Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 501891, 17 pages doi:10.1155/2012/501891 Research Article Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays Yutian Zhang College of Mathematics & Physics, Nanjing University of Information Science & Technology, Nanjing 21004, China Correspondence should be addressed to Yutian Zhang, ytzhang81@163.com Received 27 June 2011; Accepted 13 September 2011 Academic Editor: E. S. Van Vleck Copyright q 2012 Yutian Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reactiondiffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results. 1. Introduction Cellular neural networks CNNs, proposed by Chua and Yang in 1988 1, 2, have been the focus of a number of investigations due to their potential applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision 3–7. Moreover, on the ground that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in implementation of neural networks, it was followed by the introduction of the delayed cellular neural networks DCNNs so as to solve some dynamic image processing and pattern recognition problems. Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory 8, 9. Particularly, stability analysis has been a major concern in the designs and applications of the CNNs and DCNNs. The stability of CNNs and DCNNs is a subject of current interest, and considerable theoretical efforts have been put into this topic with many good results reported see, e.g., 10–13. 2 Journal of Applied Mathematics With reference to neural networks, however, it is noteworthy that the state of electronic networks is actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive effects 14, 15. For instance, according to Arbib 16 and Haykin 17, when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. As a consequence, in the past few years, scientists have become increasingly interested in the influence that impulses may have on the CNNs and DCNNs and a large number of stability criteria have been derived see, e.g. 18–22. In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs. In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu 23 formulated a mathematical model of impulsive neural networks with time-varying delays and reactiondiffusion terms described by impulsive partial differential equations and studied, via delay impulsive differential inequality, the problem of global exponential stability with some stability criteria presented. Remarkably, all of the obtained stability criteria in 23 are independent of the diffusion. In 2008, Li and song 24 investigated a class of impulsive Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By establishing a delay inequality with impulsive initial conditions and M-matrix theory, some sufficient conditions ensuring global exponential stability of the equilibrium points are given. Analogous to 23, the proposed stability criteria in 24 are also independent