Existence and Global Exponential Stability of ... - Semantic Scholar
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Existence and Global Exponential Stability of Almost Periodic Solution for Cellular Neural Networks With Variable Coefficients and Time-Varying Delays Haijun Jiang, Long Zhang, and Zhidong Teng
Abstract—In this paper, we study cellular neural networks with almost periodic variable coefficients and time-varying delays. By using the existence theorem of almost periodic solution for general functional differential equations, introducing many real parameters and applying the Lyapunov functional method and the technique of Young inequality, we obtain some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution. The results obtained in this paper are new, useful, and extend and improve the existing ones in previous literature. Index Terms—Almost periodic solution, cellular neural networks (CNNs), global exponential stability, hull equation, hull function, time-varying delays, variable coefficient, young inequality.
I. INTRODUCTION
T
HE dynamic behaviors of cellular neural networks (CNNs), delayed cellular neural networks (DCNNs), and BAM neural networks have been deeply investigated recently due to its applicability in solving some image processing, signal processing, pattern recognition, and automatic control problems. Many important results on the existence of equilibrium point, global asymptotic stability, global exponential stability, and the existence of periodic solutions and its global stability have been obtained by using the Lyapunov function (or functional) approach, directly integral method, LMI approach, the tool of nonsmooth analysis, -matrix theory and the Banach fixed point theorem, we refer to [1]–[19], and the references cited therein. In practice, although the use of constant discrete delays in the models of delayed feedback systems serve as a good approximation in simple circuits consisting of a small number of neurons, neural networks usually have a spatial extent due to the presence of an amount of parallel pathway with a variety of axon sizes and lengths. Therefore, the research on the CNNs with variable coefficients and time-varying delay also is very important and significant like on the autonomous cellular neural networks. To the best of our knowledge, a few authors have studied the boundedness, global exponential stability and the existence of periodic solution and its global exponential stability for cellular Manuscript received September 7, 2004; revised April 30, 2005. This work was supported by the National Natural Science Foundation of China under Grant 10361004 and by the Natural Science Foundation of Xinjiang University. The authors are with College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2005.857951
neural networks and BAM neural networks with variable coefficients and time-varying delay by using continuation theorem based on coincidence degree, Lyapunov functional method and inequality analysis technique, we refer to [20]–[24]. In addition, In [25], the delayed BAM neural networks with almost periodic variable coefficients and delays are studied, by using the Banach fixed point theorem and constructing suitable Lyapunov functional, authors obtained some sufficient conditions to ensure the existence, uniqueness and global stability of almost periodic solution. In [26], by combining the theory of the exponential dichotomy and Lyapunov function, the authors studied the existence and attractivity of almost periodic solution for cellular neural networks with distributed delays and variable coefficients and obtained some sufficient conditions to ensure that for the networks there exists a unique almost periodic solution, and all its solutions converge to such an almost periodic solution. In [27], using the Banach fixed point theorem, authors obtained a sufficient condition to ensure the existence of almost periodic solution and its global attractivity for shunting inhibitory cellular neural networks. In [28], several sufficient conditions are obtained ensuring the existence and global exponential stability of almost periodic solution for BAM neural networks with distributed delays based on fixed point method and some analysis technique. In this paper, we will discuss the existence, uniqueness and its global exponential stability of almost periodic solutions for the following cellular neural networks with variable coefficients and time-varying delays
(1) Here, we will not require that all nonlinear response funcand are bounded on tions . By constructing suitable Lyapunov functional, introducing ingeniously many real parameters, and applying the technique of Young inequality and the existence theorems of almost periodic solutions for general functional differential equations, we will obtain some simple and realistic sufficient conditions to ensue the existence, uniqueness and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays.
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The remainder of this paper is organized as follows. In Section II, some preliminaries and lemmas are presented. In Section III, we will give the main result. In Section IV, we will give an example to illustrate the results obtained in this paper. Finally, we make concluding remarks in Section V.
, when , then transforms into Remark 3: In the following special cases. There are constants , , , , and such that
II. PRELIMINARIES Throughout this paper, for (1) we introduce the following assumptions. , , , and • are almost periodic continuous functions defined on . is nonnegative and dif• ferentiable almost periodic functions defined on , and derivative is also uniformly continuous on and . There are nonnegative constants and • such that
for all Let by
and
.
. We denoted the Banach space of continuous functions with the norm
where and In this paper, we always assume that all solutions of (1) satisfy the following initial conditions: for all , and There are constants • , that
. ,
, , , and
, such
for all and , where is the . inverse function of Remark 1: Since and , we immediately obtain that the inverse function of exists for all . Remark 2: In assumption , when , then we can choose . In this will transform into the following special case, assumption cases. There are positive constants and such that
for all
and
.
