Exponential stabilization of Neural Networks with mixed Time-Varying Delays in state and control - Mai V.Thuan

Exponential stabilization of Neural Networks with mixed Time-Varying Delays in state and control MAI V. THUAN 1,∗ , N. T. T. HUYEN 1 and N. T. M. NGOC 2 1,∗ College of Sciences, Thainguyen university, Thainguyen, Vietnam 2 Thainguyen university of Technology, Thainguyen, Vietnam ∗ Corresponding author: maithuank1@gmail.com Abstract. This paper presents some results on the global exponential stabilization for cellular networks with various activation functions and mixed time- varying delays in state and control. Based on aug- mented time-varying Lyapunov Krasovskii functionals, new delay-dependent conditions for the global expo- nential stabilization are obtained in terms of linear matrix inequalities. Numerical examples are given to illustrate the feasibleness of our results. Key words. Cellular neutral networks, stabilization, neural networks, mixed delay, Lyapunov function, Lin- ear matrix inequalities. 1 Introduction In the area of control, signal processing, pattern recognition, image processing, and association, delayed cellular neural networks have many usedful applica- tions. Some of these applications require that the equi- librium points of the designed network be stable. In both biological and artificial neural systems, time de- lays due to integration and communication are ubiqui- tous, and often become a source of instability. The time delays in electronic neural networks are usually time- varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. Therefor, stabil- ity analysis of delayed cellular neural network is a very important issue, and many stability criteria have been developed in the literature [2, 5, 8] and the references cited therein. Recently, the stabilization issue has been an im- portant focus of research in the control society, and several feedback stabilizing control design approaches have been proposed in [3, 4]. Regarding stabilization of neural netwoks, to the best of our knowledge, only a few results are published. The papers [1, 6, 7] present some stabilization criteria for delayed neural networks. However, the results reported therein not only require the only activation function, but the system matrices are also strictly constrained. In this paper, we consider a stabilization scheme for a general class of delayed neural networks. The novel features here are that the neural networks in con- sideration are time-varying with mixed delay in state and control and with various activation function. We extend the results of [1, 2, 6, 7, 9] to exponential sta- bilization of neural networks various activation funci- tons and mixed time-varying delay in state and con- trol. Using the Lyapunov stability theory and linear matrix inequality (LMI) techniques, a control law with an appropriate gain control matrix is derived to achieve stabilization of the neural networks with mixed time- varying delayed in state and control. The stabilization criteria are obtained in terms of LMIs and hence the gain control matrix is easily determined by numerical Matlab's Control Toolbox. 