Noise-to-State Stability for Stochastic Hopfield Neural Networks with Delays

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1 Noise-to-State Stability for Stochastic Hopfield Neural Networks with Delays Yuli Fu and Xurong Mao Department of Statistics and Modelling Science,University of Strathclyde Glasgow G1 1XH,Scotland, UK Abstract It is well known that stability of Hopfield type neural networks plays a very important role in both theoretical research and applications. So, it has been kept on studying in two decades. Stochastic effectiveness to this kind of neural networks has also received a lot of attention (ref. [Liao et al, 1996 A], [Liao et al, 1996 B], [Blythe,S. et al, 2001A] and [Blythe,S. et al, 2001B]). In this paper, we intend to present some new results on robust noise-to-state stability of delayed Hopfield neural network via Razumikhin technique. Our motivation is based on the intention to discover the effect of uncertain stochastic noise to the state of the neural networks. An example shows the kind of stability could be used to design a kind of Hopfield neural networks described by cascade stochastic systems to implement associative memory. 1 Introduction Hopfield type neural networks have been applied to associative memory, since the dynamical characteristic of them can be exploited. By this reason, stability with respect to the equilibrium points of Hopfield type neural networks have been studied. For example, the attraction of the set of equilibrium points was studied in the original work of Hopfield [Hopfield, 1982] and [Hopfield, 1984], the Lyapunov stability of the equilibrium points was contributed (e.g., it can be seen in [Liao et al, 1996 A], [Liao et al, 1996 B] and [Liao, 1993]). Recently, the stochastic Hopfield neural networks have been concerned. Many contributions can be found, for example, in [Liao et al, 1996 A], [Liao et al, 1996 B] and in [Blythe,S. This work is supported by the Royal Society of UK. The author is on leave of school of electronics and information engineering, the South China University of Technology, Guangzhou, PR China et al, 2001A], [Blythe,S. et al, 2001B] for the neural networks with delay. Noise-to-state stability was proposed by some authors (see, for instance, [Deng et al, 2001], [N.S. Edgar et al, 1999] and [Stontag et al, 1994]). The original meaning of this kind of stability is to reveal the relationship between stochastic noise and the state in some dynamical systems with stochastic noise perturbations or input. In our view, for stochastic Hopfield neural networks, effect of uncertain noise can confine the state to some region or can be applied to estimate the attraction of the equilibrium points. Throughout this paper, R n denotes the n- dimensional real space, R + denotes the interval [0, + ), R m n denotes the m n matrix space. By, we stand for the appropriate norm of a vector or a matrix. For arbitrary square matrix A, by λ max (A) and λ min (A), we denote the maximal and minimal eigenvalue of it, respectively. If A is a symmetric real matrix, the expression A > 0 means the matrix is positive definite. For every t 0, denoted by L 2 F t ([ r, 0]; R n ) the family of all F t - measurable C([ r, 0]; R n )-valued random variables φ = {φ(θ) : r θ 0} such that φ 2 L := 2 sup r θ 0 E φ(θ) 2 < +. Let L 2 (Ω; R n ) denote the family of all R n -valued random variables X such that E X 2 < +. Function γ : R + R + is said to be in class K, if γ(0) = 0, and γ(s) increases to + as s + (ref., e.g., [Mao, 1997]). Consider stochastic Hopfield neural networks with delay C i dx i (t) = 1 n x i (t)dt + T ij s j (x j (t δ(t)))dt R i j=1 m m + h ik (t, x(t), x(t δ(t)))σ kl (t)db l (t), t 0, k=1 l=1 (1.1) i = 1,..., n, where C i are the input capacitances, δ(t)is the time delay mapping R + on to [0, r] continuously, r > 0. Transfer function s j ( ) depicts the feature of each neuron in the network. The connection matrix is (T ij ) m m. The each element of