of the diffusion. More recently Pan et al. 25 investigated a class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay impulsive differential inequality quoted in 23, several sufficient conditions are exploited ensuring global exponential stability of the equilibrium points. Especially, different from 23, 24, the estimate of the exponential convergence rate depends on reaction-diffusion in 25. In this paper, unlike the methods of impulsive differential inequalities and Poincaré inequality used in 25, we attempt to adopt the new techniques of the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the problem of global exponential asymptotic stability for impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. Different from the existing research, we find, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of the spatial variables do influence the stability. The rest of the paper is organized as follows. In Section 2, the model of impulsive delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary condition is outlined, and some facts and lemmas are introduced for later reference. By the new agency of the impulsive integral inequality of Gronwall-Bellman type as well as HardySobolev inequality, we discuss the global exponential asymptotic stability and develop some new criteria in Section 3. To conclude, two illustrative examples are given to verify the effectiveness of our results in Section 4. Journal of Applied Mathematics 3 2. Preliminaries Let Rn denote the n-dimensional Euclidean space, and Ω ⊂ Rm is a bounded open set containing the origin. The boundary of Ω is smooth and mes Ω > 0. Let R 0, ∞ and t0 ∈ R . We consider the following impulsive neural networks with time delays and reactiondiffusion terms: m n n ∂ ∂ui t, x ∂ui t, x Dis − ai ui t, x bij fj uj t, x cij fj uj t − τj t, x ∂t ∂xs ∂xs s 1 j 1 j 1 t ≥ t0 , t/ tk , ui tk 0, x ui tk , x Pik ui tk , x x ∈ Ω, i 1, 2, . . . , n, k 1, 2, . . . , 2.1 x ∈ Ω, k 1, 2, . . . , i 1, 2, . . . , n, 2.2 where n corresponds to the numbers of units in a neural network; x x1 , . . . , xm T ∈ Ω, ui t, x, denotes the state of the ith neuron at time t and in space x; smooth functions Dis Dis t, x, u ≥ 0 represent transmission diffusion operators of the ith unit; activation functions fj uj t, x stand for the output of the jth unit at time t and in space x; bij , cij , ai are constants: bij indicates the strength of the jth unit on the ith unit at time t and in space x, cij denotes the strength of the jth unit on the ith unit at time t − τj t and in space x, where τj t corresponds to the transmission delay along the axon of the jth unit and satisfies 0 ≤ τj t ≤ τ τ const • as well as τ j t < 1 − 1/h h > 0, and ai > 0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t and in space x. The fixed moments tk k 1, 2, . . . are called impulsive moments satisfying 0 ≤ t0 < t1 < t2 < · · · and limk → ∞ tk ∞; ui tk 0, x and ui tk − 0, x represent the right-hand and left-hand limit of ui t, x at time tk and in space x, respectively. Pik ui tk , x stands for the abrupt change of ui t, x at impulsive moment tk and in space x. Denote by ut, x ut, x; t0 , ϕ, u ∈ Rn the solution of system 2.1-2.2, satisfying the initial condition u s, x; t0 , ϕ ϕs, x, t0 − τ ≤ s ≤ t0 , x ∈ Ω 2.3 t ≥ t0 , x ∈ ∂Ω, 2.4 and Dirichlet boundary condition u t, x; t0 , ϕ 0, where the vector-valued function ϕs, x ϕ1 s, x, . . . , ϕn s, xT is such that Ω ni 1 ϕ2i s, xdx is bounded on t0 − τ, t0 and ϕi s, x i 1, 2, . . . , n is first-order continuous differentiable as to s on t0 − τ, t0 . The solution ut, x ut, x; t0 , ϕ u1 t, x; t0 , ϕ, . . . , un t, x; t0 , ϕT of problems 2.5–2.8 is, for the time variable t, a piecewise continuous function with the first kind 4 Journal of Applied Mathematics discontinuity at the points tk k 1, 2, . . ., where it is continuous from the left, that is the following relations are true: ui tk − 0, x ui tk , x, ui tk 0, x ui tk , x Pik ui tk , x. 2.5 Throughout this paper, the norm of ut, x; t0 , ϕ is governed by ⎛ ⎞1/2 n u t, x; t0 , ϕ ⎝ u2i t, x; t0 , ϕ dx⎠ . Ω i 1 2.6 Ω Before moving on, we introduce two hypotheses as follows. H1 Activation function fj uj t, x satisfies fi 0 0, and there exists constant li > 0 such that |fi y1 − fi y2 | ≤ li |y1 − y2 | holds for all y1 , y2 ∈ R and i 1, 2, . . . , n. H2 The functions Pik ui tk , x are continuous on R and Pik 0 0, i 1, 2, . . . , n, k 1, 2, . . . . According to H1 and H2, it is easy to see that problems 2.5–2.8 admits an equilibrium point u 0. Definition 2.1. The equilibrium point u 0 of problems 2.5–2.8 is said to be globally exponentially stable if there exist constants κ > 0 and M ≥ 1 such that u t, x; t0 , ϕ ≤ Mϕ e−κt−t0 , Ω Ω 2 where ϕΩ supt0 −τ≤s≤t0 n i 1 Ω t ≥ t0 , 2.7 ϕ2i s, xdx. Lemma 2.2 Gronwall-Bellman-type impulsive integral inequality 26. Assume that (A1) the sequence {tk } satisfies 0 ≤ t0 < t1 < t2 < · · · , with limk → ∞ tk ∞, (A2) q ∈ P C1 R , R and qt is left-continuous at tk , k 1, 2, . . ., (A3) p ∈ CR , R and for k 1, 2, . . . qt ≤ c t t0 psqsds ηk qtk , t ≥ t0 , 2.8 t ≥ t0 . 2.9 t0 <tk <t where ηk ≥ 0 and c const. Then, t psds , qt ≤ c 1 ηk exp t0 <tk <t t0 Journal of Applied Mathematics 5 Lemma 2.3 Hardy-Sobolev inequality 27. Let Ω ⊂ Rm (m ≥ 3) be a bounded open set containing the origin and u ∈ H 1 Ω {ω | ω ∈ L2 Ω, Di ω ∂ω/∂xi ∈ L2 Ω, 1 ≤ i ≤ m}. Then there exists a positive constant Cm Cm Ω such that m − 22 4 u2 Ω dx ≤ |x|2 Ω |∇u|2 dx Cm 2.10 u2 dσ. ∂Ω Lemma 2.4. If a > 0 and b > 0, then ab ≤ 1/εa2 εb2 holds for any ε > 0. 