(2) where and . Definition 1 [29]: System (1) is said to be globally exponenand such that tially stable, if there are constants and with the initial functions for any two solutions and , respectively, one has
Definition 2 [30]: A function defined on is said to , there is a be almost periodic, if for any real sequence subsequence such that exists . uniformly for all of an almost periodic Definition 3 [30]: The hull is a set of functions defined on , such that for function any function there is a real sequence such that uniformly for all . is an almost periodic function on , Suppose that is a differentiable almost periodic function on and the derivative is uniformly continuous on . Further let . We have following result. and Lemma 1: Assume and are the inverse functions of and , respectively. Then the following results hold. (a) If there is a sequence such that and uniformly , then we have for uniformly for . and are both almost (b) Functions periodic functions and .
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Proof: First, since is a differentiable almost periodic is uniformly continuous function on and the derivative on , by the properties of almost periodic functions (see [30, also is an almost periodic function. ch. 1]) we know that uniformly for and Since is uniformly converges for as , by the properties of uniform convergence of function sequences (see of is [31, Chapter 4]) we obtain that the derivative and uniformly existent for all . Therefore, if , then we also for . Consequently, the inverse function have of exists for all . , then we have . Since Let , we further have . Since , we obtain
This completes the proof of conclusion (a) of Lemma 1. Conclusion (b) of Lemma 1 can be obtain directly from conclusion (a). The proof of Lemma 1 is completed. , , Let functions , , such that there is a time sequence satisfies , , , and uniformly for as . Thus, we obtain hull equations of (1) as follows.
(5)
(3) On the other hand, since , we obtain that of exists and is differentiable the inverse function and are also increasing functions on . Moreover, , and , by the average value theon . For any , orem of the derivative and the derivative rule of inverse function, such that there must be a
(4)
From have
, we have
. Hence, we
From this and by (4), we further obtain
For hull equations (5), we have following lemma. Lemma 2: – , then hull equations (5) also (a) If (1) satisfies – , and all parameters , , , , satisfies , , , , and are universal and independent of the choice of hull equation (5). , (b) , , and . Proof: Since is almost periodic function and is uniformly continuous on , by the properties of almost periodic functions (see [30, ch. 1]) we know also is almost periodic function. Therefore, without that converges uniloss of generality, we can assume that formly for as . Since uniformly for , by the properties of uniform convergence of function sequences (see [31, ch. 4]) we obtain that the derivaof exists for all , tive uniformly for and . Furthermore, , then we also have if . Let function , then we have that the inverse of exists for all . function We first prove conclusion (a) of Lemma 2. From the properties of the almost periodic functions (see [30, ch. 1]) and the above discussion, we easily prove that, if (1) satisfies assump– , then hull equation (5) of (1) also satisfies astions – . sumptions Since functions , and satisfy the all assumptions of Lemma 1, from conclusion (a) of Lemma 1, we immediately obtain
Hence, by (3) we have and Since
, we finally obtain uniformly for all . Further from conclusion (b) , , of Lemma 1, we have that
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and odic functions, and
are
also
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almost
periand
. Hence, we further have
On the Dini derivative of the absolute value of a differentiable function, we have the following lemma. is a differentiable function deLemma 6: Assume that fined on . Then for any the Dini upper right derivaof function exists and has the expression as tive follows:
where the function if if if
or or and
and and .
In order to study the boundedness of solutions of hull equation (5), we need to introduce the following results on the boundedness of solutions for general functional differential equations. We consider the following equation (6)
for all and . This shows that hull equation (5) satisfies . In addition, from above proof, we directly see that all parameters , , , , , , , and are universal and independent of the choice of hull equation (5). Conclusion (b) can be obtained by directly using the properties of almost periodic functions (see [30, ch. 1]). This completes the proof. On the existence of almost periodic solution for (1), we have the following lemma. such that each Lemma 3: If there is a positive constant defined on and hull equation of (1) has a unique solution satisfies for all , then (1) has a unique almost periodic solution. This lemma can been proved by using the same method given in Theorem 3.2 in [30, ch. 3]. On the Young inequality we have the following lemma. Lemma 4: Assume that , , , with , then the following inequality hold.
is continuous where functional with respect to ( , ) and satisfies the local Lipschitz condition . For any and with respect to , where , let be the solution of (6) satisfying the for all . initial condition Definition 4 [29]: System (6) is said to be uniformly ultisuch that for any mately bounded, if there is a constant constants , there is a such that for any and with , one has
Let functions ( , 2, 3, 4) be continand as . uous and increasing, is conFurther, let functional and satisfies the local Lipschitz tinuous with respect to condition with respect to . On the ultimate boundedness of solutions for (6), we have the following result. and Lemma 7: Suppose that there are functional ( , 2, 3, 4) such that functions
(7) and (8)
The proof of Lemma 4 can be found in [19], [22]. Let function be defined on and be bounded and uniformly continuous on , we have the following useful lemma. (or , then Lemma 5: If as (or ).