2 the connection matrix denotes the weight of connection between the ith and jth neuron. As usual, the transfer function s j (u) of the neuron is assumed to be a sigmoid function saturating at +1 and 1, with maximum slope at u = 0. That means, each s j (u) is nondecreasing and satisfies s j (u) min{1, β j u }, for u (1.2) constant β j denotes the finite slope of transfer function s j (u) at u = 0.The functions h ik : R + R n R n R 1, i = 1,..., n, l = 1,..., m, are continuous. σ(t) := (σ kl (t)) m m is a uncertain matrixvalue function from R + to R m m. The information of it is uncertain, usually, may be only nonnegative definite for each t, Borel measurable and even bounded in some sense (see, e.g., [Mao, 1997], [Deng et al, 2001]). More over, vector B(t) := (B 1 (t),..., B m (t)) denotes an m-dimensional standard Brownian motion defined on a complete probability space (Ω, F t, P ) with the natural filtration {F t } t 0. σ(t)σ T (t) is the covariance matrix of the stochastic noise γ(t)db(t). We consider the initial value x i (θ) = ξ i (θ), for θ [ r, 0] (1.3) associated with the neural network (1.1). From [Mao, 1994] or [Mohammed, 1986], we find that the solution to the system (1.1) with (1.3) exits uniquely. And obviously, if h ik (0, 0) = 0, the origin is an equilibrium point of neural network (1.1). For simplicity, the neural network (1.1) with initial value (1.3) will be re-written in vector-matrix form: dx(t) = A 1 x(t)dt + A 2 S(x(t δ(t)))dt+ h(t, x(t), x(t δ(t)))σ(t)db(t), t 0, (1.4) x(θ) = ξ(θ), θ [ r, 0] (1.5) where x(t) = (x 1 (t),..., x n (t)) T, A 1 := diag(c 1 1 R 1 1,..., C 1 n Rn 1 ) A 2 := (T ij C 1 i ) n n S(x(t)) := (s 1 (x 1 (t δ(t)),..., s n (x n (t δ(t)))) h(t,, ) := (h ik (t,, )) n m In general, the initial value ξ(θ) = (ξ 1 (θ),..., ξ n (θ)) T is assumed to be in L 2 F ([ r, 0], R n ). The unknown stochastic noise σ(t)db(t) could be characterized by its uncertain covariance σ(t)σ T (t). However, by some partially observed information, the uncertainty could be estimated roughly. For example, σ(t) g(t) for some function g : R + R + (it could be a nonnegative constant). The known function g(t) is said to be the robust bound of the unknown stochastic noise. In practice, the norm of σ usually is taken to be the Frobenius norm σ(t) F, Obviously, if g(t) 0, the problem is reduced to the determinate case. From now on, we assume that the robust bound g(t) does not identically equal to zero. Now, we define robustness of noise-to-state stability of the stochastic neural network (1.1)with initial value (1.3) (or equivalently (1.4) with (1.5)). The solution to the initial value problem (1.4) with (1.5)((1.1) with (1.3)) is denoted by x(t; ξ). Definition 1.1 The neural network (1.1) with initial value (1.3) is said to be robust noise-to-state exponentially stable in mean square, with respect to the robust bound g(t), if there are positive constants λ, T 0, η 1 and η 2 such that E x(t; ξ) 2 η 1 e λt E ξ 2 L 2 + η 2 for all t T sup 0 θ t g(θ), (1.6) From this definition, we find that, the asymptotic behavior of the neural network can be estimated explicitly by the robust bound of stochastic noise ultimately (i.e., for time t T ). That is, the expectation of the state gain is limited terminally by the robust bound. 