3. Main Results Theorem 3.1. Provided that 2 1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m s 1 xs < β. In addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0. Denote Dm − 22 /2β χ, 2 Pik ui tk , x −θik ui tk , x, 0 ≤ θik ≤ 2, 3 there exists a constant γ satisfying γ λ hρeγτ > 0 as well as λ hρeγτ < 0, where λ maxi 1,...,n −χ − 2ai nj 1 bij2 cij2 ρ, ρ n maxi 1,...,n li 2 , then, the equilibrium point u 0 of problems (2.5–2.8) is globally exponentially stable with convergence rate – λ hρeγτ /2. Proof. Multiplying both sides of 2.1 by ui t, x and integrating with respect to spatial variable x on Ω, we get d Ω m ui 2 t, xdx ∂ ∂ui t, x 2 ui t, x u2i t, xdx Dis dx − 2ai dt ∂x ∂x s s Ω s 1 Ω n ui t, xfj uj t, x dx 2 bij Ω j 1 n 2 cij j 1 Ω ui t, xfj uj t − τj t, x dx tk , k 1, 2, . . . . t ≥ t0 , t / 3.1 Regarding the right-hand part of 3.1, the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.1 m m ∂ ∂ui t, x ∂ui t, x 2 ui t, x Dis dx Dis dx −2 2 ∂xs ∂xs ∂xs s 1 Ω s 1 Ω Dm − 22 ≤− 2 u2i t, x Ω |x|2 Dm − 22 dx ≤ − 2β 3.2 Ω u2i t, xdx −χ Ω u2i t, xdx. 6 Journal of Applied Mathematics Moreover, we derive from H1 that n 2 bij j 1 n bij ui t, xfj uj t, x dx ≤ 2 |ui t, x|fj uj t, x dx Ω Ω j 1 n ≤2 Ω j 1 ≤ lj bij |ui t, x|uj t, xdx n Ω bij2 u2i t, x lj2 u2i t, x dx, j 1 n n 2 cij ui t, xfj uj t − τj t, x dx ≤ 2 cij |ui t, x|fj uj t − τj t, x dx j 1 Ω 3.3 Ω j 1 n ≤2 lj cij |ui t, x|uj t − τj t, x dx Ω j 1 ≤ n Ω j 1 cij2 u2i t, x lj2 u2i t − τj t, x dx. Consequently, substituting 2.10–3.14 into 3.1 produces d Ω u2i t, xdx ≤ −χ u2i t, xdx − 2ai u2i t, xdx dt Ω Ω n bij2 u2i t, x lj2 u2i t, x dx j 1 3.4 Ω n j 1 Ω cij2 u2i t, xlj2 u2i t−τj t, x dx for t ≥ t0 , t / tk , k 1, 2, . . . . We define a Lyapunov function Vi t as Vi t Ω u2i t, xdx. It is easy to find that Vi t is a piecewise continuous function with points of discontinuity of the first kind tk k 1, 2, . . ., where it is continuous from the left, that is, Vi tk − 0 Vi tk k 1, 2, . . .. In addition, due to Vi t0 0 ≤ Vi t0 and the following estimate derived from condition 2 of Theorem 3.1 u2i tk 0, x −θik ui tk , x ui tk , x2 1 − θik 2 u2i tk , x ≤ u2i tk , x k 1, 2, . . ., 3.5 we have Vi tk 0 ≤ Vi tk , k 0, 1, 2, . . . . 3.6 Journal of Applied Mathematics 7 holds for t tk k 0, 1, 2, . . .. Put t ∈ tk , tk1 , k 0, 1, 2, . . . . Then for the derivative dVi t/dt of Vi with respect to problems 2.5–2.8, it results from 3.4 that dVi t ≤ −χ dt Ω u2i t, xdx − 2ai n j 1 Ω Ω u2i t, xdx n j 1 2 cij2 u2i t, x lj2 uj t − τj t, x Ω bij2 u2i t, x lj2 u2i t, x dx ⎛ dx ≤ ⎝−χ−2ai bij2 n j 1 n n max li2 Vj t max li 2 Vj t − τj t i 1,...,n n i 1,...,n j 1 ⎞ cij2 ⎠Vi t j 1 t ∈ tk , tk1 , k 0, 1, 2, . . . . j 1 3.7 Choose V t of the form V t n i 1 Vi t. From 3.7, one then reads ⎛ ⎞ ⎞ ⎛ n dV t ⎝ 2 2 2 bij cij ⎠ n max li ⎠V t ≤ max ⎝−χ − 2ai i 1,...,n i 1,...,n dt j 1 n n max li 2 Vj t − τj i 1,...,n λV t ρ 3.8 j 1 n Vj t − τj t t ∈ tk , tk1 , k 0, 1, 2, . . . . j 1 Construct V ∗ t eγt−t0 V t, where γ satisfies γ λ hρeγτ > 0 and λ hρeγτ < 0. Evidently, V ∗ t is also a piecewise continuous function with points of discontinuity of the first kind tk k 1, 2, . . ., in which it is continuous from the left, that is V ∗ tk − 0 V ∗ tk k 1, 2, . . .. Moreover, at t tk k 0, 1, 2, . . ., we find by use of 3.6 V ∗ tk 0 ≤ V ∗ tk , k 0, 1, 2, . . . . 