di (t) − ε) − α˙i (t) − αj (t)|aji (t)|Hji i=1 j=1 n −1 n Z ! X |bji (ψji (t))| X ∞ −1 − Kji eετ αj (ψji (t)) −1 − Lji kji (s)αj (t + s)|cji (t + s)|eεs ds j=1 1 − τ˙ji (ψji (t)) j=1 0 dV (t, zt ) aP n Thus, ≤− zi (t) ≤ 0 for all t ≥ 0. So we obtain V (t, zt ) ≤ V (0, z0 ) for all dt 2 i=1 n P t ≥ 0. From (3) we have V (t, zt ) ≥ α0 zi (t) for all t ≥ 0 where α0 = min inf t≥0 αi (t), i=1 i=1,...,n 41
TRAN THI LOAN, DUONG ANH TUAN αi = supt≥0 αi (t). Moreover, putting cij = supt≥0 |cij (t)| then n X X n n X Z 0 −1 |bij (ψij (s))| −1 V (0, z0 ) = αi (0)zi (0) + Kij −1 αi (ψij (s)) −1 zj (s)eε(s+τij (ψij (s)) ds i=1 i=1 j=1 −τij (0) 1− τ˙ij (ψij (s)) Xn Xn Z ∞ Z 0 + Lij kij (s) αi (u + s)|cij (u + s)|eε(u+s) zj (u)duds i=1 j=1 0 −s X n ≤P sup |φi (s) − ψi (s)| i=1 s∈[−∞,0] where P does not depend on the solutions of (1). It follows that there exists a positive M >1 such that n X n X |xi (t) − yi (t)| ≤ M sup |φi (s) − ψi (s)|e−εt for all t ≥ 0, i=1 i=1 s∈[−∞,0] that is|x(t) − y(t)| ≤ M kφ − ψke−εt for all t ≥ 0. This completes the proof of theorem 1. Corollary 3.2. Assume that (H1 ), (H2 ), (H3 ) hold and there exists positive constants αi > 0, i = 1, ..., n such that n X n X X n bji di αi − αj aji Hji − Kji αj − Lji αj cji ≥ a > 0, inf t≥0 (1 − τ˙ji (t)) j=1 j=1 j=1 then the system (1) is globally exponentially stable, where di = inf t≥0 di (t), aij = supt≥0 |aij (t)|, bij = supt≥0 |bij (t)|, cij = supt≥0 |cij (t)|. We consider the following autonomous neural networks Xn Xn dxi (t) = − di xi (t) + aij fij (xj (t)) + bij gij (xj (t − τij )) dt j=1 j=1 Xn Z t + cij kij (t − s)hij (xj (s))ds + Ii , (i = 1, n), (4) j=1 −∞ Corollary 3.3. Assume that (H2 ), (H3 ) hold and there exists positive constants αi > 0, i = 1, ..., n such that n X n X n X di αi − αj aji Hji − Kji αj bji − Lji αj cji ≥ a > 0, (5) j=1 j=1 j=1 then the system (4) is globally exponentially stable, Remark 3.4. In this article, we can replace fij , gij , hij , τij by fijl , gijl , hijl , τijl and our results improve and extend [1] and its refs. Firstly, in [1], M.Rehim studied CN Ns with bounded delays but we consider CN Ns with both bounded and unbounded delays. Sec- ondly, most articles derive the conditions which demand αi (t), i = 1, ..., n are contants. These demands are too strict to find the constants αi which satisfy their theorems. In our article, we only demand that there exist the functions αi (t) which satisfy our assumptions. 42
Global exponential stability for nonautonomous cellular neural networks with unbounded delay 4 Periodic solutions. In this section, we study the existence and global exponential stability of periodic solutions of system (1) by using Mawhin's continuation theorem. Lemma 4.1.