2 A General Lemma In this section, in mathematical point of view, for more general delay system we present a Razumikhin type theorem of robust exponential noise-to-state stability. Consider stochastic nonlinear delay differential equations dx(t) = f(t, x(t), x(t τ(t)))dt+ +h(t, x(t), x(t τ(t)))σ(t)db(t), t 0, (2.1) associated with the initial value (1.5), where τ(t) maps R + on to [0, r] continuously, function f : R + R n R n R n is continuous, h, σ and B(t) are defined as before. Assume that the system (2.1) with initial value (1.5) satisfies some conditions guaranteeing the existence and uniqueness of the solution, (refer [Mao, 1994] or [Mohammed, 1986]). Lemma 2.1 If there exist constants λ > 0, α 1 > 0, α 2 > 0 and µ > 1. Assume that there is a twice differentiable function V : R n R + satisfies: 1) α 1 x 2 V (x) α 2 x 2 for all x R n ; 2) For those X, Y L 2 (Ω, R n ) satisfying EV (Y ) µev (X), (2.2) and continuous robust bounded g(t) defined in the last section, it follows that E(LV (t, X, Y )) λev (X) + g(t), (2.3) where operator LV : R + R n R n R 1 is defined by V (X) LV (t, X, Y ) := f(t, X, Y )+ X trace[σt (t)h(t, X, Y ) 2 V (X) X 2 h(t, X, Y )σ(t)]

3 V (x) V (x) V (x) := (,..., ), x x 1 x n 2 V (x) x 2 := ( 2 V (x) x i x j ) n n Then, the system (2.1) is robust exponentially noiseto-state stable with respect to the robust bound g(t). Proof Let x(t) := x(t; ξ) be the solution to system (2.1) with its initial value (1.5), the fixed time T 0 guarantee that g(t ) > 0. Set a function u(t) := e λt V (x(t)) t 0 e λw g 1 (w)dw, for t r where λ > 0 and continuous function g 1 (t) are selected to satisfy following conditions. If g(0) > 0, g 1 (t) := If g(0) = 0, { g 1 (t) := 0 < λ min{ logµ r, λ} { g(t), for t 0 g(0), for r t 0 g(t), for t 0 2g(T )t/( r) for r t 0 Letting ε 0 := max{g(0), g(t )} > 0, we claim that Eu(t) V 0 := sup r θ 0 EV (ξ(θ)) + ε 0, for t 0, (2.4) Otherwise, there exists a t 0 such that Eu(t) E(u(t )) = V 0, for 0 t t and there is a sequence t n 0 (i.e., is decreasing and tends to zero ) as n +, such that E(u(t + t n )) > V 0, (2.5) From the definition of g 1, it is easy to find out that if t [ r, 0], Eu(t) = e λt EV (x(t)) 0 EV (x(t)) + r t 0 e λw g 1 (w)dw g 1 (w)dw = 0 EV (x(t)) + g 1 (w))dw V 0 r Thus, it follows Eu(t τ(t )) Eu(t ). By computation, we have EV (x(t τ(t ))) e λτ(t ) EV (x(t )) t e λ(t τ(t )) e λw g 1 (w)dw t τ(t ) e λr EV (x(t )) Recalling the condition on λ, we obtain EV (x(t τ(t ))) µev (x(t )) The second condition of this lemma is satisfied automatically. From EV (x(t )) ε 0 > 0, one derives the strict inequality ELV (x(t ), x(t τ(t ))) λev (x(t )) + g(t )) < λev (x(t )) + g(t ) Because of the continuity of the solution x(t) functions V, g and τ, one can select a sufficiently small constant ζ > 0, such that ELV (x(t), x(t τ(t))) λev (x(t)) + g(t) holds for all t [t, t + ζ]. Employing Itô s formula, recalling t 0, we have E(u(t + ζ)) E(u(t )) = = t +ζ t t +ζ t ELu(t)dt e λt {ELV (x(t), x(t τ(t)))+ + λev (x(t)) g(t)}dt 0 which leads to a contradiction of inequality (2.5). Therefore, inequality (2.4) holds for all t 0. From the first condition of this lemma and the property of function g(t), we obtain EV (x(t)) e λt V0 + e λt t 0 e λw g(w)dw E x(t) 2 α 2 e λt {E ξ 2 L α + ε 0 }+ 1 sup g(θ) 2 1 α 2 λα 1 0 θ t α 2 α 1 e λt E ξ 2 L 2 + λ +1 λα 1 sup g(θ) 0 θ t for all t T. This completes the proof. We can see that, if the robust bound satisfies g(0) > 0, the time T can be fixed at origin. So, the estimation will hold for all t 0 in this case. Applying this lemma, we can get more results of robust exponential noise-to-state stability for stochastic Hopfield neural network. 