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Global Exponential Stability of Bidirectional Associative Memory Neural Networks With Time Delays Xin-Ge Liu, Ralph R. Martin, Min Wu, and Mei-Lan Tang
Abstract—In this paper, we consider delayed bidirectional associative memory (BAM) neural networks (NNs) with Lipschitz continuous activation functions. By applying Young’s inequality and Hölder’s inequality techniques together with the properties of monotonic continuous functions, global exponential stability criteria are established for BAM NNs with time delays. This is done -matrix. through the use of a new Lyapunov functional and an The results obtained in this paper extend and improve previous results. Index Terms—Bidirectional associative memory (BAM) neural networks (NNs), global exponential stability, Lyapunov functionals, Young’s inequality.
I. INTRODUCTION
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EURAL networks (NNs) have many important applications. Delayed versions of NNs have also proved to be important for solving certain classes of motion-related optimization problems. Various results concerning the dynamical behavior of NNs with delays have been reported in [1]–[9]. Since Kosto [10] introduced bidirectional associative memory (BAM) NNs, researchers have paid particular attention to the stability analysis of BAM NNs with time delays, as such NNs have been shown to be a useful network model for applications in pattern recognition, optimization, and automatic control (see, for example, [11]–[18]). Various sufficient conditions have been presented for the stability of BAM NNs, most of which require that the activation functions are bounded and Lipschitz continuous. Arik [19] studied global asymptotic stability of BAM NNs with time delays, and presented a sufficient condition for uniqueness and global asymptotic stability of the equilibrium point. However, his analysis requires that the activation functions and signal propagation functions are bounded and Lipschitz continuous. Furthermore, he did not study global exponential stability of BAM NNs. Manuscript received February 2, 2006; revised January 19, 2007; accepted July 10, 2007. X.-G. Liu is with the School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410083, China and also with the School of Computer Science, Cardiff University, Cardiff CF24 3AA, U.K. (e-mail: [email protected]). R. R. Martin is with the School of Computer Science, Cardiff University, Cardiff CF24 3AA, U.K. M. Wu is with the School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China. M.-L. Tang is with the School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410083, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNN.2007.908633
Global asymptotic stability of a BAM NN only guarantees that the solution of the BAM NN converges to the equilibrium point and says nothing about the rate of convergence of the solution to the equilibrium point. Global exponential stability ensures exponential convergence of the BAM NN to its equilibrium point. In practice, applications of NNs with time delays often require that the network has a unique equilibrium point which is globally exponentially stable, if the network is to be suitable for solving problems in real time. Furthermore, in the case of globally exponentially stable BAM NNs with delays, it is easier to make a quantitative analysis, and thus to determine the convergence behavior of the delayed BAM NN, such as its convergence rate and its precision. This paper thus considers global exponential stability of BAM NNs with constant time delays. We do not assume that the activation functions and signal propagation functions are bounded. Several new sufficient criteria are derived for the existence, uniqueness, and global exponential stability of the equilibrium point of BAM NNs, by constructing a suitable Lyapunov functional, introducing a real parameter, and applying Young’s inequality, Hölder’s inequality, and the intermediate value theorem of continuous functions. Our new results extend and improve earlier results in [19]–[21]. II. PRELIMINARIES A. Definitions and Assumptions The dynamic behavior of a BAM NN with constant time delays is described by the following set of differential equations:
(1) where and The BAM NN model given in (1) can be regarded as an NN with two layers. is the number of neurons in the first layer and is the number of neurons in the second layer. stands and represent the activations (i.e., states) for time. of the th neuron in the first layer and th neuron in the second and are the synaptic conlayer at time , respectively. nection strengths between the neurons in the two layers, while and represent the activation functions of the neurons and the signal propagation functions, respectively, as in [19]. In the of neuron is dependent on an external first layer, the state
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constant input and other inputs determined by the outputs of the neurons in the second layer via activation functions ; in of neuron depends on an exthe second layer, the state ternal constant input and inputs determined by the outputs of the neurons in the first layer via the activation functions . and are differentiable real functions with positive derivatives and repredefining the neuron charging time. sent constant time delays; and . The initial conditions of the BAM NN given in (1) are assumed to be
is said to be a nonsingular -matrix if the real part of every eigenvalue of is positive. Definition 3: Suppose we are given the BAM NN in (1). is said to be an A vector equilibrium point if it satisfies
(2)
in which the initial value functions and are continand , respectively. uous functions on In order to allow comparison of our results with previous results in the literature, we introduce the following assumptions. Assumption 1: and are bounded real functions, i.e., there exist positive constants and such that for all
Assumption 2: The activation functions and the signal propagation functions are Lipschitz continuous, i.e., there exist posand such that for all itive constants
Definition 4: The aforementioned equilibrium point is said such to be globally exponentially stable if we can find an and such that for any that there exist constants
Definition 5: A map is a homeomorphism of onto itself, if , is one-to-one, is onto, and is the inverse map , where represents the set of all to . continuous functions from B. Basic Lemmas
In this paper, we assume
Let be the Banach space of continuous into with the topology of unifunctions which maps , we define the -norm of form convergence. For any to be
where is a constant. For simplicity, we usually denote the -norm just by . A similar -norm can be defined on . the Banach space We now explain other basic concepts needed in this paper. be any continuous function. Definition 1: Let is defined to be The upper right Dini-derivative
Definition 2: Let satisfying ,
be a real , and
matrix with elements for . Then,
In this section, we start with some lemmas related to nonsinmatrix with nonpositive gular -matrices. Let be an off-diagonal element. is a nonsingular -matrix in only the following cases [22]. Lemma 1: is a nonsingular -matrix if there exists a pos(i.e., with every element positive) such that itive vector . Lemma 2: is a nonsingular -matrix if there exists a pos, such that itive diagonal matrix , i.e., with each is strictly row diagonally dominant, i.e.,
Lemma 3: is a nonsingular -matrix if there exists a positive diagonal matrix such that is strictly row diagonally dominant. If we set to a unit matrix in Lemmas 2 and 3, we readily obtain Fact 1. Fact 1: If is strictly row (or column) diagonally dominant, then is a nonsingular -matrix. To prove our main result, the next four lemmas are needed. and satisfies the following condiLemma 4: If tions: is injective on ; 1) ; 2) then
is a homeomorphism of
[23].
