global exponential stability of discrete time multidirectional associative memory neural network with variable delays

International Scholarly Research Network ISRN Discrete Mathematics Volume 2012, Article ID 831715, 10 pages doi:10.5402/2012/831715 Research Article Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Variable Delays Min Wang, Tiejun Zhou, and Xiaolan Zhang College of Science, Hunan Agricultural University, Hunan, Changsha 410128, China Correspondence should be addressed to Tiejun Zhou, hntjzhou@126.com Received July 2012; Accepted 20 September 2012 Academic Editors: C.-K Lin and W F Smyth Copyright q 2012 Min Wang et al 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 A discrete-time multidirectional associative memory neural networks model with varying time delays is formulated by employing the semidiscretization method A sufficient condition for the existence of an equilibrium point is given By calculating difference and using inequality technique, a sufficient condition for the global exponential stability of the equilibrium point is obtained The results are helpful to design global exponentially stable multidirectional associative memory neural networks An example is given to illustrate the effectiveness of the results Introduction The multidirectional associative memory MAM neural networks were first proposed by the Japanese scholar M Hagiwara in 1990 The MAM neural networks have found wide applications in areas of speech recognition, image denoising, pattern recognition, and other more complex intelligent information processing So they have attracted the attention of many researchers 2–7 In , we proposed a mathematical model of multidirectional associative memory neural network with varying time delays as follows, which consists of the m fields, and there are nk neurons in the field k k 1, 2, , m : dxki dt Iki − aki xki t m np ki ki wpj fpj xpj t − τpj t , 1.1 p 1,p / k j where k 1, 2, , m, i 1, 2, , nk , xki t denote the membrane voltage of the ith neuron in the field k at time t, aki > denote the decay rate of the ith neuron in the field k, fpj · is ISRN Discrete Mathematics ki is the connection weight a neuronal activation function of the ith neuron in the field k, wpj from the jth neuron in the field p to the ith neuron in the field k, Iki is the external input of ki t is the time delay of the synapse from the j neuron in the ith neuron in the field k, and τpj the field p to the ith neuron in the field k at time t We studied the existence of an equilibrium point by using Brouwer fixed-point theorem and obtained a sufficient condition for the global exponential stability of an equilibrium point by constructing a suitable Lyapunov function However, discrete-time neural networks are more important than their continuoustime counterparts in applications of neural networks One can refer to 8–10 in order to find out the research significance of discrete-time neural networks To the best of our knowledge, few studies have considered the stability of discrete-time MAM neural networks In this paper, we first formulate a discrete-time analogues of the continuous-time network 1.1 , and in a next study the existence and the global exponential stability of an equilibrium point for the discrete-time MAM neural network Discrete-Time MAM Neural Network Model and Some Notations In this section we formulate a discrete-time MAM neural network model with time-varying delays by employing the semidiscretization technique m {1, 2, 3, }, Z0 {0, 1, 2, }, and Let N k nk , Z denote the integers set, Z Z a, b {a, a 1, , b} where a, b ∈ Z Let h be a fixed positive real number denoting a uniform discretionary step size and u denote the integer part of the real number u If t ∈ nh, n h n ∈ Z0 , then t/h n By