2 Preliminaries The following notation will be used in this paper: R+ denotes the set of all real non-negative numbers; Rn denotes the n−dimensional space with the scalar product 〈., .〉 and the vector norm ‖ . ‖; Mn×r denotes the space of all matrices of (n× r)−dimensions. AT denotes the transpose of matrix A; A is symmet- ric if A = AT ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λ max (A) = max{Reλ;λ ∈ λ(A)}. xt := {x(t + s) : s ∈ [−h, 0]}, ‖ xt ‖= sups∈[−h,0] ‖ x(t+s) ‖; C([0, t], Rn) denotes the set of all Rn−valued continuous functions on [0, t]; L2([0, t], Rm) denotes the 1 Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên set of all the Rm−valued square integrable functions on [0, t]; Matrix A is called semi-positive definite (A ≥ 0) if 〈Ax, x〉 ≥ 0, for all x ∈ Rn;A is positive definite (A > 0) if 〈Ax, x〉 > 0 for all x 6= 0;A > B means A−B > 0. The notation diag{. . .} stands for a block- diagonal matrix. The symmetric term in a matrix is denoted by ∗. Consider the following cellular neural networks with mixed time-varying delays in state and control of the form x˙(t) = −Ax(t) +W0f(x(t)) +W1g(x(t− τ1(t))) +W2 ∫ t t−τ2(t) h(x(s)) ds+B0u(t) +B1u(t− τ3(t)) +B2 ∫ t t−τ4(t) u(s) ds x(t) = φ(t), t ∈ [−d, 0], d = max{τ1, τ2, τ3, τ4}, (2.1) where x(t) = [x1(t), x2(t), . . . , xn(t)]T ∈ Rn is the state vector of the neural networks; u(t) ∈ L2([0, s], Rm),∀s > 0, is the control input vector of the neural networks; n is the number of neurals, and f(x(t)) = [f1(x1(t)), f2(x2(t)), . . . , fn(xn(t))]T , g(x(t)) = [g1(x1(t)), g2(x2(t)), . . . , gn(xn(t))]T , h(x(t)) = [h1(x1(t)), h2(x2(t)), . . . , hn(xn(t))]T are the neural activation functions. The diagonal matrix A = diag(a1, a2, . . . , an) repre- sents the self-feedback term and W0,W1, W2, B0, B1, B2 are given real constant matrices with appropriate dimensions. The time-varying delay func- tions τ1(t), τ2(t), τ3(t), τ4(t) satisfy the condition 0 ≤ τ1(t) ≤ τ1, τ˙1(t) ≤ δ1 < 1, 0 ≤ τ2(t) ≤ τ2, 0 ≤ τ3(t) ≤ τ3, τ˙3(t) ≤ δ2 < 1, 0 ≤ τ4(t) ≤ τ4. The initial functions φ(t) ∈ C([−d, 0], Rn), d = max{τ1, τ2, τ3, τ4}, with the uniform norm ‖ φ ‖= maxt∈[−d,0] ‖ φ(t) ‖ . We assume that the activation functions f(x), g(x), h(x) are Lipschitz with the Lips- chitz constains fi, gi, hi > 0 : | fi(ξ1)− fi(ξ2) |≤ fi | ξ1 − ξ2 |, | gi(ξ1)− gi(ξ2) |≤ gi | ξ1 − ξ2 |, | hi(ξ1)− hi(ξ2) |≤ hi | ξ1 − ξ2 |, ∀i = 1, 2, . . . , n, ∀ξ1, ξ2 ∈ R. (2.2) Definition 