for some constant . Then (6) is uniformly ultimately bounded. Remark 4: The proof of Lemma 7 has been given by Burton in [29, Theorem 4.2.10]. Further, from the proof of Lemma 7, we can see that ultimate bound obtained in Lemma 7 depend ( , 2, 3, 4), constant and delay . only on functions
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III. MAIN RESULT
Thus, from the above inequalities we obtain
In this section, we will state and prove the following main results on the existence and uniqueness of almost periodic solution and its global exponential stability for (1). Theorem 1: Suppose that – hold. Then, (1) has a unique almost periodic solution which is globally exponentially stable. Proof: We divide the proof into the following several steps. First, we prove the ultimate boundedness of solutions be of hull equation (5). Let any solution of hull equation (5). We consider the Lyapunov functional as follows:
We let
,
and
This shows that satisfies (7) of Lemma 7. Calculating further the Dini upper right derivative of with respect to time , by Lemma 6, we have
Using assumption
, we obtain
From conclusion (b) of Lemma 2, we directly obtain
and
We further choose functions ( , , , 2, 3, 4) as follows, , and . From conclusion (a) of ( , 2, 3, 4) are independent Lemma 2, we have that of the choice of hull equation (5). By directly calculating, we obtain (9)
and
, 2) , to be To this end, we have two cases: 1) , estimating the right-hand side (RHS) considered. When of (9) by using Young inequality, we have
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(10) and
(12)
From assumption
and conclusion (a) of Lemma 2, we have
(11) Therefore, from (9)–(11) and assumptions obtain
and
, we
for all obtain
and
. Hence, by (12) we finally
(13)
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, where
Obviously, from conclusion (b) of Lemma 2 we directly have
Let , then by conclusion (a) of Lemma is independent of the choice of hull equation 2, we see that (5). From (13) and (15) we finally have
and
Furthermore, from conclusion (b) of Lemma 2, we directly obtain
Next, when
, then we must have . Directly estimating the RHS (9), we have
This shows that satisfy (8) of Lemma 7. Hence, by Lemma 7, we obtain that all solutions of hull equation (5) are and are uniformly ultimately bounded. Let is defined on the ultimate bound, then, from Remark 4, we know that ultimate ( , 2, 3, 4), constant bound depends only on functions and delay . Therefore, is independent of the choice of hull equation (5) and is only relevant to (1). Second, we prove that hull equation (5) has a bounded with bound . By the propersolution defined on ties of almost periodic functions, there exists a time se, and , such that quence , , , and uniformly for . Let be a solution of hull equation (5). Then, from the ultimate boundedness of (1), we have
Further, we obtain (16) (14) Let From assumption
for all
and
, then each function and satisfies
and conclusion (a) of Lemma 2, we have
. Hence, by (14) we obtain (17)
(15) for all
, where
, is defined on
where all
. From (16), we have for , where . Hence, for any integer , the sequence is uniformly bounded . further, by (16), (17) and assumptions and on , we can obtain that there is a constant which is independent of any , such that
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for all and ( , 2) we have
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. Hence, for any
From this we obtain easily that, for any integer , the seis equicontinuous on . Using quence Ascoli-Arzela theorem, we can obtain that there is a subsequence of such that is convergent on and uniformly convergent in any bounded interval of . Let . Then is defined on function and continuous. Further, by (16), also satisfies
(20) From (17) and assumptions and , we can easily prove is a solution of hull equation (5) defined on . Thus, that each hull equation (5) has a bounded solution defined on with bound . Third, we prove the uniqueness of bounded solution with bound of hull equation (5). Let defined on and are two bounded solution dewith bound of hull equation (5). Further, let fined on , , then satisfies the following differential inequality.
For (20), using a similar argument as in (10)–(14), we finally obtain
is nonincreasing for all This show that (21) from to 0 for any , we obtain
(21) . Integrating
Hence, we have (18)
We introduce the Lyapunov function as follows: Thus, by Lemma 4, we further obtain for each . Let
as
(22) (19) and , we have By assumptions , there are sufficient large stant Obviously, is bounded on . Calculating Dini upper right with respect to , from (18), assumptions derivative of and conclusion (a) of Lemma 2, we can obtain
. For any consuch that
(23) Directly from (19) of
, we have
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Further, from (22), (23), and , when , we obtain
This shows that as . Since is nonfor all . increasing function on , we have Therefore, we finally obtain for all and . This shows that each hull equation (5) has a unique bounded solution defined on with bound . Therefore, by Lemma 3, we finally obtain that (1) has a unique almost periodic solution defined on . Last, we prove the global exponential stability of almost periodic solution of (1). Let be the almost periodic solution of (1) and further let be any solutions of (1) satisfying the for all , where initial condition . From the ultimate boundedness of (1), we obtain that is defined on and is bounded . Further, by assumption , we can choose a constant on such that
(24) for all
. We choose the Lyapunov functional as follows:
(26)
(25) Calculating the Dini upper right derivative of to , by Lemma 6 and assumption we have
with respect
, 2) , to be To this end, we have two cases: 1) , estimating the RHS of (26) by using considered. When Young inequality, similar to (10) and (11) we have
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and
, then we must have . Directly estimating the RHS of (26), by (24) we have Next, when
Thus, from (24) and (26), we obtain
(28) for all
. Therefore, from (27) and (28), we finally have for all
(29)
Directly from (25), we obtain
for all
where
and
for , and
,
Hence, by (29) we finally obtain
(27) for all
.
for all , where is a constant and independent of any solution (1). This shows that the almost periodic solution of (1) is globally exponentially stable. This finally completes the proof of Theorem 1. As direct consequence of Theorem 1, we have the following two corollaries. – and Corollary 1: Suppose that assumptions hold. Then, (1) has a uniqueness almost periodic solution which is global exponential stability.