LIU et al.: GLOBAL EXPONENTIAL STABILITY OF BAM NNS WITH TIME DELAYS
Lemma 5: Assume that , , , and . Then, the following inequality, Young’s inequality [12], holds:
Clearly, if or , this extended version of Young’s inequality still holds. and , then the following Lemma 6: If inequality, Hölder’s inequality, holds:
Lemma 7: Assume that , , , and is a continuous function. Then, the following inequality holds:
(3) where is a positive constant. and Proof: Let . No matter whether or , we have . So
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Since global exponential stability implies global asymptotic stability, we can obtain the following results. Corollary 1: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally asymptotically such that is a nonsingular stable if there exists a number -matrix (again, using the same notation). , , and Remark 1: If we set in Corollary 1, Corollary 1 becomes the same as [19, Th. 1]. However, using our methods, Corollary 1 does not require Assumption 1 to hold. Thus, [19, Th. 1] is a special case of Corollary 1 here. Furthermore, in this setting, Theorem 1 in our paper shows that the BAM NN is globally exponentially stable if it satisfies the same conditions as for [19, Th. 1]. Therefore, Theorem 1 in our paper is a stronger version of the result in [19]. Since strictly row (or column) diagonally dominant matrices are nonsingular -matrices by Fact 1, it is easy for us to obtain the following further corollary. Corollary 2: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally exponentially such that is strictly stable if there exists a number row (or column) diagonally dominant (again, using the same notation). The result of Corollary 2 is important as it allows us to impose and independently of constraint conditions on matrices each other, whereas checking whether in Theorem 1 is a nonsingular -matrix requires the establishment of a relationship between the elements of matrices and . IV. COMPARISONS AND EXAMPLES
(4)
In this section, we compare our results with previous results in the literature, which are restated in the following. Theorem 2 (From [20]): Under Assumptions 1 and 2, the equilibrium point of the NN defined by (1) with and is globally exponentially stable and such that if there exist constants
III. GLOBAL STABILITY ANALYSIS In this section, we present a new sufficient condition which guarantees the existence, uniqueness, and global exponential stability of the equilibrium point of the system given in (1). Throughout this paper, we use the notation , , , and ; is an matrix with and is an matrix with , where , , and . We write
Theorem 3 (From [21]): Under Assumptions 1 and 2, the equilibrium point of the NN defined by (1) with and is globally exponentially stable if the following conditions hold:
(5) Theorem 1: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally exponentially such that is a nonsingular stable if there exists a number -matrix (using the same notation as before). The proof of Theorem 1 is given in the Appendix.
Theorems 1–3 provide different sufficient conditions for the global exponential stability of the BAM NN in (1) even if we set and . Note that the activation functions and signal propagation functions in Theorems
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2 and 3 are required to be bounded and Lipschitz continuous. However, Theorem 1 only requires the activation functions and signal propagation functions to satisfy the Lipschitz condition. The assumption in Theorem 1 is weaker than those in Theorems 2 and 3. To further illustrate the differences between these three criteria, we analyze the stability of the BAM NN given in Example 1 using these three criteria and compare the ranges of and when we permissible synaptic connection strengths use these three criteria to design a BAM NN which is globally exponentially stable. Example 1: Consider the following first-order BAM NN:
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where , , and the activation functions and signal propagation functions are unbounded and satisfy Lipschitz conditions
and with for all . Theorems 2 and 3 cited previously and [19, Th. 1] are not directly applicable to this example unless the function is bounded or the equilibrium point exists. Arik [19] only shows that this BAM NN is globally asympunder the strict assumption totically stable when that is bounded. The criterion for global asymptotic stability derived by Arik [19] cannot be applied to analyze the global exponential stability of this BAM NN. However, using our Theorem 1, we find that this BAM NN is globally exponentially for any even if the function stable when is unbounded. Since the boundedness requirements of the activation functions and signal propagation functions are removed, Theorem 1 is not as restrictive as that in [19]. Since global exponential stability implies the global asymptotic stability, this BAM NN is also globally asymptotically stable when the function is unbounded. Therefore, our result improves the criterion for globally asymptotical stability derived by Arik [19] for this particular example. Using [20, Th. 2] and Lemma 1, the equilibrium point of this first-order BAM NN is globally exponentially stable if . Now, if , then , . The condition is weaker than the i.e., . Note that Theorems 1 and 2 provide condition different sufficient conditions. For global exponential stability of this first-order BAM NN, [21, Th. 3] requires that and . Obviously, the condition is weaker than and . Again, note that Theorems the conditions 1 and 3 give different sufficient conditions. Next, we consider the stability and numerical simulation of the following BAM NN with unbounded activation functions and signal propagation functions. Example 2: Consider the following BAM NN:
Since the activation functions and signal propagation functions in this BAM NN are unbounded, the conditions in Theorems 2 and 3 cannot be satisfied; so, these two global exponential stability criteria cannot be applied to analyze the global exponential stability of this BAM NN. Theorem 1 in [19] also cannot be applied to the global asymptotic stability analysis of this BAM NN. The Lipschitz constants for activation functions and signal propagation functions can be determined to , , , and . Since and be are differentiable functions, we have , , , and . such that the following Using Theorem 1, we may set matrix:
is a nonsingular -matrix. In fact, the eigenvalues of the matrix in Theorem 1 are 7.4257, 19.1020, 37.9625, and 55.8849—they are all positive. Therefore, the BAM NN given in Example 2 has a unique equilibrium point which is globally exponentially stable. For numerical simulation, let and the delay parameter . The initial conditions of the BAM NN given in Example 2 are assumed to be and for . Fig. 1 depicts the time responses of state variables with step . It confirms that the proposed criterion leads to the unique and globally exponentially stable equilibrium point Remark 2: The BAM NN model, an extension of the unidirectional autoassociator of Hopfield [24], was first introduced by Kosko [10]. Although the BAM NN model given in (1) can be mathematically regarded as a Hopfield-type NN with dimenor a Cohen–Grossberg NN with dimension , sion it has many special properties due to the particular structure of the connection weights, and has practical applications in storing
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Fig. 1. Transient response of state variable u and z .