2.1 [9] Given α > 0. The system (2.1) is α−exponentially stable if there exist a positive number N > 0 such that every solution x(t, φ) satisfies the following condition: ‖ x(t, φ) ‖≤ Ne−αt ‖ φ ‖, ∀t ≥ 0. Definition 2.2 [9] Given α > 0. The system (2.1) is globally α−exponentially stabilizable if there is a feedback control u(t) = Kx(t), such that the closed- loop time-delay system x˙(t) = −[A0 −B0K]x(t) +W0f(x(t)) +W1g(x(t− τ1(t))) +B1Kx(t− τ3(t)) +W2 ∫ t t−τ2(t) h(x(s)) ds +B2 ∫ t t−τ4(t) Kx(s) ds x(t) = φ(t), t ∈ [−d, 0], d = max{τ1, τ2, τ3, τ4} (2.3) is α−exponentially stable. We introduce the following technical well-known propositions, which will be used in the proof of our results. Proposition 2.1 [9] Let P,Q be matrices of appro- priate dimensions and Q is symmetric positive definite. Then 2〈Py, x〉 − 〈Qy, y〉 ≤ 〈PQ−1PTx, x〉, ∀(x, y). Proposition 2.2 [9] For any symmetric positive def- inite matrix M > 0, scalar γ > 0 and vector function ω : [0, γ] → Rn such that the integrations concerned are well defined, the following inequality holds (∫ γ 0 ω(s) ds )T M (∫ γ 0 ω(s) ds ) ≤ γ (∫ γ 0 ωT (s)Mω(s) ds ) Proposition 2.3 (Schur complement lemma) [8] Given constant symmetric matrices X,Y, Z with ap- propriate dimensions satisfying X = XT , Y = Y T > 0. Then X + ZTY −1Z < 0 if and only if( X ZT Z −Y ) < 0 or (−Y Z ZT X ) < 0. 2 Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên 3 Main result Let us denote Σ =−AP − PAT − (B0Y + Y TBT0 ) + 2αP +W0D0WT0 + (1− δ1)−1e2ατ1W1D1WT1 + (1− δ2)−1e2ατ3B1BT1 + τ2e2ατ2W2D2WT2 + τ4e2ατ4B2BT2 , G = diag{gi, i = 1, 2, . . . , n}, H = diag{hi, i = 1, 2, . . . , n}, F = diag{fi, i = 1, 2, . . . , n}, g2 = max{g2i , i = 1, 2, . . . , n}, h2 = max{h2i , i = 1, 2, . . . , n},K = −Y P−1, θ = 1 + τ4, λ1 = λmin(P−1), λ2 = λmax(P−1) + λmax(D−11 )g 2τ1 + λmax(D−12 )h 2τ22 + (τ3 + 1 2 τ24 )λmax(K TK). Theorem 3.1. Given α > 0. System (2.1) is α−exponentially stabilizable if there exist a symmet- ric positive definite matrix P ∈ Rn×n, three diagonal positive matrices Di, i = 0, 1, 2 and a matrix Y with ap- propriate dimension such that the following LMI holds: Ξ =  Σ PF PG τ2PH θY T FP −D0 0 0 0 GP 0 −D1 0 0 τ2HP 0 0 −τ2D2 0 θY 0 0 0 −θIm  < 0. (3.1) The stabilizing feedback control is given by u(t) = −Y P−1x(t), and the solution x(t, φ) of the system satisfies ‖ x(t, φ) ‖≤ √ λ2 λ1 e−αt ‖ φ ‖, ∀t ≥ 0. proof. Let us denote X = P−1,K = −Y P−1. With the feedback control u(t) = −Y P−1x(t), we consider the Lyapunov-Krasovskii functional for closed-loop sys- tem (2.3) V (t, xt) = 5∑ i=1 Vi(t, xt), where V1 = xT (t)Xx(t), V2 = ∫ t t−τ1(t) e2α(s−t)gT (x(s))D−11 g(x(s)) ds, V3 = ∫ 0 −τ2 ∫ t t+s e2α(θ−t)hT (x(θ))D−12 h(x(θ)) dθ ds, V4 = ∫ t t−τ3(t) e2α(s−t)xT (s)KTKx(s) ds, V5 = ∫ 0 −τ4 ∫ t t+s e2α(θ−t)xT (θ)KTKx(θ) dθ ds. It easy