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Corollary 2: Suppose that assumptions – and hold. Then (1) has a uniqueness almost periodic solution which is global exponential stability. IV. EXAMPLE We consider the following two-dimensional (2-D) CNNs with time-varying coefficients and delays
(30) where ously,
,
,( , 2). Obviand satisfy ( , 2). that is,
are unbounded on ,
Comparing with the corresponding results given in [25]–[28], we find that the results obtained in this paper very strongly improve and extend those results in many aspects. First, we see that the method introduced in this paper is new and completely different from the methods given in [25]–[28]. Second, we see that the criterion on the existence of almost periodic solutions obtained in this paper is completely different from the corresponding results given in [25]–[28]. Third, in this paper, we and do not assume that the nonlinear response functions are bounded on . Fourth, we see that main assumpintroduced in this paper which ensures the existence, tion uniqueness, and global exponential stability is very weaker than the corresponding assumptions given in [25]–[28]. In particular, we can choose many parameters, for exin assumption , and , such that holds. In addiample, parameters tion, in assumption we also have not to use the above and , and below bounds of the coefficients of (1). Therefore, we finally have that the main results given in this paper are more general than those in [25]–[28].
. , ,
For (30), we take , ,
,
, . We directly have and . Obviously, – fied. Further choose the parameters ( , 2) and , then we have
, , ,
,
ACKNOWLEDGMENT The authors would like to thank the referees and the Associate Editor for suggesting improvements in presentation.
and
REFERENCES are satis-
and
for all . Hence, also holds. Therefore, by Theorem 1, (30) has a unique almost periodic solution which is globally exponentially stable. But under the conditions above, the condition given in [25]–[28] is not satisfied. V. CONCLUSION In this paper, we study a class of the cellular neural networks with almost periodic coefficients and time-varying delays. By using the existence theorem of almost periodic solutions for general almost periodic differential equations, introducing many real parameters and applying the Lyapunov function method and Young inequality technique, under the assumpand may be tions that nonlinear response functions unbounded on . We establish some sufficient and realistic conditions to ensure the existence and global exponential stability of almost periodic solutions.
[1] S. Arik, “Global asymptotic stability of a larger class of neural networks with constant time delay,” Phys. Lett. A, vol. 311, pp. 504–511, 2003. [2] J. Cao, “On exponential stability and periodic solutions of CNN’s with delay,” Phys. Lett. A, vol. 267, pp. 312–318, 2000. , “Periodic oscillation and exponential stability of delayed cellular [3] neural networks,” Phys. Rev. E, vol. 60, no. 3, pp. 3244–3248, 1999. [4] , “Global exponential stability and periodic solutions of delayed cellular neural networks,” J. Comput. Syst. Sci., vol. 60, pp. 38–46, 2000. [5] J. Cao and Q. Li, “On the exponential stability and periodic solutions of delayed cellular neural networks,” J. Math. Anal. Appl., vol. 252, pp. 50–64, 2000. [6] 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, 2002. [7] Q. Dong, K. Matsui, and X. Huang, “Existence and stability of periodic solutions for Hopfield neural network equations with periodic input,” Nonlin. Anal., vol. 49, pp. 471–479, 2002. [8] M. Dong, “Global exponential stability and existence of periodic solutions of CNN’s with delays,” Phys. Lett. A, vol. 300, pp. 49–57, 2002. [9] C. Feng and R. Plamondon, “Stability analysis of bidirectional associative memory networks with time delays,” IEEE Trans. Neural Netw., vol. 14, no. 6, pp. 1560–1565, 2003. [10] H. Huang, J. Cao, and J. Wang, “Global exponential stability and periodic solutions of recurrent neural networks with delays,” Phys. Lett. A, vol. 298, no. 5–6, pp. 393–404, 2002. [11] X. Liao and J. Wang, “Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 2, pp. 268–275, 2003. [12] X. Liao, K. K. Wong, and Z. Wu, “Asymptotic stability criteria for a two-neuron network with different time delays,” IEEE Trans. Neural Netw., vol. 14, no. 1, pp. 222–227, 2003. [13] S. Mohamad and K. Gopalsamy, “Exponential stability of continuous-time and discrete-time cellular neural networks with delays,” Appl. Math. Comput., vol. 135, pp. 17–38, 2003. [14] H. Qi and L. Qi, “Deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis,” IEEE Trans. Neural Netw., vol. 15, no. 1, pp. 99–109, 2004. [15] T. Roska and L. O. Chua, “Cellular neural networks with nonlinear and delay-type template,” J. Circuit Theory Appl., vol. 20, pp. 469–481, 1992.