paired patterns or memories. By constructing proper vector Lyapunov functions, using -matrix theory and qualitative property of the differential inequalities, Zhang [25] established the sufficient conditions for global exponential stability of the equilibrium point for Cohen–Grossberg NNs in which the activation , Theorem 1 is functions are unbounded. However, when different from the result obtained by combining [25, Th. 2] with [25, Th. 1]: Theorem 1 is not a corollary of [25, Th. 1 and 2]. Remark 3: Using fixed-point theorem in Banach space and differential inequality techniques, Zhao [26] obtained new sufficient conditions ensuring the global exponential stability and existence of periodic solutions for cellular NNs with variable delays. By employing homeomorphism theory and the inequality with , , , and , Zhao and Cao [27] derived some important conditions ensuring the existence and uniqueness of the equilibrium point, and its global exponential stability, for cellular NNs with constant delays and without assuming boundedness for the signal functions. Furthermore, using the same inequality in [27] and further ideas, Zhao [28] presented other new criteria ensuring global exponential stability and existence of periodic oscillatory solutions of BAM NN with constant delays. By introducing ingenious real parameters, employing suitable Lyapunov functionals, and applying contracting mapping, Zhao and Wang [29] gave a set of novel sufficient conditions ensuring the existence, uniqueness, and global exponential stability of periodic oscillatory solutions of a class of reaction–diffusion NNs with constant delays and time-varying coefficients. In this paper, however, we have only investigated the existence, uniqueness
of equilibrium point, and global exponential stability of BAM NNs using an -matrix, Young’s inequality, and Hölder’s inequality, instead of the inequality used in [27]. The sufficient conditions obtained in this paper are different from those in [27]. V. CONCLUSION The main contribution of this paper is a result that ensures the existence, uniqueness, and global exponential stability of the equilibrium point of BAM NNs with time delays, without assuming that the activation functions and signal propagation functions are bounded. New sufficient conditions for ascertaining global exponential stability, i.e., Theorem 1 and Corollary 2, have been derived for BAM NNs, improving and extending previous work. APPENDIX PROOF OF THEOREM 1 We here prove Theorem 1. Proof: We consider here the case when , using a sequence of three steps. The case can be directly proved in three similar steps, but in which we use a simple computation (omitted here) instead of Young’s inequality. is an equilibrium point of the BAM NN given in If (1), then satisfies (2). Let (6)
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where
with As
where and Obviously, the solution of is the equilibrium point of the BAM NN given in (1). Hence, the existence and the uniqueness of the equilibrium of the BAM NN given in (1) is a homeomorcan be reduced to proving that the map . phism of Step 1) We prove that is an injective map on by showing that assuming otherwise leads to a contradiction. and exist in We suppose that values such that while , so or . is a nonsingular -matrix, by Lemma Since 1, then there exists a positive vector such that , so
and
(9) Next, we bound the value of the left-hand side of (9). By Assumption 2, using the extended version of Young’s inequality, we obtain
(7) where Since
, or
and ,
, , , we can see that
, and
(8) Since and entiable functions, there exist such that
and
are differand
(10) Furthermore
From (9) and (10), we find that (11) which contradicts (8), and hence implies that is an injective map on .
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Step 2) We prove that
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where
Since , , , it suffices to show that
, , and
are constants,
Clearly,
. Thus
where and
(13) Let In fact, there exist
and
such that Using Hölder’s inequality gives
(14) Combining (13) and (14) gives
so
and, therefore
(12)
From Steps 1) and 2) and Lemma 4, the map is a homeomorphism of . Thus, the BAM NN given in (1) has a unique equilibrium point. Let us . denote the unique equilibrium point by Step 3) We now prove that this unique equilibrium point is globally exponentially stable. In order to simplify our proof, using translation and , we
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transform the system given in (1) to the following system:
(15) where
and
Since the transformation is a translation, we only need to prove that the origin (0, 0) of the transformed system is globally exponentially stable. We consider the following Lyapunov function:
where
(16) where the positive real number will be determined later. The upper right Dini-derivative along the trajectories of the BAM NN given in (15) are given by
so
(17) Now
(18)
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Hence (20) and , and Since . Since and are monotonic increasing and continuous functions of in
(19) Denote
using the intermediate value theorem of continuous , functions, there exist positive values , and , , such that , , and , . Let
Since each for Noting that
Setting
, then . Noting that , we have that , and , . to in (20) gives
,
by Lemma 7, we have (21) Setting
in (16) gives
(22) Combining (21) and (22) gives (23) where tively, given by
Let
and
and
are, respec-
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Clearly,
. From (16), it is clear that
(24) It follows that
for any , where is a constant. This implies that the solution of (15), at the origin, is globally exponentially stable. This completes the proof of Theorem 1. REFERENCES [1] S. Arik, “Global asymptotic stability of a class of dynamical neural networks,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 4, pp. 568–571, Apr. 2000. [2] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 2, pp. 168–171, Feb. 1998. [3] S. Arik and V. Tavsanoglu, “On global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 4, pp. 571–574, Apr. 2000. [4] J. Cao, “Periodic-solutions and exponential stability in delayed cellular neural networks,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 60, no. 3, pp. 3244–3248, Sep. 1999. [5] J. Cao and J. Wang, “Global exponential stability and periodicity of recurrent neural networks with time delays,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 5, pp. 920–931, May 2005. [6] J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Trans. Neural Netw., vol. 18, no. 2, pp. 416–430, Mar. 2007. [7] S. Guo and L. Huang, “Stability analysis of Cohen-Grossberg neural networks,” IEEE Trans. Neural Netw., vol. 17, no. 1, pp. 106–117, Jan. 2006. [8] H. Lu and S.-I. Amari, “Global exponential stability of multitime scale competitive neural networks with nonsmooth functions,” IEEE Trans. Neural Netw., vol. 17, no. 5, pp. 1152–1164, Sep. 2006. [9] Z. Zeng and J. Wang, “Improved conditions for global exponential stability of recurrent neural networks with time-varying delays,” IEEE Trans. Neural Netw., vol. 17, no. 3, pp. 623–635, May 2006. [10] B. Kosto, “Bidirectional associative memories,” IEEE Trans. Syst. Man Cybern., vol. SMC-18, no. 1, pp. 49–60, Jan./Feb. 1988. [11] J. Cao and L. Wang, “Exponential stability and periodic oscillatory solution in BAM networks with delays,” IEEE Trans. Neural Netw., vol. 13, no. 2, pp. 457–463, Mar. 2002. [12] J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Phys. Lett. A, vol. 307, no. 2–3, pp. 136–147, Jan. 2003. [13] K. Gopalsamy and X. Z. He, “Delay-dependent stability in bidirectional associative memory neural networks,” IEEE Trans. Neural Netw., vol. 5, no. 6, pp. 998–1002, Nov. 1994. [14] X. Liao and J. Yu, “Qualitative analysis of bi-directional associative memory with time delay,” Int. J. Circuit. Theory Appl., vol. 26, no. 3, pp. 219–229, May/Jun. 1998. [15] X. Liao, J. Yu, and G. Chen, “Novel stability criteria for bidirectional associative memory neural networks with time delays,” Int. J. Circuit Theory Appl., vol. 30, no. 5, pp. 519–546, Sep./Oct. 2002. [16] S. Mohamad, “Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks,” Physica D, vol. 159, no. 3–4, pp. 233–251, Nov. 2001.