(see [5],[6])Let Ω ∩ X be an open bounded set and let N : X → Y be a continuous operator which is L− compact on Ω. Assume (a) for each λ ∈ (0, 1), x ∈ ∂Ω∩DomL, Lx 6= λN x (b) for each x ∈ ∂Ω∩KerL, QN x 6= 0 (c) deg(JQN, Ω∩KerL, 0)6= 0. Then Lx = N x has at least one solution in Ω∩DomL. On the existence and global exponential stability of periodic solutions, we add (H5 ) assumption. (H5 ). Functions di (t), aij (t), bij (t), cij (t), τij (t), Ii (t), i, j = 1, ..., n are continuous and ω−periodic, inf t≥0 (1 − τ˙ij (t)) > 0. The functions fij (t), gij (t), hij (t), i, j = 1, ..., n are bounded. Theorem 4.2. Assume that H2 , (H3 ), (H5 ) hold, then the system (1) has at least one ω -periodic solution. Proof. We put di = inf t≥0 di (t), d = min di , aij = supt≥0 |aij (t)|, a = max aij , i=1,...,n i,j=1,...,n bij = supt≥0 |bij (t)|, b = max bij , cij = supt≥0 |cij (t)|, c = max cij , Ii = supt≥0 |Ii (t)|, i,j=1,...,n i,j=1,...,n I = max Ii . The functions fij , gij , hij are bounded by the constants Mf , Mg , Mh for all i=1,...,n i, j = 1, ..., n. To use the continuation theorem of coincidence degree theory to establish the existence of an ω− periodic solution of (1), we take X = Y = {u ∈ C(R, Rn ) : u(t + ω) = Pn u(t)} and kuk = maxt∈[0.,ω] |ui (t)|, then X is a Banach space. Set L : DomL → X, i=1 ˙ , where DomL = {u ∈ X ∩ C 1 (R, Rn )}. For all x ∈ X, x(t) = (x1 (t), ..., xn (t)), Lu = u(t) set N : X → X, N x(t) = y(t) where y(t) = (y1 (t), ..., yn (t)), yi (t) = −di (t)xi (t) + Pn n P n P R0 aij (t)fij (xj (t))+ bij (t)gij (xj (t−τij (t)))+ cij (t) −∞ kij (−s)hij (xj (t+s))ds+Ii (t), j=1 j=1 j=1 i = 1, ..., n. Define two projectors P and Q as Z 1 ω P u = Qu = u(t)dt. ω 0 n Rω Clearly, KerL=R , ImL = {x ∈ X : 0 xi (t)dt = 0, i = 1, ..., n} is closed in X and dimKerL =codimImL=n. Hence, L is a Fredholm mapping of index zero. Furthermore, we can easily show that N is L− compact on Ω with any open bounded set Ω ⊂ X. Corresponding to operator equation Lu = λN u, λ ∈ (0, 1), we have dxi (t) Xn Xn =λ − di (t)xi (t) + aij (t)fij (xj (t)) + bij (t)gij (xj (t − τij (t))) dt j=1 j=1 n X Z 0 + cij (t) kij (−s)hij (xj (t + s))ds + Ii (t) , (i = 1, n). (6) j=1 −∞ 43
TRAN THI LOAN, DUONG ANH TUAN Suppose that x∈X is a solution of (6) for some λ ∈ (0, 1), x(t) = (x1 (t), ..., xn (t)). Let ηi ∈ [0, ω] such that xi (ηi ) = maxt∈[0,ω] xi (t), i = 1, ..., n, then n X n X di (ηi )xi (ηi ) = aij (ηi )fij (xj (ηi )) + bij (ηi )gij (xj (ηi − τij (ηi ))) j=1 j=1 Xn Z 0 + cij (ηi ) kij (−s)hij (xj (ηi + s))ds + Ii (ηi ). j=1 −∞ Hence, n X n X di (ηi )xi (ηi ) ≤ |aij (ηi )fij (xj (ηi ))| + |bij (ηi )gij (xj (ηi − τij (ηi )))| j=1 j=1 Xn Z 0 + |cij (ηi ) kij (−s)hij (xj (ηi + s))ds| + |Ii (ηi )| j=1 −∞ for all i = 1, ..., n. Therefore, 1h i xi (ηi ) ≤ n.a.Mf + nb.Mg + nc.Mh + I d for all i = 1, ..., n. A similar argument as used above, let ηi ∈ [0, ω] such that xi (ηi ) = mint∈[0,ω] xi (t), we also have 1h i n.a.Mf + nb.Mg + nc.Mh + I xi (ηi ) ≥ − d 1 h i for all i = 1, ..., n. Denote C = n.a.Mf + nb.Mg + nc.Mh + I + T , where T > 0. Then d C independent of