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[16] T. Roska, C. W. Wu, and L. O. Chua, “Stability of cellular neural networks with dominant nonlinear and delay-type template,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 40, no. 4, pp. 270–272, 1993. [17] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability and dynamics of delay-type general and cellular neural networks,” IEEE Trans. Circuits Syst., vol. 39, no. 6, pp. 487–490, 1992. [18] V. Singh, “A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Neural Netw., vol. 15, no. 1, pp. 223–225, 2004. [19] C. Sun and C. Feng, “Exponential periodicity and stability of delays neural networks,” Math. Compu. Simul., vol. 66, pp. 469–478, 2004. [20] S. Guo, L. Huang, B. Dai, and Z. Zhang, “Global existence of periodic solutions of BAM neural networks with variable coefficients,” Phys. Lett. A, vol. 317, pp. 97–106, 2003. [21] H. Jiang, Z. Li, and Z. Teng, “Boundedness and stability for nonautonomous cellular neural networks with delay,” Phys. Lett. A, vol. 306, pp. 313–325, 2003. [22] J. Liang and J. Cao, “Boundedness and stability for recurrent neural networks with variable coefficient and time-varying delays,” Phys. Lett. A, vol. 318, pp. 53–64, 2003. [23] Z. Liu and L. Liao, “Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays,” J. Math. Anal. Appl., vol. 290, pp. 247–262, 2004. [24] J. Zhou, Z. Liu, and G. Chen, “Dynamics of periodic delayed neural networks,” Neural Netw., vol. 17, pp. 87–101, 2004. [25] A. Chen, L. Huang, and J. Cao, “Existence and stability of almost periodic solution for BAM neural networks with delays,” Appl. Math. Comput., vol. 137, pp. 177–193, 2003. [26] A. Chen and J. Cao, “Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients,” Appl. Math. Comput., vol. 134, pp. 125–140, 2002. [27] , “Almost periodic solution of shunting inhibitory CNN’s with delays,” Phy. Lett. A, vol. 298, pp. 161–170, 2002. [28] Z. Liu, A. Chen, J. Cao, and L. Huang, “Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays,” Phy. Lett. A, vol. 319, pp. 305–316, 2003. [29] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations. New York: Academic, 1985.
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[30] C. He, Almost Periodic Differential Equations. Beijing, China: Higher Education, 1992. [31] G. Klambauer, Mathematical Analysis. New York: Marcel Dekker, 1975.
Haijun Jiang was born in Hunan, China, in 1968. He received the B.S. degree from the Mathematics Department, Yili Teacher College, Yili, Xinjiang, China, the M.S. degree from the Mathematics Department, East China Normal University, Shanghai, China, and the Ph.D. degree from the College of Mathematics and System Sciences, Xinjiang University, China, in 1990, 1994, and 2004, respectively. He is an Associate Professor with the College of Mathematics and System Sciences, Xinjiang University. His present research interests include nonlinear systems, mathematical biology, empdemiology, and dynamics of neural networks.
Long Zhang was born in Xinjiang, China, in 1978. He received the M.S. degree in mathematics from Xinjiang University in 2004. He is currently working toward the Ph.D. degree. His present research interests include nonlinear systems and dynamics of neural networks.
Zhidong Teng was born in Xinjiang, China, in 1960. He received the B.S. degree in mathematics from Xinjiang University in 1982 and the Ph.D. degree in applied mathematics and mechanics from Kazakhstan State University, Kazakhstan, in 1995. He is a Professor and Doctoral Advisor of Mathematics and System Sciences with Xinjiang University. His current research interests include nonlinear dynamics, delay differential equations, dynamics of neural networks, mathematical biology, and empdemiology.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 6, NOVEMBER 2005
Existence and Global Exponential Stability of Almost Periodic Solution for Cellular Neural Networks With Variable Coefficients and Time-Varying Delays Haijun Jiang, Long Zhang, and Zhidong Teng
Abstract—In this paper, we study cellular neural networks with almost periodic variable coefficients and time-varying delays. By using the existence theorem of almost periodic solution for general functional differential equations, introducing many real parameters and applying the Lyapunov functional method and the technique of Young inequality, we obtain some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution. The results obtained in this paper are new, useful, and extend and improve the existing ones in previous literature. Index Terms—Almost periodic solution, cellular neural networks (CNNs), global exponential stability, hull equation, hull function, time-varying delays, variable coefficient, young inequality.