[17] J. Zhang and Y. Yang, “Global stability analysis of bidirectional associative memory neural networks with time delay,” Int. J. Circuit Theory Appl., vol. 29, no. 2, pp. 185–196, Mar./Apr. 2001. [18] H. Zhao, “Global stability of bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 297, no. 3–4, pp. 182–190, May 2002. [19] S. Arik, “Global asymptotic stability of bidirectional associative memory neural networks with time delays,” IEEE Trans. Neural Netw., vol. 16, no. 3, pp. 580–586, May 2005. [20] J. Cao and M. Dong, “Exponential stability of delayed bi-directional associative memory networks,” Appl. Math. Comput., vol. 135, no. 1, pp. 105–112, Feb. 2003. [21] X. Liao and K. Wong, “Global exponential stability of hybrid bi-directional associative memory neural networks with discrete delays,” Phys. Rev. E, vol. 67, pp. 042 901–042 904, Apr. 2003. [22] R. A. Horn and C. R. Johnson, Topics in Matrix Analyis. Cambridge, U.K.: Cambridge Univ. Press, 1991. [23] J. Zhang and X. S. Jin, “Global stability analysis in delayed Hopfield neural network models,” IEEE Trans. Neural Netw., vol. 13, no. 3, pp. 745–753, Jun. 2000. [24] J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neutons,” Proc. Nat. Acad. Sci. USA, vol. 81, pp. 3088–3092, 1984. [25] J. Zhang, Y. Suda, and H. Komine, “Global exponential stability of Cohen-Grossberg neural networks with variable delays,” Phys. Lett. A, vol. 338, no. 1, pp. 44–50, Apr. 2005. [26] H. Zhao, “Global exponential stability and periodicity of cellular neural networks with variable delays,” Phys. Lett. A, vol. 336, no. 4–5, pp. 331–341, Mar. 2005. [27] H. Zhao and J. Cao, “New conditions for global exponential stability of cellular neural networks with delays,” Neural Netw., vol. 18, no. 10, pp. 1332–1340, Dec. 2005. [28] H. Zhao, “Exponential stability and periodic oscillatory of bi-directional associative memory neural network involving delays,” Neurocomputing, vol. 69, no. 4–6, pp. 424–448, Jan. 2006. [29] H. Zhao and G. Wang, “Existence of periodic oscillatory solution of reaction diffusion neural networks with delays,” Phys. Lett. A, vol. 343, no. 15, pp. 372–383, Aug. 2005.
Xin-Ge Liu received the B.S. and M.S. degrees from Hunan Normal University, Changsha, China and the Ph.D. degree from Central South University, Changsha, China, all in mathematics, in 1991, 1994, and 2001, respectively. In May 1994, he joined the School of Mathematical Science and Computing Technology, Central South University, where he is currently an Associate Professor. From 2002 to 2004, he was a Postdoctor of Automatic Control Engineering, Central South University. From October 2004 to September 2006, he was a Postdoctor and Visiting Scholar in the School of Computer Science, Cardiff University, Cardiff, U.K. His research interests include harmonic analysis, wavelets analysis, nonlinear systems, neural networks, and stability theory.
Ralph R. Martin received the Ph.D. degree from Cambridge University, Cambridge, U.K., in 1983, for a dissertation on “Principal Patches.” He has been working in the field of geometric modeling since 1979. He has progressed from Lecturer to Professor at Cardiff University, Cardiff, U.K., taking this last post in 2000. His publications include over 170 papers and ten books covering such topics as solid modeling, surface modeling, reverse engineering, intelligent sketch input, mesh processing, vision-based geometric inspection, and geometric reasoning. Dr. Martin is a Fellow of the Institute of Mathematics and its Applications and a Member of the British Computer Society. He is on the editorial boards of Computer Aided Design, International Journal of Shape Modelling, CAD and Applications, and International Journal of CAD/CAM.
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Min Wu received the B.S. and M.S. degrees in engineering from the Central South University, Changsha, China, in 1983 and 1986, respectively, and the Ph.D. degree in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1999. In July 1986, he joined the staff of the Central South University, where he is currently a Professor of Automatic Control Engineering. He was a Visiting Scholar at the Department of Electrical Engineering, Tohoku University, Sendai, Japan, from 1989 to 1990, and a Visiting Research Scholar at the Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo, Japan, from 1996 to 1999. His current research interests are robust control and its application, process control, and intelligent control. Dr. Wu received the best paper award at the International Federation of Automatic Control (IFAC) 1999 (jointly with M. Nakano and J.-H. She). He is a member of the Nonferrous Metals Society of China and the China Association of Automation.
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Mei-Lan Tang received the B.S. degree in mathematics from Hunan Normal University, Changsha, China, in 1997 and the M.S. degree in applied mathematics from Central South University, Changsha, China, in 2004. Currently, she is a Lecturer at the School of Mathematical Science and Computing Technology, Central South University, Changsha, China. She is the author or coauthor of more than 30 journal papers. Her research interests include neural networks and functional differential equation.