λ. Now we take Ω = {u ∈ X; kuk ≤ C}. This Ω satisfies condition (a) n n in lemma 4.1. When u ∈ ∂Ω∩kerL = ∂Ω ∩ R , u is a constant vector in R with kuk = C . Then n h X n X n X T u QN u ≤ − di u2i + aij |ui fij (uj )| + bij |ui gij (uj )| i=1 j=1 j=1 Xn i + cij |ui hij (uj )| + ui Ii < 0, j=1 where C be large enough. So for anyu ∈ ∂Ω∩kerL, then QN u 6= 0. It follows that condition(b) is satisfied. Furthermore, from ImQ=kerL, we choose J = id. Let Ψ(γ; u) = −γu + (1 − γ)QN u, then for any x ∈ ∂Ω∩kerL, uT Ψ(γ; u) < 0. Hence (according to the axiom (4) of definition of mapping degree) we get deg{JQN, Ω ∩ kerL, 0} = deg{−u, Ω ∩ kerL, 0} = 6 0, that is condition (c) is satisfied. Thus, by lemma 4.1 we conclude that Lu = N u has at least one solution in X. 44
Global exponential stability for nonautonomous cellular neural networks with unbounded delay From theorem 3.1 and theorem 4.2 we have the following result: Theorem 4.3. Assume that (H2 ), (H3 ), (H4 ) and (H5 ) hold, then the system (1) has a unique ω -periodic solution. Futhermore all other solutions of the system (1) converge exponentially to it as t → ∞. We consider the system (4) under assumption that fij , gij , hij are bounded functions. We easily get the following corollary. Corollary 4.4 Assume that (H2 ), (H3 ), and (5) hold, then the system (4) has a unique equilibrium point which is globally exponentially stable. Proof. It follows from theorem 4.2 that the system (4) has a unique ω− periodic x∗ (t) which is globally exponentially stable, for all ω > 0. This shows that the solution x∗ (t) is a constant. Hence, x∗ (t) is a unique equilibrium point of the system (4) which is globally exponentially stable. 5 An example We consider the system (1) when n=2 X 2 X 2 dxi (t) = −di (t)xi (t) + aij (t)fij (xj (t)) + bij (t)gij (xj (t − τij (t))) dt j=1 j=1 2 X Z t + cij (t) kij (t − s)hij (xj (s))ds + Ii (t), (i = 1, 2). (7) j=1 −∞ Example Consider the system (7), where 1 1 10 + 2 sin t 0 2 sin t 8 (1 + cos t) D(t) = , A(t) = 1 1 , 0 9 − cos t 6 (1 + sin t) 2 cos t 2 cos 2t cos2 t cos 2t sin 2t −3t , (i, j = B(t) = 2 , C(t) = 1 2 1 2 , kij (t) = 3e cos2 t e− sin t 2 (1 + cos 2t) 5 (1 + 4 cos 2t 1, 2; t ≥ 0), fij (u) = arctan t, gij (u) = hij (u) = u2u+1 , (i, j = 1, 2, u ∈ R); τij (t) = τ (t) = 1 + 21 sint. 1 Hence, we have di = inf t≥0 di (t) = 8, aij = supt≥0 |aij (t)| = , bij = supt≥0 |bij (t)| = 1, 2 1 cij = supt≥0 |cij (t)| = 1, inf t≥0 (1 − τ (t)) ≥ 2 , Hij = Kij = Lij = 1, i, j = 1, 2. If we choose αi = 1, i = 1, 2 then 2 X 2 X X2 bji di αi − αj aji Hji − Kji αj − Lji αj cji ≥ 1. inf t≥0 (1 − τ˙ji (t)) j=1 j=1 j=1 Hence, the system (7) satisfies corollary 3.2. Thus, it also satisfies theorem 4.3. So, the system (7) has a unique ω -periodic solution and all other solutions converge exponentially to it as t → ∞. TÂM TT TNH ÊN ÀNH MÔ TON CÖC CÕA 45
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