I. INTRODUCTION
T
HE dynamic behaviors of cellular neural networks (CNNs), delayed cellular neural networks (DCNNs), and BAM neural networks have been deeply investigated recently due to its applicability in solving some image processing, signal processing, pattern recognition, and automatic control problems. Many important results on the existence of equilibrium point, global asymptotic stability, global exponential stability, and the existence of periodic solutions and its global stability have been obtained by using the Lyapunov function (or functional) approach, directly integral method, LMI approach, the tool of nonsmooth analysis, -matrix theory and the Banach fixed point theorem, we refer to [1]–[19], and the references cited therein. In practice, although the use of constant discrete delays in the models of delayed feedback systems serve as a good approximation in simple circuits consisting of a small number of neurons, neural networks usually have a spatial extent due to the presence of an amount of parallel pathway with a variety of axon sizes and lengths. Therefore, the research on the CNNs with variable coefficients and time-varying delay also is very important and significant like on the autonomous cellular neural networks. To the best of our knowledge, a few authors have studied the boundedness, global exponential stability and the existence of periodic solution and its global exponential stability for cellular Manuscript received September 7, 2004; revised April 30, 2005. This work was supported by the National Natural Science Foundation of China under Grant 10361004 and by the Natural Science Foundation of Xinjiang University. The authors are with College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of China (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2005.857951
neural networks and BAM neural networks with variable coefficients and time-varying delay by using continuation theorem based on coincidence degree, Lyapunov functional method and inequality analysis technique, we refer to [20]–[24]. In addition, In [25], the delayed BAM neural networks with almost periodic variable coefficients and delays are studied, by using the Banach fixed point theorem and constructing suitable Lyapunov functional, authors obtained some sufficient conditions to ensure the existence, uniqueness and global stability of almost periodic solution. In [26], by combining the theory of the exponential dichotomy and Lyapunov function, the authors studied the existence and attractivity of almost periodic solution for cellular neural networks with distributed delays and variable coefficients and obtained some sufficient conditions to ensure that for the networks there exists a unique almost periodic solution, and all its solutions converge to such an almost periodic solution. In [27], using the Banach fixed point theorem, authors obtained a sufficient condition to ensure the existence of almost periodic solution and its global attractivity for shunting inhibitory cellular neural networks. In [28], several sufficient conditions are obtained ensuring the existence and global exponential stability of almost periodic solution for BAM neural networks with distributed delays based on fixed point method and some analysis technique. In this paper, we will discuss the existence, uniqueness and its global exponential stability of almost periodic solutions for the following cellular neural networks with variable coefficients and time-varying delays
(1) Here, we will not require that all nonlinear response funcand are bounded on tions . By constructing suitable Lyapunov functional, introducing ingeniously many real parameters, and applying the technique of Young inequality and the existence theorems of almost periodic solutions for general functional differential equations, we will obtain some simple and realistic sufficient conditions to ensue the existence, uniqueness and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays.
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The remainder of this paper is organized as follows. In Section II, some preliminaries and lemmas are presented. In Section III, we will give the main result. In Section IV, we will give an example to illustrate the results obtained in this paper. Finally, we make concluding remarks in Section V.
, when , then transforms into Remark 3: In the following special cases. There are constants , , , , and such that
II. PRELIMINARIES Throughout this paper, for (1) we introduce the following assumptions. , , , and • are almost periodic continuous functions defined on . is nonnegative and dif• ferentiable almost periodic functions defined on , and derivative is also uniformly continuous on and . There are nonnegative constants and • such that
for all Let by
and
.
. We denoted the Banach space of continuous functions with the norm
where and In this paper, we always assume that all solutions of (1) satisfy the following initial conditions: for all , and There are constants • , that
. ,
, , , and
, such
for all and , where is the . inverse function of Remark 1: Since and , we immediately obtain that the inverse function of exists for all . Remark 2: In assumption , when , then we can choose . In this will transform into the following special case, assumption cases. There are positive constants and such that
for all
and
.
(2) where and . Definition 1 [29]: System (1) is said to be globally exponenand such that tially stable, if there are constants and with the initial functions for any two solutions and , respectively, one has
Definition 2 [30]: A function defined on is said to , there is a be almost periodic, if for any real sequence subsequence such that exists . uniformly for all of an almost periodic Definition 3 [30]: The hull is a set of functions defined on , such that for function any function there is a real sequence such that uniformly for all . is an almost periodic function on , Suppose that is a differentiable almost periodic function on and the derivative is uniformly continuous on . Further let . We have following result. and Lemma 1: Assume and are the inverse functions of and , respectively. Then the following results hold. (a) If there is a sequence such that and uniformly , then we have for uniformly for . and are both almost (b) Functions periodic functions and .
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Proof: First, since is a differentiable almost periodic is uniformly continuous function on and the derivative on , by the properties of almost periodic functions (see [30, also is an almost periodic function. ch. 1]) we know that uniformly for and Since is uniformly converges for as , by the properties of uniform convergence of function sequences (see of is [31, Chapter 4]) we obtain that the derivative and uniformly existent for all . Therefore, if , then we also for . Consequently, the inverse function have of exists for all . , then we have . Since Let , we further have . Since , we obtain
This completes the proof of conclusion (a) of Lemma 1. Conclusion (b) of Lemma 1 can be obtain directly from conclusion (a). The proof of Lemma 1 is completed. , , Let functions , , such that there is a time sequence satisfies , , , and uniformly for as . Thus, we obtain hull equations of (1) as follows.