to check that λ1 ‖ x(t) ‖2≤ V (t, xt) ≤ λ2 ‖ xt ‖2, t ∈ R+. (3.2) Taking derivative of V1 along solutions of the closed- loop system (2.3), we get V˙1 =xT (t)[−XA−ATX −X(B0Y + Y TBT0 )X]x(t) + 2xT (t)XW0f(x(t)) + 2xT (t)XW1g(x(t− τ1(t))) + 2xT (t)XB1u(t− τ3(t)) + 2xT (t)XW2 ∫ t t−τ2(t) h(x(s)) ds + 2xT (t)XB2 ∫ t t−τ4(t) u(s) ds Applying Proposition 2.1 and 2.2 gives 2xT (t)XW0f(x(t)) ≤ xT (t)XW0D0WT0 Xx(t) + fT (x(t))D−10 f(x(t)); 2xT (t)XW1g(x(t− τ1(t)) ≤ (1− δ1)−1e2ατ1xT (t)XW1D1WT1 Xx(t) + (1− δ1)e−2ατ1gT (x(t− τ1(t))D−11 g(x(t− τ1(t))); 2xT (t)XB1u(t− τ3(t)) ≤ (1− δ2)−1e2ατ3xT (t)XB1BT1 Xx(t) + (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2; 2xT (t)XW2 ∫ t t−τ2(t) h(x(s)) ds ≤ τ2e2ατ2xT (t)XW2D2WT2 Xx(t) + 1 τ2 e−2ατ2 (∫ t t−τ2(t) h(x(s)) ds )T D−12 (∫ t t−τ2(t) h(x(s)) ds ) ≤ τ2e2ατ2xT (t)XW2D2WT2 Xx(t) + e−2ατ2 ∫ t t−τ2 hT (x(s))D−12 h(x(s)) ds; 3 Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên 2xT (t)XB2 ∫ t t−τ4(t) u(s) ds ≤ τ4e2ατ4xT (t)XB2BT2 Xx(t) + 1 τ4 e−2ατ4 (∫ t t−τ4(t) u(s) ds )T(∫ t t−τ4(t) u(s) ds ) ≤ τ4e2ατ4xT (t)XB2BT2 Xx(t) + e−2ατ4 ∫ t t−τ4 ‖ u(s) ‖2 ds. Therefor V˙1(t, xt) ≤ xT (t)[−XA−ATX −X(B0Y + Y TBT0 )X]x(t) + xTX(t) [ W0D0W T 0 + (1− δ1)−1e2ατ1W1D1WT1 + (1− δ2)−1e2ατ3B1BT1 + τ2e2ατ2W2D2WT2 + τ4e2ατ4B2BT2 ] Xx(t) + fT (x(t))D−10 f(x(t)) + (1− δ1)e−2ατ1gT (x(t− τ1(t)))D−11 g(x(t− τ1(t))) + (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2 + e−2ατ2 ∫ t t−τ2 hT (x(s))D−12 h(x(s)) ds + e−2ατ4 ∫ t t−τ4 ‖ u(s) ‖2 ds (3.3) Next, the derivatives of Vk, k = 2, . . . , 5 give V˙2(t, xt) ≤ −2αV2 + (g(x(t)))TD−11 (g(x(t))) − (1− δ1)e−2ατ1gT (x(t− τ1(t)))D−11 gT (x(t− τ1(t))); V˙3(t, xt) ≤ −2αV3 + τ2hT (x(t))D−12 h(x(t)) − e−2ατ2 ∫ t t−τ2 hT (x(s))D−12 h(x(s)) ds; V˙4(t, xt) ≤ −2αV4(t, xt) + xT (t)XY TY Xx(t) − (1− δ2)e−2ατ3 ‖ u(t− τ3(t)) ‖2; V˙5(t, xt) ≤ −2αV5(t,Xt) + τ4xT (t)XY TY Xx(t) − e−2ατ4 ∫ t t−τ4 ‖ u(s) ‖2 ds. (3.4) From (3.3)− (3.4), we obtain V˙ + 2αV ≤ xT (t) [ −XA−ATX −X(B0Y + Y TBT0 )X + 2αX +XW0D0WT0 X + (1− δ1)−1e2ατ1XW1D1WT1 X + (1− δ2)−1e2ατ3XB1BT1 X + τ2e2ατ2XW2D2WT2 X + τ4e2ατ4XB2BT2 X + (1 + τ4)XY TY X ] x(t) + fT (x(t))D−10 f(x(t)) + g T (x(t))D−11 g(x(t)) + τ2hT (x(t))D−12 h(x(t)). (3.5) Using the condition (2.2) and since the matrices Di > 0, i = 0, 1, 2 are diagonal, we have fT (x(t))D−10 f(x(t)) ≤ xT (t)FD−10 Fx(t), gT (x(t))D−11 g(x(t)) ≤ xTGD−11 Gx(t), τ2h T (x(t))D−12 h(x(t)) ≤ τ2xT (t)HD−12 Hx(t). (3.6) Since (3.5) and (3.6), we obtain V˙ + 2αV ≤ xT (t)Ωx(t), where Ω =−XA−ATX −X(B0Y + Y TBT0 )X + 2αX +XW0D0WT0 X + (1− δ1)−1e2ατ1XW1D1WT1 X + (1− δ2)−1e2ατ3XB1BT1 X + τ2e2ατ2XW2D2WT2 X + τ4e2ατ4XB2BT2 X + (1 + τ4)XY TY X + FD−10 F +GD −1 1 G+ τ2HD −1 2 H. We have, Ω < 0 if and only if PΩP < 0. And by Schur complement lemma, PΩP < 0 if and only if Ξ < 0. Thus V˙ (t, xt) + 2αV (t, xt) ≤ 0 and hence V (xt) ≤ V (φ)e−2αt ≤ λ2 ‖ φ ‖2 e−2αt, t ≥ 0. Taking (3.2) into account we obtain ‖ x(t, φ) ‖≤ √ λ2 λ1 e−αt ‖ φ ‖, t ≥ 0, which completes the proof. 