3 Main results Let us return to the stochastic Hopfield neural networks (1.4) with the initial value (1.5). We obtain a result of exponential noise-to-state stability as follows Theorem 3.1 Suppose that there exist constants λ > 0, µ > 1, a class K function γ, and a positive definite symmetric matrix P and a continuous robust bound g(t), such that γ( σ(t) 2 F ) g(t) E(2X T P ( A 1 X + A 2 S(Y ))+

4 +trace(σ T (t)h T (X, Y )P h(x, Y )σ(t))) λe(x T P X) + γ( σ(t) 2 F ) (3.1) whenever all of those X, Y L 2 (Ω; R n ) satisfying E(Y T P Y ) µe(x T P X) Then, the neural network (1.4) is robust exponentially noise-to-state stable with respect to the robust bound g(t). The proof of this theorem can be completed from Lemma 2.1 directly by using the Lyapunov function V (X) = X T P X. Concretely, we give some algebraic conditions as corollaries. Corollary 3.1 There exist positive symmetric matrices P and R satisfying the matrix inequality Assume that Q := A T 1 P + P A 1 P A 2 R 1 A T 2 P > 0 h T (X, Y )h(x, Y ) M for some constant M > 0, there is a continuous nonnegative function p(t) satisfying σ(t) 2 F p(t). The inequality β 2 < λ min(q) λ min (P ) λ max (P ) λ max (R) holds, where β > 0, such that S T (w)s(w) β 2 w T w. Then, for any class K function γ satisfying γ(w) Mλ max (P )w, for w 0 the neural network (1.4) is robust exponentially noise-to-state stable with respect to the robust bound g(t) := γ(p(t)). Proof By using the Lyapunov function V (x) = x T P x, we obtain LV (X) = X T (A T 1 P + P A 1 )X + 2X T P A 2 S(Y )+ +trace(σ T (t)h T (X, Y )P h(x, Y )σ(t)) X T (A T 1 P + P A 1 P A 2 R 1 A T 2 P )X+ +S T (Y )RS(Y ) + Mλ max (P ) σ(t) 2 F X T QX + λ max(r)β 2 λ min (P ) Y T P Y + +Mλ max (P ) σ(t) 2 F where X, Y L 2 (Ω, R n ). If where µ > 1 satisfies we have E(Y T P Y ) µe(x T P X) β 2 < µβ 2 < λ min(q) λ min (P ) λ max (P ) λ max (R) ELV (t, X, Y ) λ minq λ max (P ) EXT P X+ + λ max(r)β 2 λ min (P ) (µext P X) + γ( σ(t) 2 F ) ( λ min(q) λ max (P ) λ max(r)β 2 µ )EX T P X +γ( σ(t) 2 F ) λ min (P ) = λex T P X +γ(q(t)) λex T P X +γ( σ(t) 2 F ) where λ = ( λ min(q) λ max (P ) λ max(r)β 2 µ ) > 0 λ min (P ) Thereby, from theorem 3.1, taking the robust bound g(t) = γ(p(t)), the proof is completed. Corollary 3.2 If there exist constants θ 1, θ 2 > 0 such that trace(σ T (t)h T (X, Y )h(x, Y )σ(t)) θ 1 X T X + θ 2 Y T Y + γ( σ(t) 2 F ) γ is in class K. A continuous and nonnegative function g(t) is assumed to be an estimation of the uncertainty: γ( σ(t) 2 F ) g(t). And, there is a positive definite matrix R such that Q 1 := A T 1 + A 2 A 2 R 1 A T 2 θ 1 I n > 0 I n is the n n identity matrix. Assume that max{β 2, θ 2 } < λ min (Q 1 )/(λ max (R) + 1). Then the neural network (1.4) is robust exponentially noiseto-state stable with respect to the robust bound g(t). Proof Taking the Lyapunov function V (x) = x T x, if EY T Y µex T X, µ > 1 satisfies max{β 2, θ 2 } < µmax{β 2, θ 2 } < λ min (Q 1 )/(λ max (R) + 1), we have ELV (t, X, Y ) E[ X T (A T 1 + A 1 A 2 R 1 A T 2 )X + S T (Y )RS(Y )+ +(θ 1 X T X + θ 2 Y T Y )] + γ( σ(t) 2 F ) E(X T Q 1 X λ max (R)β 2 µx T X)+ +(θ 1 + θ 2 µ)e(x T X) + γ( σ(t) 2 F ) (λ mim (Q 1 ) β 2 λ max (R)µ θ 2 µ)e(x T X) + γ( σ(t) 2 F ) By the condition of this corollary, we have ELV (X, Y ) λe(x T X) + γ( σ(t) 2 F ) λ = λ mim (Q 1 ) β 2 λ max (R)µ θ 2 µ > 0, whenever EY T Y µex T X. For the robust bound g(t), by using theorem 3.1, the proof can be completed.