(5)
(3) On the other hand, since , we obtain that of exists and is differentiable the inverse function and are also increasing functions on . Moreover, , and , by the average value theon . For any , orem of the derivative and the derivative rule of inverse function, such that there must be a
(4)
From have
, we have
. Hence, we
From this and by (4), we further obtain
For hull equations (5), we have following lemma. Lemma 2: – , then hull equations (5) also (a) If (1) satisfies – , and all parameters , , , , satisfies , , , , and are universal and independent of the choice of hull equation (5). , (b) , , and . Proof: Since is almost periodic function and is uniformly continuous on , by the properties of almost periodic functions (see [30, ch. 1]) we know also is almost periodic function. Therefore, without that converges uniloss of generality, we can assume that formly for as . Since uniformly for , by the properties of uniform convergence of function sequences (see [31, ch. 4]) we obtain that the derivaof exists for all , tive uniformly for and . Furthermore, , then we also have if . Let function , then we have that the inverse of exists for all . function We first prove conclusion (a) of Lemma 2. From the properties of the almost periodic functions (see [30, ch. 1]) and the above discussion, we easily prove that, if (1) satisfies assump– , then hull equation (5) of (1) also satisfies astions – . sumptions Since functions , and satisfy the all assumptions of Lemma 1, from conclusion (a) of Lemma 1, we immediately obtain
Hence, by (3) we have and Since
, we finally obtain uniformly for all . Further from conclusion (b) , , of Lemma 1, we have that
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and odic functions, and
are
also
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almost
periand
. Hence, we further have
On the Dini derivative of the absolute value of a differentiable function, we have the following lemma. is a differentiable function deLemma 6: Assume that fined on . Then for any the Dini upper right derivaof function exists and has the expression as tive follows:
where the function if if if
or or and
and and .
In order to study the boundedness of solutions of hull equation (5), we need to introduce the following results on the boundedness of solutions for general functional differential equations. We consider the following equation (6)
for all and . This shows that hull equation (5) satisfies . In addition, from above proof, we directly see that all parameters , , , , , , , and are universal and independent of the choice of hull equation (5). Conclusion (b) can be obtained by directly using the properties of almost periodic functions (see [30, ch. 1]). This completes the proof. On the existence of almost periodic solution for (1), we have the following lemma. such that each Lemma 3: If there is a positive constant defined on and hull equation of (1) has a unique solution satisfies for all , then (1) has a unique almost periodic solution. This lemma can been proved by using the same method given in Theorem 3.2 in [30, ch. 3]. On the Young inequality we have the following lemma. Lemma 4: Assume that , , , with , then the following inequality hold.
is continuous where functional with respect to ( , ) and satisfies the local Lipschitz condition . For any and with respect to , where , let be the solution of (6) satisfying the for all . initial condition Definition 4 [29]: System (6) is said to be uniformly ultisuch that for any mately bounded, if there is a constant constants , there is a such that for any and with , one has
Let functions ( , 2, 3, 4) be continand as . uous and increasing, is conFurther, let functional and satisfies the local Lipschitz tinuous with respect to condition with respect to . On the ultimate boundedness of solutions for (6), we have the following result. and Lemma 7: Suppose that there are functional ( , 2, 3, 4) such that functions
(7) and (8)
The proof of Lemma 4 can be found in [19], [22]. Let function be defined on and be bounded and uniformly continuous on , we have the following useful lemma. (or , then Lemma 5: If as (or ).
for some constant . Then (6) is uniformly ultimately bounded. Remark 4: The proof of Lemma 7 has been given by Burton in [29, Theorem 4.2.10]. Further, from the proof of Lemma 7, we can see that ultimate bound obtained in Lemma 7 depend ( , 2, 3, 4), constant and delay . only on functions
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III. MAIN RESULT
Thus, from the above inequalities we obtain
In this section, we will state and prove the following main results on the existence and uniqueness of almost periodic solution and its global exponential stability for (1). Theorem 1: Suppose that – hold. Then, (1) has a unique almost periodic solution which is globally exponentially stable. Proof: We divide the proof into the following several steps. First, we prove the ultimate boundedness of solutions be of hull equation (5). Let any solution of hull equation (5). We consider the Lyapunov functional as follows:
We let
,
and
This shows that satisfies (7) of Lemma 7. Calculating further the Dini upper right derivative of with respect to time , by Lemma 6, we have
Using assumption
, we obtain
From conclusion (b) of Lemma 2, we directly obtain
and
We further choose functions ( , , , 2, 3, 4) as follows, , and . From conclusion (a) of ( , 2, 3, 4) are independent Lemma 2, we have that of the choice of hull equation (5). By directly calculating, we obtain (9)
and
, 2) , to be To this end, we have two cases: 1) , estimating the right-hand side (RHS) considered. When of (9) by using Young inequality, we have
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(10) and
(12)
From assumption
and conclusion (a) of Lemma 2, we have
(11) Therefore, from (9)–(11) and assumptions obtain
and
, we
for all obtain
and
. Hence, by (12) we finally
(13)
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, where
Obviously, from conclusion (b) of Lemma 2 we directly have