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Global Exponential Stability of Bidirectional Associative Memory Neural Networks With Time Delays Xin-Ge Liu, Ralph R. Martin, Min Wu, and Mei-Lan Tang
Abstract—In this paper, we consider delayed bidirectional associative memory (BAM) neural networks (NNs) with Lipschitz continuous activation functions. By applying Young’s inequality and Hölder’s inequality techniques together with the properties of monotonic continuous functions, global exponential stability criteria are established for BAM NNs with time delays. This is done -matrix. through the use of a new Lyapunov functional and an The results obtained in this paper extend and improve previous results. Index Terms—Bidirectional associative memory (BAM) neural networks (NNs), global exponential stability, Lyapunov functionals, Young’s inequality.
I. INTRODUCTION
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EURAL networks (NNs) have many important applications. Delayed versions of NNs have also proved to be important for solving certain classes of motion-related optimization problems. Various results concerning the dynamical behavior of NNs with delays have been reported in [1]–[9]. Since Kosto [10] introduced bidirectional associative memory (BAM) NNs, researchers have paid particular attention to the stability analysis of BAM NNs with time delays, as such NNs have been shown to be a useful network model for applications in pattern recognition, optimization, and automatic control (see, for example, [11]–[18]). Various sufficient conditions have been presented for the stability of BAM NNs, most of which require that the activation functions are bounded and Lipschitz continuous. Arik [19] studied global asymptotic stability of BAM NNs with time delays, and presented a sufficient condition for uniqueness and global asymptotic stability of the equilibrium point. However, his analysis requires that the activation functions and signal propagation functions are bounded and Lipschitz continuous. Furthermore, he did not study global exponential stability of BAM NNs. Manuscript received February 2, 2006; revised January 19, 2007; accepted July 10, 2007. X.-G. Liu is with the School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410083, China and also with the School of Computer Science, Cardiff University, Cardiff CF24 3AA, U.K. (e-mail: [email protected]). R. R. Martin is with the School of Computer Science, Cardiff University, Cardiff CF24 3AA, U.K. M. Wu is with the School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China. M.-L. Tang is with the School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410083, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNN.2007.908633
Global asymptotic stability of a BAM NN only guarantees that the solution of the BAM NN converges to the equilibrium point and says nothing about the rate of convergence of the solution to the equilibrium point. Global exponential stability ensures exponential convergence of the BAM NN to its equilibrium point. In practice, applications of NNs with time delays often require that the network has a unique equilibrium point which is globally exponentially stable, if the network is to be suitable for solving problems in real time. Furthermore, in the case of globally exponentially stable BAM NNs with delays, it is easier to make a quantitative analysis, and thus to determine the convergence behavior of the delayed BAM NN, such as its convergence rate and its precision. This paper thus considers global exponential stability of BAM NNs with constant time delays. We do not assume that the activation functions and signal propagation functions are bounded. Several new sufficient criteria are derived for the existence, uniqueness, and global exponential stability of the equilibrium point of BAM NNs, by constructing a suitable Lyapunov functional, introducing a real parameter, and applying Young’s inequality, Hölder’s inequality, and the intermediate value theorem of continuous functions. Our new results extend and improve earlier results in [19]–[21]. II. PRELIMINARIES A. Definitions and Assumptions The dynamic behavior of a BAM NN with constant time delays is described by the following set of differential equations:
(1) where and The BAM NN model given in (1) can be regarded as an NN with two layers. is the number of neurons in the first layer and is the number of neurons in the second layer. stands and represent the activations (i.e., states) for time. of the th neuron in the first layer and th neuron in the second and are the synaptic conlayer at time , respectively. nection strengths between the neurons in the two layers, while and represent the activation functions of the neurons and the signal propagation functions, respectively, as in [19]. In the of neuron is dependent on an external first layer, the state
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constant input and other inputs determined by the outputs of the neurons in the second layer via activation functions ; in of neuron depends on an exthe second layer, the state ternal constant input and inputs determined by the outputs of the neurons in the first layer via the activation functions . and are differentiable real functions with positive derivatives and repredefining the neuron charging time. sent constant time delays; and . The initial conditions of the BAM NN given in (1) are assumed to be
is said to be a nonsingular -matrix if the real part of every eigenvalue of is positive. Definition 3: Suppose we are given the BAM NN in (1). is said to be an A vector equilibrium point if it satisfies
(2)
in which the initial value functions and are continand , respectively. uous functions on In order to allow comparison of our results with previous results in the literature, we introduce the following assumptions. Assumption 1: and are bounded real functions, i.e., there exist positive constants and such that for all
Assumption 2: The activation functions and the signal propagation functions are Lipschitz continuous, i.e., there exist posand such that for all itive constants
Definition 4: The aforementioned equilibrium point is said such to be globally exponentially stable if we can find an and such that for any that there exist constants
Definition 5: A map is a homeomorphism of onto itself, if , is one-to-one, is onto, and is the inverse map , where represents the set of all to . continuous functions from B. Basic Lemmas
In this paper, we assume
Let be the Banach space of continuous into with the topology of unifunctions which maps , we define the -norm of form convergence. For any to be
where is a constant. For simplicity, we usually denote the -norm just by . A similar -norm can be defined on . the Banach space We now explain other basic concepts needed in this paper. be any continuous function. Definition 1: Let is defined to be The upper right Dini-derivative
Definition 2: Let satisfying ,
be a real , and
matrix with elements for . Then,
In this section, we start with some lemmas related to nonsinmatrix with nonpositive gular -matrices. Let be an off-diagonal element. is a nonsingular -matrix in only the following cases [22]. Lemma 1: is a nonsingular -matrix if there exists a pos(i.e., with every element positive) such that itive vector . Lemma 2: is a nonsingular -matrix if there exists a pos, such that itive diagonal matrix , i.e., with each is strictly row diagonally dominant, i.e.,
Lemma 3: is a nonsingular -matrix if there exists a positive diagonal matrix such that is strictly row diagonally dominant. If we set to a unit matrix in Lemmas 2 and 3, we readily obtain Fact 1. Fact 1: If is strictly row (or column) diagonally dominant, then is a nonsingular -matrix. To prove our main result, the next four lemmas are needed. and satisfies the following condiLemma 4: If tions: is injective on ; 1) ; 2) then
is a homeomorphism of
[23].