4 Numerical example Let us consider system (2.1) with τ1(t) = sin 20.5t, τ3(t) = sin20.6t, and{ τ2(t) = 0.8 sin2 t if t ∈ I = ∪k≥0[2kpi, (2k + 1)pi] τ2(t) = 0 if t ∈ R+ \ I, 4 Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên { τ4(t) = 0.5 cos2 t if t ∈ I = ∪k≥0[2kpi, (2k + 1)pi] τ4(t) = 0 if t ∈ R+ \ I, A = ( 32 0 0 21 ) ,W0 = ( 1 0.15 1 2 ) ,W1 = ( 0 1 1 0 ) , W2 = ( 0.1 0.4 0.5 0.2 ) , B0 = ( 0 3 ) , B1 = ( 2 3 ) , B2 = ( 1 2 ) , F = ( 0.1 0 0 0.4 ) , G = ( 0.5 0 0 0.8 ) , H = ( 0.1 0 0 0.6 ) . We see that the time delay function τ2(t), τ4(t) are bounded but non-differentiable and τ1 = 1, τ2 = 0.8, τ3 = 1, τ4 = 0.5, δ1 = 0.5, δ2 = 0.6. For α = 0.5, us- ing MATLABs LMI Toolbox, the LMI (3.1) is feasible with the following matrices: P = ( 6.4893 2.2340 2.2340 15.5236 ) , D0 = ( 47.8388 0 0 47.6199 ) , D1 = ( 47.6449 0 0 49.2631 ) , D2 = ( 52.2809 0 0 52.3712 ) , Y = [ 0.1316 −0.0040] , and accordingly the feedback control is u(t) =[−0.0214 0.0033]x(t). The solution of the closed-loop system satisfy ‖ x(t, φ) ‖≤ 2.9851 ‖ φ ‖ e−0.5t, ∀t ≥ 0. 5 Conclusions In this paper, the problem of designing a feedback control law to exponentially stabilization a class neu- ral networks with various activation and mixed time- varying delay in state and control has been studied. By using augmented Lyapunov-Krasovskii functionals, a new delay-dependent condition for the global expo- nentially stabilization have been established in terms of linear matrix inequalities which allows to compute simultaneously the two bounds that characterize the exponential stability of the solution. A numerical ex- ample is given to show the effectiveness of the obtained results. T½nh ên ành ho¡ ÷ñc d¤ng mô cho h» nìron th¦n kinh câ tr¹ tr¶n tr¤ng th¡i v  i·u khiºn Tâm t­t. Trong b i b¡o n y, chóng tæi ÷a ra mët k¸t qu£ nghi¶n cùu mîi cho t½nh ên ành ho¡ ÷ñc d¤ng mô cho lîp h» nìron th¦n kinh vîi c¡c h m k½nh ho¤t kh¡c nhau v  câ tr¹ tr¶n tr¤ng th¡i v  i·u khiºn. B¬ng vi»c sû döng h m Lyapunov-Krasovskii c£i ti¸n, chóng tæi ¢ ÷a ra ÷ñc mët i·u ki»n õ mîi cho t½nh ên ành ho¡ ÷ñc d¤ng mô cho lîp h» ÷ñc nâi ¸n ð tr¶n. ffii·u ki»n õ cõa chóng tæi ÷ñc biºu di¹n d÷îi d¤ng c¡c b§t ¯ng thùc ma trªn tuy¸n t½nh cho ph²p x¡c ành ÷ñc c¡c ch¿ sè ên ành mô v  cæng thùc i·u khiºn ng÷ñc. Cuèi còng, v½ dö sè ÷ñc ÷a ra º minh ho¤ cho t½nh hi»u qu£ cho k¸t qu£ cõa chóng tæi. T i li»u [1] J. Cao, S. Zhong, and Y. Wu, " Global stability analysis for a class of neural networks with varying delays and control input", Applied Math Comput. , vol. 189, pp. 1480-1490, 2007. [2] B. Chen and J. 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