5 4 Application and examples Now, we apply our result to some neural network described by the cascade systems [Sepulchre et al, 1997]. Consider a couple of neural network systems dz(t) = {φ(e( ξ(t) 2 ))Az(t) + DS(z(t 1))}dt +Cz(t 1)dB 1 (t), dξ(t) = ( A 1 ξ(t) + A 2 S(ξ(t 1)))dt +σ(t)db 2 (t), (4.1) where z R nz, ξ R n ξ, n z, n ξ are positive integers, A, D, C, A 1, A 2 are appropriate matrices, B 1 (t), B 2 (t) are the Brownian motions in appropriate dimensional. The function S is the transfer function of a neuron defined before, here, we set it as s i (w) = b i e w e w e w + e w with n i=1 b2 i b = const.. The robust bound of σ(t) can be treated as an input signal. Function φ : R + R 1 is continuous. The subsystems of (4.1)are interconnected in cascading way. Our design goal is to setup some φ and σ(t) such that E( ξ(t) 2 ) s = const., t T > 0 implies lim t + E( z(t) 2 ) = 0. If a pattern has been memorized at the equilibrium of the z subsystem, in the view of associative memory, the design goal means the pattern could be retrieved asymptotically in mean square when the ξ subsystem is noise-to-state stable with respect to some robust bound of σ(t), that could be specified by designer. Thus, the z subsystem can be treated as a storage of the pattern, while the ξ subsystem will yield a stimulation to lead the retrieval (in mean square) of the pattern asymptotically from the z subsystem. Now, we give some conditions to realize it. Example 4.1 If the function φ(s) satisfies φ(s) < λ min(dd T + µ 1 ( β)i z) λ min (A) (4.2) for 0 < s < s = constant, and A 1 +A 2 A T 2 +µ 2 βi ξ is negative definite, σ(t) 2 < s for t T = const., where µ 1, µ 2 > 1 are constants, I z, I ξ denote the identity matrix of dimensional n z and n ξ, respectively, then E( z(t) 2 ) tends to zero as t. By the Lyapunov functions V 1 (z) = 1 2 zt z, V 2 = 1 2 ξt ξ, for EV 1 (ẑ) < µ 1 EV (z), EV 2 (ˆξ) < µ 2 EV 2 (ξ), we have ELV 2 (ξ, ˆξ) E(ξ T ( A 1 + A T 2 A 2 + µ 2 bi ξ )ξ)+ + σ(t) 2 /2 (4.3) ELV 1 (z) E(z T (φ(e( ξ(t) 2 ))A+ +DD T + µ 1 (b )I z)z) (4.4) From the inequality (4.3), we obtain ELV 2 (ξ, ˆξ) λ min (A 1 A T 2 A 2 µ 2 bi ξ )EV 2 (ξ) + s /2 for t T. Our theorem implies that there exists a time T > 0 such that for t T E( ξ(t) 2 ) s From the inequality (4.4), we have ELV 1 (z) Ez T (φ(e( ξ(t) 2 ))A+ +DD T + (b )µ 1I z )z If t T, we can get a constant λ > 0 such that ELV 1 (z) λev 1 (z) By using the typical Razumikhin type theorem (see [Mao, 1997]), the proof can be completed. Numerically, we assume that 5 0 A = D = A 1 = A 2 = 1 1 C = I z, µ 1 = µ 2 = 2, and b = 2. Thus, a := λ min(dd T + µ 2 ( b)) = λ min (A) A 1 + A T A 2 + µ 1 bi xi = 3 9 is negative definite. Let φ(s) = s (s + a ). If the robust bound of σ(t) satisfies σ(t) 2 s e T t, by the theorem, we have a time T > 0 such that E( ξ(t) 2 ) < s for t T (implies φ(e( ξ(t) 2 )) < a ). Therefore, E( z(t) 2 ) tends to the equilibrium (zero) asymptotically. 5 Conclusion In this paper, employing the well known Razumikhin technique, we present some new results of robust noise-to-state exponential stability for stochastic Hopfield neural networks with delays. By this conception, the relationship between the state and uncertain stochastic noise can be revealed in a explicit form. We also proposed a kind of Hopfield neural networks depicted by cascade system. A memorized pattern can be retrieved asymptotically in mean square from one of the subsystem, under some specified stimulation yielded by another subsystem of it.

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