replacing the time t of the network 1.1 with nh, we can formulate the following approximation of the network 1.1 : dxki dt for t ∈ nh, n m Iki − aki xki t np ⎛ ki n ∈ Z0 Denote τpj nh /h dxki dt ki nh τpj h ⎤ ⎞⎞ ⎦h⎠⎠ 2.1 κki pj n , we rewrite 2.1 as follows: np m aki xki t ⎡ ki wpj × fpj ⎝xpj ⎝nh − ⎣ p 1,p / k j 1h ⎛ ki wpj fpj xpj nh − κki pj n h Iki 2.2 p 1,p / k j Multiplying both sides of 2.2 by eaki t , we obtain d xki eaki t dt ⎛ eaki t ⎝Iki m ⎞ np p 1,p / k j ki wpj × fpj xpj nh − κki pj n h ⎠ 2.3 ISRN Discrete Mathematics Integrating the 2.3 over nh, t t ≤ n xki t eaki t − xki nh eaki nh h , we have eaki t − eaki nh aki ⎛ × ⎝Iki ⎞ np m xpj nh − ki wpj fpj κki pj n h 2.4 ⎠ p 1,p / k j Letting t → n h in 2.4 , we obtain xki n − e−aki h aki xki nh e−aki h 1h ⎛ × ⎝Iki ⎞ np m ki wpj fpj xpj nh − κki pj n h 2.5 ⎠ p 1,p / k j If we adopt the notations xki n ki wpj ki wpj xki nh , e−aki h , αki n − e−aki h , aki Iki Iki − e−aki h , aki 2.6 then we obtain the discrete-time analogue of the continuous-time network 1.1 as follows: m xki n Iki np ki wpj fpj αki xki n xpj n − κki pj n 2.7 p 1,p / k j for n ∈ Z0 , k ∈ Z 1, m , i ∈ Z 1, nk Obviously, < αki < Throughout this paper, for any k ∈ Z 1, m , p ∈ Z 1, m p / k , i ∈ Z 1, nk , j ∈ Z 1, np , we assume that the neuronal activation functions fki and the time delays sequences κki pj n satisfy the following conditions, respectively: H1 There exist Lki > such that |fki x − fki y | ≤ Lki |x − y| for each x, y ∈ R, H2 < κki pj supn∈Z κki pj n < ∞ The initial conditions associated with 2.7 are of the form xki n ϕki n , 2.8 where k ∈ Z 1, m , i ∈ Z 1, nk , n ∈ Z −κki , , κki max1≤p≤m,p / k max1≤j≤np κki pj b11 , , b1n1 , b21 , , b2n2 , , bm1 , , bmnm T , x For convenience sake, set col bki col fki xki For any matrixes U uij and V vij , we use the notation col xki , f x ISRN Discrete Mathematics |uij | Let matrix A diag − α11 , , − U ≥ V to mean that uij ≥ vij for all i, j, and |U| α1n1 , − α21 , , − αm1 , , − αmnm , L diag L11 , , L1n1 , L21 , , Lm1 , , Lmnm , ⎛ W O11 W12 · · · W1m ⎞ ⎜ ⎟ ⎜ W21 O22 · · · W2m ⎟ ⎜ ⎟ ⎜ ⎟, ⎜ ··· ··· ··· ··· ⎟ ⎝ ⎠ 2.9 Wm1 Wm2 · · · Omm where ⎛ Wkp k1 wp1 k1 k1 wp2 · · · wpn p ⎞ ⎟ ⎜ ⎜ wk2 wk2 · · · wk2 ⎟ ⎜ p1 pnp ⎟ p2 ⎟, ⎜ ⎜ ··· ··· ··· ··· ⎟ ⎟ ⎜ ⎠ ⎝ knk knk knk wp1 wp2 · · · wpnp 2.10 Okk are zero matrixes k, p ∈ Z 1, m The Existence and Global Exponential Stability of an Equilibrium Point In this section, we will give two theorems about the existence and the global exponential stability of an equilibrium point of the discrete-time MAM neural network 2.7 Lemma 3.1 If < b < 1, ≤ x ≤ − b, then bex ≤ x b Proof Define a function g t − t ln t < t ≤ From g t − t /t > for < t < 1, we know that the function g t is increasing on the interval 0, Therefore, g b < g for < b < So we have − b < − ln b bex − 1, Define a function h x bex − x − b for x ≥ again Obviously h x x be From h x > 0, the h x is increasing Because there exists an unique critical h x for ≤ x ≤ − ln b It shows that h x is number x0 − ln b, we know that h x < h − ln b In view of − b < − ln b, we deceasing on 0, − ln b So we have h x bex − x − b ≤ h obtain bex ≤ x b for ≤ x ≤ − b With a similar method of , we can prove the following Theorem 3.2 Theorem 3.2 Suppose that all the neuronal activation functions fki · k ∈ Z 1, m , i ∈ Z 1, nk are continuous and the condition (H1) holds If B A − |W|L is a nonsingular M matrix, then there exists an equilibrium point of the discrete-time MAM neural network 2.7 Proof Let βki β col βki , r m p 1, / k col rki np j ki |wpj fpj |/ − αki A − |W|L −1 |Iki |/ − αki Obviously, βki ≥ Denote Aβ Because Aβ ≥ and B A − |W|L is a nonsingular ISRN Discrete Mathematics M matrix, by Lemma A3 in 11 , we have r A − |W|L −1 Aβ ≥ That is rki ≥ for any k ∈ Z 1, m , i ∈ Z 1, nk