3.9 Set t ∈ tk , tk1 , k 0, 1, 2, . . .. By virtue of 3.8, one has ⎞ ⎛ n dV ∗ t γt−t0 γt−t0 dV t γt−t0 γe ≤ γe V t e V t ⎝λV t ρ Vj t − τj t ⎠eγt−t0 dt dt j 1 n γ λ V ∗ t ρeγt−t0 Vj t − τj t t ∈ tk , tk1 , k 0, 1, 2, . . . . j 1 3.10 8 Journal of Applied Mathematics Choose small enough ε > 0. Integrating 3.10 from tk ε to t gives ∗ ∗ V t ≤ V tk ε γ λ t γs−t0 ρe tk ε t V ∗ sds tk ε 3.11 n Vj s−τj s ds, t ∈ tk , tk1 , k 0, 1, 2, . . . j 1 which yields after letting ε → 0 in 3.11 V ∗ t ≤ V ∗ tk 0 γ λ t V ∗ sds tk t γs−t0 ρe n Vj s−τj s ds, t ∈ tk , tk1 , k 0, 1, 2, . . . . tk 3.12 j 1 We now proceed to estimate the value of V ∗ t at t tk1 , k 0, 1, 2, . . . . For small enough ε > 0, we put t tk1 − ε. Now an application of 3.12 leads to, for k 0, 1, 2, . . ., ∗ ∗ V tk1 − ε ≤ V tk 0 γ λ tk1 −ε ∗ V sds tk1 −ε tk ρeγs−t0 tk n Vj s − τj s ds. j 1 3.13 If we let ε → 0 in 3.13, there results ∗ ∗ V tk1 − 0 ≤ V tk 0 γ λ tk1 V ∗ sds tk tk1 γs−t0 ρe tk n Vj s − τj s ds, 3.14 k 0, 1, 2, . . . . j 1 Note that V ∗ tk1 − 0 V ∗ tk1 is applicable for k 0, 1, 2, . . . . Thus, V ∗ tk1 ≤ V ∗ tk 0 γ λ tk1 V ∗ sds tk1 tk tk ρeγs−t0 n Vj s − τj s ds 3.15 j 1 holds for k 0, 1, 2, . . . . By synthesizing 3.12 and 3.15, we then arrive at ∗ ∗ V t ≤ V tk 0 γ λ t V ∗ sds tk γs−t0 ρe tk t n Vj s − τj s ds j 1 3.16 t ∈ tk , tk1 , k 0, 1, 2, . . . . Journal of Applied Mathematics 9 This, together with 3.9, results in t V ∗ t ≤ V ∗ tk γ λ V ∗ sds t tk n ρeγs−t0 Vj s−τj s ds tk 3.17 j 1 for t ∈ tk , tk1 , k 0, 1, 2, . . . . • Recalling assumptions that 0 ≤ τj t ≤ τ and τ j t < 1 − 1/h h > 0, we have t ρeγs−t0 tk n t−τj t n Vj s − τj s ds j 1 j 1 ≤ hρe tk −τj tk γτ ρeγθτj s−t0 Vj θ n t−τj t j 1 γθ−t0 e tk −τj tk 1 • 1 − τ j s dθ 3.18 Vj θdθ. Hence, V ∗ t ≤ V ∗ tk γ λ t V ∗ sds hρeγτ tk n t−τj t tk −τj tk j 1 eγs−t0 Vj sds 3.19 t ∈ tk , tk1 , k 0, 1, 2, . . . . By induction argument, we reach ∗ ∗ V tk ≤ V tk−1 γ λ tk ∗ V sds hρe γτ tk−1 n tk −τj tk j 1 tk−1 −τj tk−1 eγs−t0 Vj sds, .. . V ∗ t2 ≤ V ∗ t1 γ λ t2 V ∗ sds hρeγτ t1 V ∗ t1 ≤ V ∗ t0 γ λ t1 n t2 −τj t2 t1 −τj t1 j 1 V ∗ sds hρeγτ t0 n t1 −τj t1 t0 −τj t0 j 1 eγs−t0 Vj sds, eγs−t0 Vj sds. Therefore, ∗ ∗ V t ≤ V t0 γ λ t ∗ V sds hρe γτ t0 ∗ ≤ V t0 γ λ t t0 n t−τj t j 1 ∗ V sds hρe γτ t0 −τj t0 n t j 1 t0 −τj t0 eγs−t0 Vj sds eγs−t0 Vj sds 3.20 10 Journal of Applied Mathematics V ∗ t0 γ λ hρeγτ t V ∗ sds t0 hρeγτ n t0 j 1 t0 −τj t0 eγs−t0 Vj sds t ∈ tk , tk1 , k 0, 1, 2, . . . . 3.21 Since hρeγτ n t0 j 1 t0 −τj t0 hρeγτ t0 t0 −τ eγs−t0 Vj sds ≤ hρeγτ n t0 j 1 ⎛ ⎝ Vj sds 3.22 ⎞ n j 1 t0 −τ Ω 2 ϕ2j s, xdx⎠ds ≤ τhρeγτ ϕΩ , we claim 2 V ∗ t ≤ V ∗ t0 τhρeγτ ϕΩ γ λ hρeγτ t V ∗ sds t ∈ tk , tk1 , k 0, 1, 2 . . . . t0 3.23 According to Lemma 2.2, we claim ∗ V t ≤ 2 V t0 τhρe ϕ Ω exp γ λ hρeγτ t − t0 , ∗ γτ t ≥ t0 3.24 which reduces to u t, x; t0 , ϕ ≤ 1 τhρeγτ ϕ exp Ω Ω λ hρeγτ t − t0 , 2 t ≥ t0 . 