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Lemma 5: Assume that , , , and . Then, the following inequality, Young’s inequality [12], holds:
Clearly, if or , this extended version of Young’s inequality still holds. and , then the following Lemma 6: If inequality, Hölder’s inequality, holds:
Lemma 7: Assume that , , , and is a continuous function. Then, the following inequality holds:
(3) where is a positive constant. and Proof: Let . No matter whether or , we have . So
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Since global exponential stability implies global asymptotic stability, we can obtain the following results. Corollary 1: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally asymptotically such that is a nonsingular stable if there exists a number -matrix (again, using the same notation). , , and Remark 1: If we set in Corollary 1, Corollary 1 becomes the same as [19, Th. 1]. However, using our methods, Corollary 1 does not require Assumption 1 to hold. Thus, [19, Th. 1] is a special case of Corollary 1 here. Furthermore, in this setting, Theorem 1 in our paper shows that the BAM NN is globally exponentially stable if it satisfies the same conditions as for [19, Th. 1]. Therefore, Theorem 1 in our paper is a stronger version of the result in [19]. Since strictly row (or column) diagonally dominant matrices are nonsingular -matrices by Fact 1, it is easy for us to obtain the following further corollary. Corollary 2: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally exponentially such that is strictly stable if there exists a number row (or column) diagonally dominant (again, using the same notation). The result of Corollary 2 is important as it allows us to impose and independently of constraint conditions on matrices each other, whereas checking whether in Theorem 1 is a nonsingular -matrix requires the establishment of a relationship between the elements of matrices and . IV. COMPARISONS AND EXAMPLES
(4)
In this section, we compare our results with previous results in the literature, which are restated in the following. Theorem 2 (From [20]): Under Assumptions 1 and 2, the equilibrium point of the NN defined by (1) with and is globally exponentially stable and such that if there exist constants
III. GLOBAL STABILITY ANALYSIS In this section, we present a new sufficient condition which guarantees the existence, uniqueness, and global exponential stability of the equilibrium point of the system given in (1). Throughout this paper, we use the notation , , , and ; is an matrix with and is an matrix with , where , , and . We write
Theorem 3 (From [21]): Under Assumptions 1 and 2, the equilibrium point of the NN defined by (1) with and is globally exponentially stable if the following conditions hold:
(5) Theorem 1: The BAM NN given in (1), under Assumption 2, has a unique equilibrium point which is globally exponentially such that is a nonsingular stable if there exists a number -matrix (using the same notation as before). The proof of Theorem 1 is given in the Appendix.
Theorems 1–3 provide different sufficient conditions for the global exponential stability of the BAM NN in (1) even if we set and . Note that the activation functions and signal propagation functions in Theorems
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2 and 3 are required to be bounded and Lipschitz continuous. However, Theorem 1 only requires the activation functions and signal propagation functions to satisfy the Lipschitz condition. The assumption in Theorem 1 is weaker than those in Theorems 2 and 3. To further illustrate the differences between these three criteria, we analyze the stability of the BAM NN given in Example 1 using these three criteria and compare the ranges of and when we permissible synaptic connection strengths use these three criteria to design a BAM NN which is globally exponentially stable. Example 1: Consider the following first-order BAM NN:
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where , , and the activation functions and signal propagation functions are unbounded and satisfy Lipschitz conditions
and with for all . Theorems 2 and 3 cited previously and [19, Th. 1] are not directly applicable to this example unless the function is bounded or the equilibrium point exists. Arik [19] only shows that this BAM NN is globally asympunder the strict assumption totically stable when that is bounded. The criterion for global asymptotic stability derived by Arik [19] cannot be applied to analyze the global exponential stability of this BAM NN. However, using our Theorem 1, we find that this BAM NN is globally exponentially for any even if the function stable when is unbounded. Since the boundedness requirements of the activation functions and signal propagation functions are removed, Theorem 1 is not as restrictive as that in [19]. Since global exponential stability implies the global asymptotic stability, this BAM NN is also globally asymptotically stable when the function is unbounded. Therefore, our result improves the criterion for globally asymptotical stability derived by Arik [19] for this particular example. Using [20, Th. 2] and Lemma 1, the equilibrium point of this first-order BAM NN is globally exponentially stable if . Now, if , then , . The condition is weaker than the i.e., . Note that Theorems 1 and 2 provide condition different sufficient conditions. For global exponential stability of this first-order BAM NN, [21, Th. 3] requires that and . Obviously, the condition is weaker than and . Again, note that Theorems the conditions 1 and 3 give different sufficient conditions. Next, we consider the stability and numerical simulation of the following BAM NN with unbounded activation functions and signal propagation functions. Example 2: Consider the following BAM NN:
Since the activation functions and signal propagation functions in this BAM NN are unbounded, the conditions in Theorems 2 and 3 cannot be satisfied; so, these two global exponential stability criteria cannot be applied to analyze the global exponential stability of this BAM NN. Theorem 1 in [19] also cannot be applied to the global asymptotic stability analysis of this BAM NN. The Lipschitz constants for activation functions and signal propagation functions can be determined to , , , and . Since and be are differentiable functions, we have , , , and . such that the following Using Theorem 1, we may set matrix:
is a nonsingular -matrix. In fact, the eigenvalues of the matrix in Theorem 1 are 7.4257, 19.1020, 37.9625, and 55.8849—they are all positive. Therefore, the BAM NN given in Example 2 has a unique equilibrium point which is globally exponentially stable. For numerical simulation, let and the delay parameter . The initial conditions of the BAM NN given in Example 2 are assumed to be and for . Fig. 1 depicts the time responses of state variables with step . It confirms that the proposed criterion leads to the unique and globally exponentially stable equilibrium point Remark 2: The BAM NN model, an extension of the unidirectional autoassociator of Hopfield [24], was first introduced by Kosko [10]. Although the BAM NN model given in (1) can be mathematically regarded as a Hopfield-type NN with dimenor a Cohen–Grossberg NN with dimension , sion it has many special properties due to the particular structure of the connection weights, and has practical applications in storing
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Fig. 1. Transient response of state variable u and z .