From the definition of r, we have E − A−1 |W|L r β Therefore, ⎡ ⎣ − αki p m ⎤ np Lpj rpj ⎦ ki wpj βki rki 3.1 1,p / k j Let Ω {x col xki | xki ∈ −rki , rki } with a norm x max1≤k≤m max1≤i≤nk {|xki |} Obviously, Ω is a bounded closed compact subset Define a function F : Ω → RN as F x col Fki x , where Fki x ⎡ ⎣ − αki p m ⎤ np Iki ⎦ ki wpj fpj xpj 3.2 1,p / k j From the condition H1 and 3.1 , we have |Fki ⎡ ⎣ x |≤ − αki p ⎡ ⎣ ≤ − αki p ⎡ ⎣ − αki p m ⎤ np ki wpj Lpj xpj |Iki |⎦ fpj 1,p / k j m ⎤ np ki wpj Lpj rpj |Iki |⎦ fpj 3.3 1,p / k j m np ⎤ ki wpj Lpj rpj ⎦ βki rki 1,p / k j Thus F x is a self-map from Ω to Ω By Brouwer fixed-point theorem, there exists at least a x∗ That is x∗ ∈ Ω, such that F x∗ ∗ xki Iki ∗ αki xki m np ki ∗ wpj fpj xpj 3.4 p 1,p / k j Therefore x∗ is an equilibrium point of the MAM neural network 2.7 Next we prove the global exponential stability of the equilibrium point of the discretetime MAM neural network 2.7 Theorem 3.3 Suppose that all the neuronal activation functions fki · k ∈ Z 1, m , i ∈ Z 1, nk are continuous and the conditions (H1) and (H2) hold If B A − |W|L is a nonsingular M matrix, then the equilibrium point of the MAM neural network 2.7 is global exponential stable ∗ be an equilibrium point for the MAM neural network 2.7 , x n Proof Let x∗ col xki col uki n , where col xki n , be an arbitrary solution of 2.7 Set u n uki n ∗ xki n − xki 3.5 ISRN Discrete Mathematics for k ∈ Z 1, m , i ∈ Z 1, nk Define functions Fpj z fpj z ∗ ∗ xpj − fpj xpj , where p ∈ Z 1, m , j ∈ Z 1, np Obviously, Fpj Fpj z 3.6 and from the condition H1 , we have ≤ Lpj |z| 3.7 for any z ∈ R By 3.5 , the MAM neural network 2.7 is reduced to the form m uki n np ki wpj Fpj upj n − κki pj n αki uki n , 3.8 p 1,p / k j where k ∈ Z 1, m , i ∈ Z 1, nk Obviously, there exists an equilibrium point u∗ of the system 3.8 From 2.8 and 3.5 , the initial conditions associated with 3.8 are of the form uki n ∗ , ϕki n − xki ψki n 3.9 where k ∈ Z 1, m , i ∈ Z 1, nk , n ∈ Z −κki , Let ψ col ψki n , ψ maxk∈Z 1,m maxi∈Z 1,nk sup−κki ≤n≤0 |ψki n | Because B A − |W|L is a nonsingular M matrix, then there exist constants ξki > i ∈ Z 1, nk , k ∈ Z 1, m such that − αki ξki − m np ki wpj Lpj ξpj > 3.10 p 1,p / k j Define the functions Hki λ np m − αki − λ ξki e−λ − ki ki wpj Lpj ξpj eλκpj , 3.11 p 1,p / k j where λ ∈ R , k ∈ Z 1, m , i ∈ Z 1, nk Apparently Hki λ is strictly monotone decreasing and continuous function In view of 3.10 , it is clear that Hki > 0, Hki 1−αki < Therefore k ∈ Z 1, m , i ∈ Z 1, nk Taking α there exist λki ∈ 0, − αki such that Hki λki mink∈Z 1,m mini∈Z 1,nk {λki } < mink∈Z 1,m mini∈Z 1,nk {1 − αki }, we have Hki α − αki − α ξki e−α − m np p 1,p / k j for i ∈ Z 1, nk , k ∈ Z 1, m ki ki wpj Lpj ξpj eακpj ≥ 3.12 ISRN Discrete Mathematics eαn uki n By calculating Δ|yki n | Set yki n of system 3.8 , we have Δ yki n eα n ≤ 1 | − eαn |uki n | |uki n αki − e−α α n e αki |uki n eαn αki | 3.13 np m × |yki n |−|yki n | along the solutions ki Fpj upj n − κki wpj pj n p 1,p / k j By using the inequality 3.7 , we have Δ yki n ≤ αki − e−α α n e αki × |uki n 1| eαn αki np m ki wpj Lpj upj n − κki pj n p 1,p / k j αki − e−α yki n αki ki × Lpj eακpj n αki p np m ki wpj 1,p / k j ypj n − κki pj n αki − e−α ≤ yki n αki αki p 3.14 np m ki wpj 1,p / k j ki × Lpj eακpj ypj n − κki pj n ≤ αki eα − yki n αki eα ki × Lpj eακpj 1 αki p m np ki wpj 1,p / k j ypj s sup n−κki pj ≤s≤n By Lemma 3.1, we have Δ yki n ≤ α αki − yki n αki eα × Lpj e ακki pj sup n−κki pj ≤s≤n ypj s αki p m np 1,p / k j ki wpj 3.15 ISRN Discrete Mathematics Let ξ mink∈Z 1,m mini∈Z 1,nk {ξki }, ξ maxk∈Z 1,m maxi∈Z 1,nk {ξki } and l0 where δ is a positive constant Therefore, when s ∈ Z −κki , , we have yki s ≤ ψ < ξki l0 eαs ψki s δ ψ /ξ, 3.16 for i ∈ Z 1, nk , k ∈ Z 1, m We assert that yki n < ξki l0 3.17 for n ∈ Z , i ∈ Z 1, nk and k ∈ Z 1, m If the assertion is false, then there exist k, i, and a minimum time t0 ∈ Z such that |yki t0 | ≥ ξki l0 , Δ|yki t0 | ≥ 0, and |ypj n | ≤ ξpj l0 when n ∈ Z −κki , t0 From 3.12 and 3.15 , and noticed that α αki − < 0, we obtain Δ yki t0 ⎡ ⎣ ≤ α αki αki − ξki e m −α ⎤ np ki wpj Lpj ξpj e ακki pj ⎦ l0 < 3.18 p 1,p / k j It conflicts with Δ|yki t0 | ≥ Therefore, yki n < ξki l0 3.19 for n ∈ Z Then we have |uki n | < ξki l0 e−αn ≤ δ ξ ξ ψ e−αn M ψ e−αn , 3.20 where M δ ξ/ξ > So the zero solution of the system 3.8 is global exponential stable; thus the equilibrium point of the discrete-time MAM neural network 2.7 is global exponential stable An Example Consider the following discrete-time MAM neural network with three fields: x11 n I11 α11 x11 n 11 f21 x21 n − κ11 w21 21 n 11 w31 f31 x31 t − κ11 31 x12 n I12 α12 x12 n , 12 f21 x21 n − κ12 w21 21 n 12 w31 f31 x31 t − κ12 31 , ISRN Discrete Mathematics The state trajectories of the 1th neuron on the first field with three initial values −5 10 20 30 40 The state trajectories of the 2th neuron on the first field with three initial values −5 10 20 30 40 The state trajectories of the 1th neuron on the first field with three initial values −5 10 20 30 40 The state trajectories of the 1th neuron on the first field with three initial values −5 10 20 30 40 Figure 1: The globally exponential stability of the equilibrium of the MAM network 4.1 with 10 cases random initial values x21 n I21 α21 x21 n 21 f11 x11 n − κ21 w11 11 n 21 w12 f12 x12 n − κ21 12 n x31 n I31 α31 x31 n 21 w31 f31 x31 n − κ21 31 n , 31 f11 x11 n − κ31 w11 11 n 31 w12 f12 x12 n − κ31 12 n 31 w21 f21 x21 n − κ31 21 n , 4.1 where the neuronal signal decay rates α11 0.1, α12 0.2, α21 0.2, α31 0.1, the external 11 11 12 0.2, w31 0.3, w21 0.2, inputs I11 1, I21 1, I21 1, I31 1, and connection weights w21 31 31 31 31 21 21 21 w21 0.4, w11 0.25, w12 −0.2, w31 0.3, w11 −0.2, w12 0.2, w21 0.3 The neuronal x, the time delays κki sin π/2 n activation functions fki x pj n Obviously, signal transfer functions fki x are continuous, and they satisfy the condition H1 , and the constant Lki The time delays κki pj n satisfy the condition H2 , and κki By calculating, we have ⎛ B A − |W|L 0.9 −0.2 ⎜ 0.8 −0.2 ⎜ ⎝−0.25 −0.2 0.8 −0.2 −0.2 −0.3 ⎞ −0.3 −0.4⎟ ⎟ −0.3⎠ 0.9 4.2 10 ISRN Discrete Mathematics It is easy to verify that the matrix B is a nonsingular M matrix Then by Theorems 3.2 and 3.3, there exists an equilibrium point which is globally exponentially stable for the MAM neural network 4.1 The numerical simulation is given in Figure in which ten cases of initial values are taken at random From Figure 1, we can know that the MAM neural network 4.1 converges ∗ ∗ ∗ ∗ T , x12 , x21 , x31 ≈ 2.1735, 2.68, 1.9635, globally exponentially to the equilibrium point x11 1.8782 T no matter what it starts from the initial states Conclusions In this paper, we have formulated a discrete-time analogue of the continuous-time multidirectional associative memory neural network with time-varying delays by using semidiscretization method Some sufficient conditions for the existence and the global exponential stability of an equilibrium point have been obtained Our results have shown that the discrete-time analogue inherits the existence and global exponential stability of equilibrium point for the continuous-time MAM neural network References M Hagiwara, “Multidirectional associative memory,” in Proceedings of the International Joint 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nonlinear perturbations,” IEEE Transactions on Circuits and Systems I, vol 48, no 8, pp 1011–1019, 2001 Copyright of ISRN Discrete Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use