3.25 This completes the proof. Remark 3.2. According to the conditions of Theorem 3.1, we see that the reaction-diffusion term do influence the stability of problem 2.5–2.8. Moreover, besides the reactiondiffusion coefficients, the dimension of the space and the boundary of spatial variables have also an effect on the stability of the equilibrium point u 0. Theorem 3.3. Providing that 2 1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exist constants β such that |x|2 m s 1 xs < β, in addition, there exists constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote Dm − 22 /2β χ, √ √ 2 Pik ui tk , x −θik ui tk , x, 1 − 1 α ≤ θik ≤ 1 1 α, α ≥ 0, 3 infk 1,2... tk − tk−1 > μ, Journal of Applied Mathematics 11 4 there exists constant γ which satisfies γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0, where λ maxi 1,...,n −χ − 2ai nj 1 bij2 cij2 ρ and ρ n maxi 1,...,n li 2 , then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with convergence rate –1/2λ hρeγτ ln1 α/μ. Proof. Define a Lyapunov function V of the form V t ni 1 Vi t, where Vi t Ω u2i t, xdx. Obviously, V t is a piecewise continuous function with points of discontinuity of the first kind tk , k 1, 2, . . ., where it is continuous from the left, that is, V1 tk −0 V1 tk k 1, 2, . . .. Furthermore, for t tk k 0, 1, 2, . . ., it follows from condition 2 of Theorem 3.3 that u2i tk 0, x − u2i tk , x 1 − θik 2 u2i tk , x − u2i tk , x ≤ αu2i tk , x. 3.26 V tk 0 ≤ αV tk V tk , k 0, 1, 2, . . . . 3.27 Thereby, Construct another Lyapunov function defined by V ∗ t eγt−t0 V t, where γ satisfies γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0. Then, V ∗ t is also a piecewise continuous function with points of discontinuity of the first kind tk , k 1, 2, . . ., where it is continuous from the left, that is V ∗ tk − 0 V ∗ tk k 1, 2, . . .. And for t tk k 0, 1, 2, . . ., it results from 3.27 that V ∗ tk 0 ≤ αV ∗ tk V ∗ tk , k 0, 1, 2, . . . . 3.28 Set t ∈ tk , tk1 , k 0, 1, 2, . . . . Following the same procedure as in Theorem 3.1, we get ∗ ∗ V t ≤ V tk 0 γ λ t V ∗ sds hρeγτ tk × n t−τj t tk −τj tk j 1 eγθ−t0 Vj θdθ t ∈ tk , tk1 , k 0, 1, 2, . . . . 3.29 The relations 3.28 and 3.29 yield V ∗ t − V ∗ tk ≤ αV ∗ tk γ λ t V ∗ sds hρeγτ tk × n t−τj t j 1 tk −τj tk eγθ−t0 Vj θdθ t ∈ tk , tk1 , k 0, 1, 2, . . . . 3.30 12 Journal of Applied Mathematics By induction argument, we arrive at ∗ ∗ ∗ V tk − V tk−1 ≤ αV tk−1 γ λ tk ∗ V sds hρe γτ tk−1 n tk −τj tk j 1 tk−1 −τj tk−1 eγθ−t0 Vj θdθ, .. . ∗ ∗ ∗ t2 V t2 − V t1 ≤ αV t1 γ λ ∗ V sds hρe γτ n t2 −τj t2 t1 t1 V ∗ t1 − V ∗ t0 ≤ αV ∗ t0 γ λ j 1 V ∗ sds hρeγτ t1 −τj t1 n t1 −τj t1 t0 j 1 t0 −τj t0 eγθ−t0 Vj θdθ, eγθ−t0 Vj θdθ. 3.31 Hence, V ∗ t − V ∗ t0 ≤ αV ∗ t0 γ λ t V ∗ sds t0 hρeγτ n t−τj t j 1 t0 −τj t0 eγθ−t0 Vj θdθ α V tk t0 <tk <t ∗ ≤ αV t0 γ λ hρe γτ t V ∗ sds t0 n t0 hρeγτ j 1 t0 −τj t0 eγθ−t0 Vj θdθ α V tk t ∈ tk , tk1 , k 0, 1, 2, . . . . t0 <tk <t 3.32 2 t Introducing hρeγτ nj 1 t00 −τj t0 eγθ−t0 Vj θdθ ≤ τhρeγτ ϕΩ as shown in the proof of Theorem 3.1 into 3.32, the expression becomes 2 V ∗ t − V ∗ t0 ≤ αV ∗ t0 τhρeγτ ϕΩ γ λ hρeγτ α V tk t V ∗ sds t0 3.33 t ∈ tk , tk1 , k 0, 1, 2 . . . . t0 <tk <t It then results from Lemma 2.2 that V ∗ t ≤ 2 α 1V ∗ t0 τhρeγτ ϕΩ 2 α 1V ∗ t0 τhρeγτ ϕ Ω 1 α exp γ λ hρeγτ t − t0 t0 <tk <t 1 αk exp γ λ hρeγτ t − t0 , 3.34 t ≥ t0 . Journal of Applied Mathematics 13 On the other hand, since infk 1,2,... tk − tk−1 > μ, one has k < tk − t0 /μ. Thereby, 1 αk < exp ln1 α tk − t0 μ < exp ln1 α t − t0 . μ 3.35 And 3.34 can be rewritten as V ∗ t ≤ 2 ln1 α γ λ hρeγτ α 1V ∗ t0 τhρeγτ ϕΩ exp t − t0 μ 3.36 which implies 1 ln1 α γτ γτ ϕ exp u t, x; t0 , ϕ ≤ α 1 τhρe λ hρe − t , t 0 Ω Ω 2 μ t ≥ t0 . 3.37 The proof is completed. Remark 3.4. Theorem 3.1 is in fact the special case of Theorem 3.3 by choosing α 0. Due to Lemma 2.4, we know n n ui t, xf uj t, x dx ≤ 2 bij j 1 n 2 cij j 1 Ω Ω ui t, xf uj t − τj , x dx ≤ j 1 Ω n j 1 ε1 bij2 u2i t, x Ω ε2 cij2 u2i t, x lj2 ε1 u2i t, x lj2 ε2 dx, u2i t − τj , x dx 3.38 hold for any ε1 , ε2 > 0. In the sequel, we follow the same procedures as in Theorems 3.1 and 3.3 to find the following theorems. Theorem 3.5. Provided that 2 1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m s 1 xs < β. in addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote Dm − 22 /2β χ, 2 Pik ui tk , x −θik ui tk , x, 0 ≤ θik ≤ 2, 3 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ < 0, where λ maxi 1,...,n −χ − 2ai nj 1 ε1 bij2 ε2 cij2 n/ε1 maxi 1,...,n li2 and ρ n/ε2 maxi 1,...,n li2 , then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with convergence rate –λ hρeγτ /2. Theorem 3.6. Assume that 2 1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m s 1 xs < β; In addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0; denote Dm − 22 /2β χ, 14 Journal of Applied Mathematics 2 Pik ui tk , x −θik ui tk , x, 1 − √ 1 α ≤ θik ≤ 1 √ 1 α, α ≥ 0, 3 infk 1,2,... tk − tk−1 > μ, 4 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0, where λ maxi 1,...,n −χ − 2ai nj 1 ε1 bij2 ε2 cij2 n/ε1 maxi 1,...,n li2 and ρ n/ε2 maxi 1,...,n li2 . Then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with convergence rate −1/2λ hρeγτ ln1 α/μ. 2 < α−1/2 and α ≥ 1, Further, on the condition that |Pik ui tk , x| ≤ θik |ui tk , x|, where θik we obtain, for t tk (k 1, 2, . . .), u2i tk 0, x − u2i tk , x Pik ui tk , x ui tk , x2 − u2i tk , x ≤ 2ui tk , x2 2Pik ui tk , x2 − u2i tk , x ≤ 2 2θ2ik ui tk , x2 − u2i tk , x ≤ αu2i tk , x. 3.39 Identical with the proof of Theorem 3.3, we present the theorem as follows. Theorem 3.7. Assume that 2 1 for x x1 , . . . , xm T ∈ Ω(m ≥ 3), there exists a constant β such that |x|2 m s 1 xs < β. in addition, there exists a constant D > 0 such that Dis Dis t, x, u ≥ D > 0. Denote Dm − 22 /2β χ, 2 2 |Pik ui tk , x| ≤ θik |ui tk , x|, where θik ≤ α − 1/2 and α ≥ 1, 3 infk 1,2,... tk − tk−1 > μ, 4 there exist constants γ and ε1 , ε2 > 0 such that γ λ hρeγτ > 0 and λ hρeγτ ln1 α/μ < 0, where λ maxi 1,...,n −χ − 2ai nj 1 ε1 b2 ij ε2 cij2 n/ε1 maxi 1,...,n li 2 and ρ n/ε2 maxi 1,...,n li 2 . Then, the equilibrium point u 0 of problem (2.5–2.8) is globally exponentially stable with convergence rate −1/2λ hρeγτ ln1 α/μ. Remark 3.8. Different from Theorems 3.1–3.6, the impulsive part in Theorem 3.7 could be nonlinear and this will be of more applicability. Actually, Theorems 3.1–3.6 can be regarded as the special cases of Theorem 3.7. 4. Examples Example 4.1. Consider the following impulsive reaction-diffusion delayed neural network: m n n ∂ui t, x ∂ ∂ui t, x Dis − ai ui t, x bij fj uj t, x cij fj uj t − τj t, x ∂t ∂xs ∂xs s 1 j 1 j 1 t ≥ 0, t/ tk , x ∈ Ω, k 1, 2, . . . , i 1, . . . , n 4.1 Journal of Applied Mathematics 15 with the impulsive effects characterized by u1 tk 0, x u1 tk , x 1.343u1 tk , x, u2 tk 0, x u2 tk , x 1.343u2 tk , x k 1, 2, . . . , x∈Ω 4.2 and initial condition 2.3 and Dirichlet condition 2.4, n 2, m 4, Ω 1.2 where T 4 2 1.3 2.3 2.5 3.1 {x1 , . . . , x4 | i 1 xi < 4}, a1 a2 6.5, Dis 2×4 1.8 3.2 2.7 3.4 , bij 2×2 −0.23 −0.1 3 , −0.1 −0.2 • cij 2×2 0.1 −0.3 , fj uj 1/4|uj 1| − |uj − 1|, 0 ≤ τj t ≤ 0.5, and τ j t < 0. For β 4 and D 1.2, we compute χ 0.6. This, together with the chosen li 1/2, yields ⎛ λ max ⎝−χ − 2ai i 1,...,n n ⎞ bij2 cij2 ⎠ n max li2 −4, j 1 i 1,...,n 1 ρ n max li2 . i 1,...,n 2 4.3 By selecting γ 2.6, τ 0.5 and h 1, we estimate 1 γ λ hρeγτ 2.6 − 4 e1.3 > 0, 2 1 λ hρeγτ −4 e1.3 < 0. 2 4.4 According to Theorem 3.1, we therefore conclude that the system in Example 4.1 is globally exponential stable. Example 4.2. Consider the following impulsive reaction-diffusion delayed neural network: m n n ∂ui t, x ∂ ∂ui t, x Dis − ai ui t, x bij fj uj t, x cij fj uj t − τj t, x ∂t ∂x ∂x s s s 1 j 1 j 1 t ≥ 0, t/ tk , x ∈ Ω, k 1, 2, . . . , i 1, . . . , n 4.5 with the impulsive effects featured by u1 tk 0, x u1 tk , x arctan0.5u1 tk , x, u2 tk 0, x u2 tk , x arctan0.5u2 tk , x k 1, 2, . . . , x ∈ Ω 4.6 and initial condition 2.3 and Dirichlet condition 2.4, n 2, m 4, Ω where −0.23 1.3 , 2.3 2.5 3.1 , b {x1 , . . . , x4 T | 4i 1 xi2 < 4}, a1 a2 6.5, Dis 2×4 1.2 ij 2×2 −0.1 3 1.8 3.2 2.7 3.4 −0.1 −0.2 • τ cij 2×2 u 1/4|u 1| − |u − 1|, 0 ≤ τ t ≤ 0.5, t < 0, and , f j j j j j j 0.1 −0.3 16 Journal of Applied Mathematics infk 1,2,... tk − tk−1 > 1. For β 4 and D 1.2, we computeχ 0.6. This, together with li 1/2 and ε1 ε2 1, yields 1 n ρ max li2 , ε2 i 1,...,n 2 ⎛ λ max ⎝−χ − 2ai n i 1,...,n j 1 ⎞ n ε1 bij2 ε2 cij2 ⎠ max li2 −4. ε1 i 1,...,n 4.7 Select α 1.5 by setting θik 0.5. 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