paired patterns or memories. By constructing proper vector Lyapunov functions, using -matrix theory and qualitative property of the differential inequalities, Zhang [25] established the sufficient conditions for global exponential stability of the equilibrium point for Cohen–Grossberg NNs in which the activation , Theorem 1 is functions are unbounded. However, when different from the result obtained by combining [25, Th. 2] with [25, Th. 1]: Theorem 1 is not a corollary of [25, Th. 1 and 2]. Remark 3: Using fixed-point theorem in Banach space and differential inequality techniques, Zhao [26] obtained new sufficient conditions ensuring the global exponential stability and existence of periodic solutions for cellular NNs with variable delays. By employing homeomorphism theory and the inequality with , , , and , Zhao and Cao [27] derived some important conditions ensuring the existence and uniqueness of the equilibrium point, and its global exponential stability, for cellular NNs with constant delays and without assuming boundedness for the signal functions. Furthermore, using the same inequality in [27] and further ideas, Zhao [28] presented other new criteria ensuring global exponential stability and existence of periodic oscillatory solutions of BAM NN with constant delays. By introducing ingenious real parameters, employing suitable Lyapunov functionals, and applying contracting mapping, Zhao and Wang [29] gave a set of novel sufficient conditions ensuring the existence, uniqueness, and global exponential stability of periodic oscillatory solutions of a class of reaction–diffusion NNs with constant delays and time-varying coefficients. In this paper, however, we have only investigated the existence, uniqueness
of equilibrium point, and global exponential stability of BAM NNs using an -matrix, Young’s inequality, and Hölder’s inequality, instead of the inequality used in [27]. The sufficient conditions obtained in this paper are different from those in [27]. V. CONCLUSION The main contribution of this paper is a result that ensures the existence, uniqueness, and global exponential stability of the equilibrium point of BAM NNs with time delays, without assuming that the activation functions and signal propagation functions are bounded. New sufficient conditions for ascertaining global exponential stability, i.e., Theorem 1 and Corollary 2, have been derived for BAM NNs, improving and extending previous work. APPENDIX PROOF OF THEOREM 1 We here prove Theorem 1. Proof: We consider here the case when , using a sequence of three steps. The case can be directly proved in three similar steps, but in which we use a simple computation (omitted here) instead of Young’s inequality. is an equilibrium point of the BAM NN given in If (1), then satisfies (2). Let (6)
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where
with As
where and Obviously, the solution of is the equilibrium point of the BAM NN given in (1). Hence, the existence and the uniqueness of the equilibrium of the BAM NN given in (1) is a homeomorcan be reduced to proving that the map . phism of Step 1) We prove that is an injective map on by showing that assuming otherwise leads to a contradiction. and exist in We suppose that values such that while , so or . is a nonsingular -matrix, by Lemma Since 1, then there exists a positive vector such that , so
and
(9) Next, we bound the value of the left-hand side of (9). By Assumption 2, using the extended version of Young’s inequality, we obtain
(7) where Since
, or
and ,
, , , we can see that
, and
(8) Since and entiable functions, there exist such that
and
are differand
(10) Furthermore
From (9) and (10), we find that (11) which contradicts (8), and hence implies that is an injective map on .
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Step 2) We prove that
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where
Since , , , it suffices to show that
, , and
are constants,
Clearly,
. Thus
where and
(13) Let In fact, there exist
and
such that Using Hölder’s inequality gives
(14) Combining (13) and (14) gives
so
and, therefore
(12)
From Steps 1) and 2) and Lemma 4, the map is a homeomorphism of . Thus, the BAM NN given in (1) has a unique equilibrium point. Let us . denote the unique equilibrium point by Step 3) We now prove that this unique equilibrium point is globally exponentially stable. In order to simplify our proof, using translation and , we
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transform the system given in (1) to the following system:
(15) where
and
Since the transformation is a translation, we only need to prove that the origin (0, 0) of the transformed system is globally exponentially stable. We consider the following Lyapunov function:
where
(16) where the positive real number will be determined later. The upper right Dini-derivative along the trajectories of the BAM NN given in (15) are given by
so
(17) Now
(18)
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Hence (20) and , and Since . Since and are monotonic increasing and continuous functions of in
(19) Denote
using the intermediate value theorem of continuous , functions, there exist positive values , and , , such that , , and , . Let
Since each for Noting that
Setting
, then . Noting that , we have that , and , . to in (20) gives
,
by Lemma 7, we have (21) Setting
in (16) gives
(22) Combining (21) and (22) gives (23) where tively, given by
Let
and
and
are, respec-
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Clearly,
. From (16), it is clear that
(24) It follows that
for any , where is a constant. This implies that the solution of (15), at the origin, is globally exponentially stable. This completes the proof of Theorem 1. REFERENCES [1] S. Arik, “Global asymptotic stability of a class of dynamical neural networks,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 4, pp. 568–571, Apr. 2000. [2] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 2, pp. 168–171, Feb. 1998. [3] S. Arik and V. Tavsanoglu, “On global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 4, pp. 571–574, Apr. 2000. [4] J. Cao, “Periodic-solutions and exponential stability in delayed cellular neural networks,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 60, no. 3, pp. 3244–3248, Sep. 1999. [5] J. Cao and J. Wang, “Global exponential stability and periodicity of recurrent neural networks with time delays,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 5, pp. 920–931, May 2005. [6] J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Trans. Neural Netw., vol. 18, no. 2, pp. 416–430, Mar. 2007. [7] S. Guo and L. Huang, “Stability analysis of Cohen-Grossberg neural networks,” IEEE Trans. Neural Netw., vol. 17, no. 1, pp. 106–117, Jan. 2006. [8] H. Lu and S.-I. Amari, “Global exponential stability of multitime scale competitive neural networks with nonsmooth functions,” IEEE Trans. Neural Netw., vol. 17, no. 5, pp. 1152–1164, Sep. 2006. [9] Z. Zeng and J. Wang, “Improved conditions for global exponential stability of recurrent neural networks with time-varying delays,” IEEE Trans. Neural Netw., vol. 17, no. 3, pp. 623–635, May 2006. [10] B. Kosto, “Bidirectional associative memories,” IEEE Trans. Syst. Man Cybern., vol. SMC-18, no. 1, pp. 49–60, Jan./Feb. 1988. [11] J. Cao and L. Wang, “Exponential stability and periodic oscillatory solution in BAM networks with delays,” IEEE Trans. Neural Netw., vol. 13, no. 2, pp. 457–463, Mar. 2002. [12] J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Phys. Lett. A, vol. 307, no. 2–3, pp. 136–147, Jan. 2003. [13] K. Gopalsamy and X. Z. He, “Delay-dependent stability in bidirectional associative memory neural networks,” IEEE Trans. Neural Netw., vol. 5, no. 6, pp. 998–1002, Nov. 1994. [14] X. Liao and J. Yu, “Qualitative analysis of bi-directional associative memory with time delay,” Int. J. Circuit. Theory Appl., vol. 26, no. 3, pp. 219–229, May/Jun. 1998. [15] X. Liao, J. Yu, and G. Chen, “Novel stability criteria for bidirectional associative memory neural networks with time delays,” Int. J. Circuit Theory Appl., vol. 30, no. 5, pp. 519–546, Sep./Oct. 2002. [16] S. Mohamad, “Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks,” Physica D, vol. 159, no. 3–4, pp. 233–251, Nov. 2001.
[17] J. Zhang and Y. Yang, “Global stability analysis of bidirectional associative memory neural networks with time delay,” Int. J. Circuit Theory Appl., vol. 29, no. 2, pp. 185–196, Mar./Apr. 2001. [18] H. Zhao, “Global stability of bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 297, no. 3–4, pp. 182–190, May 2002. [19] S. Arik, “Global asymptotic stability of bidirectional associative memory neural networks with time delays,” IEEE Trans. Neural Netw., vol. 16, no. 3, pp. 580–586, May 2005. [20] J. Cao and M. Dong, “Exponential stability of delayed bi-directional associative memory networks,” Appl. Math. Comput., vol. 135, no. 1, pp. 105–112, Feb. 2003. [21] X. Liao and K. Wong, “Global exponential stability of hybrid bi-directional associative memory neural networks with discrete delays,” Phys. Rev. E, vol. 67, pp. 042 901–042 904, Apr. 2003. [22] R. A. Horn and C. R. Johnson, Topics in Matrix Analyis. Cambridge, U.K.: Cambridge Univ. Press, 1991. [23] J. Zhang and X. S. Jin, “Global stability analysis in delayed Hopfield neural network models,” IEEE Trans. Neural Netw., vol. 13, no. 3, pp. 745–753, Jun. 2000. [24] J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neutons,” Proc. Nat. Acad. Sci. USA, vol. 81, pp. 3088–3092, 1984. [25] J. Zhang, Y. Suda, and H. Komine, “Global exponential stability of Cohen-Grossberg neural networks with variable delays,” Phys. Lett. A, vol. 338, no. 1, pp. 44–50, Apr. 2005. [26] H. Zhao, “Global exponential stability and periodicity of cellular neural networks with variable delays,” Phys. Lett. A, vol. 336, no. 4–5, pp. 331–341, Mar. 2005. [27] H. Zhao and J. Cao, “New conditions for global exponential stability of cellular neural networks with delays,” Neural Netw., vol. 18, no. 10, pp. 1332–1340, Dec. 2005. [28] H. Zhao, “Exponential stability and periodic oscillatory of bi-directional associative memory neural network involving delays,” Neurocomputing, vol. 69, no. 4–6, pp. 424–448, Jan. 2006. [29] H. Zhao and G. Wang, “Existence of periodic oscillatory solution of reaction diffusion neural networks with delays,” Phys. Lett. A, vol. 343, no. 15, pp. 372–383, Aug. 2005.
Xin-Ge Liu received the B.S. and M.S. degrees from Hunan Normal University, Changsha, China and the Ph.D. degree from Central South University, Changsha, China, all in mathematics, in 1991, 1994, and 2001, respectively. In May 1994, he joined the School of Mathematical Science and Computing Technology, Central South University, where he is currently an Associate Professor. From 2002 to 2004, he was a Postdoctor of Automatic Control Engineering, Central South University. From October 2004 to September 2006, he was a Postdoctor and Visiting Scholar in the School of Computer Science, Cardiff University, Cardiff, U.K. His research interests include harmonic analysis, wavelets analysis, nonlinear systems, neural networks, and stability theory.
Ralph R. Martin received the Ph.D. degree from Cambridge University, Cambridge, U.K., in 1983, for a dissertation on “Principal Patches.” He has been working in the field of geometric modeling since 1979. He has progressed from Lecturer to Professor at Cardiff University, Cardiff, U.K., taking this last post in 2000. His publications include over 170 papers and ten books covering such topics as solid modeling, surface modeling, reverse engineering, intelligent sketch input, mesh processing, vision-based geometric inspection, and geometric reasoning. Dr. Martin is a Fellow of the Institute of Mathematics and its Applications and a Member of the British Computer Society. He is on the editorial boards of Computer Aided Design, International Journal of Shape Modelling, CAD and Applications, and International Journal of CAD/CAM.
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Min Wu received the B.S. and M.S. degrees in engineering from the Central South University, Changsha, China, in 1983 and 1986, respectively, and the Ph.D. degree in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1999. In July 1986, he joined the staff of the Central South University, where he is currently a Professor of Automatic Control Engineering. He was a Visiting Scholar at the Department of Electrical Engineering, Tohoku University, Sendai, Japan, from 1989 to 1990, and a Visiting Research Scholar at the Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo, Japan, from 1996 to 1999. His current research interests are robust control and its application, process control, and intelligent control. Dr. Wu received the best paper award at the International Federation of Automatic Control (IFAC) 1999 (jointly with M. Nakano and J.-H. She). He is a member of the Nonferrous Metals Society of China and the China Association of Automation.
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Mei-Lan Tang received the B.S. degree in mathematics from Hunan Normal University, Changsha, China, in 1997 and the M.S. degree in applied mathematics from Central South University, Changsha, China, in 2004. Currently, she is a Lecturer at the School of Mathematical Science and Computing Technology, Central South University, Changsha, China. She is the author or coauthor of more than 30 journal papers. Her research interests include neural networks and functional differential equation.