Let , then by conclusion (a) of Lemma is independent of the choice of hull equation 2, we see that (5). From (13) and (15) we finally have
and
Furthermore, from conclusion (b) of Lemma 2, we directly obtain
Next, when
, then we must have . Directly estimating the RHS (9), we have
This shows that satisfy (8) of Lemma 7. Hence, by Lemma 7, we obtain that all solutions of hull equation (5) are and are uniformly ultimately bounded. Let is defined on the ultimate bound, then, from Remark 4, we know that ultimate ( , 2, 3, 4), constant bound depends only on functions and delay . Therefore, is independent of the choice of hull equation (5) and is only relevant to (1). Second, we prove that hull equation (5) has a bounded with bound . By the propersolution defined on ties of almost periodic functions, there exists a time se, and , such that quence , , , and uniformly for . Let be a solution of hull equation (5). Then, from the ultimate boundedness of (1), we have
Further, we obtain (16) (14) Let From assumption
for all
and
, then each function and satisfies
and conclusion (a) of Lemma 2, we have
. Hence, by (14) we obtain (17)
(15) for all
, where
, is defined on
where all
. From (16), we have for , where . Hence, for any integer , the sequence is uniformly bounded . further, by (16), (17) and assumptions and on , we can obtain that there is a constant which is independent of any , such that
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for all and ( , 2) we have
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. Hence, for any
From this we obtain easily that, for any integer , the seis equicontinuous on . Using quence Ascoli-Arzela theorem, we can obtain that there is a subsequence of such that is convergent on and uniformly convergent in any bounded interval of . Let . Then is defined on function and continuous. Further, by (16), also satisfies
(20) From (17) and assumptions and , we can easily prove is a solution of hull equation (5) defined on . Thus, that each hull equation (5) has a bounded solution defined on with bound . Third, we prove the uniqueness of bounded solution with bound of hull equation (5). Let defined on and are two bounded solution dewith bound of hull equation (5). Further, let fined on , , then satisfies the following differential inequality.
For (20), using a similar argument as in (10)–(14), we finally obtain
is nonincreasing for all This show that (21) from to 0 for any , we obtain
(21) . Integrating
Hence, we have (18)
We introduce the Lyapunov function as follows: Thus, by Lemma 4, we further obtain for each . Let
as
(22) (19) and , we have By assumptions , there are sufficient large stant Obviously, is bounded on . Calculating Dini upper right with respect to , from (18), assumptions derivative of and conclusion (a) of Lemma 2, we can obtain
. For any consuch that
(23) Directly from (19) of
, we have
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Further, from (22), (23), and , when , we obtain
This shows that as . Since is nonfor all . increasing function on , we have Therefore, we finally obtain for all and . This shows that each hull equation (5) has a unique bounded solution defined on with bound . Therefore, by Lemma 3, we finally obtain that (1) has a unique almost periodic solution defined on . Last, we prove the global exponential stability of almost periodic solution of (1). Let be the almost periodic solution of (1) and further let be any solutions of (1) satisfying the for all , where initial condition . From the ultimate boundedness of (1), we obtain that is defined on and is bounded . Further, by assumption , we can choose a constant on such that
(24) for all
. We choose the Lyapunov functional as follows:
(26)
(25) Calculating the Dini upper right derivative of to , by Lemma 6 and assumption we have
with respect
, 2) , to be To this end, we have two cases: 1) , estimating the RHS of (26) by using considered. When Young inequality, similar to (10) and (11) we have
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and
, then we must have . Directly estimating the RHS of (26), by (24) we have Next, when
Thus, from (24) and (26), we obtain
(28) for all
. Therefore, from (27) and (28), we finally have for all
(29)
Directly from (25), we obtain
for all
where
and
for , and
,
Hence, by (29) we finally obtain
(27) for all
.
for all , where is a constant and independent of any solution (1). This shows that the almost periodic solution of (1) is globally exponentially stable. This finally completes the proof of Theorem 1. As direct consequence of Theorem 1, we have the following two corollaries. – and Corollary 1: Suppose that assumptions hold. Then, (1) has a uniqueness almost periodic solution which is global exponential stability.
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Corollary 2: Suppose that assumptions – and hold. Then (1) has a uniqueness almost periodic solution which is global exponential stability. IV. EXAMPLE We consider the following two-dimensional (2-D) CNNs with time-varying coefficients and delays
(30) where ously,
,
,( , 2). Obviand satisfy ( , 2). that is,
are unbounded on ,
Comparing with the corresponding results given in [25]–[28], we find that the results obtained in this paper very strongly improve and extend those results in many aspects. First, we see that the method introduced in this paper is new and completely different from the methods given in [25]–[28]. Second, we see that the criterion on the existence of almost periodic solutions obtained in this paper is completely different from the corresponding results given in [25]–[28]. Third, in this paper, we and do not assume that the nonlinear response functions are bounded on . Fourth, we see that main assumpintroduced in this paper which ensures the existence, tion uniqueness, and global exponential stability is very weaker than the corresponding assumptions given in [25]–[28]. In particular, we can choose many parameters, for exin assumption , and , such that holds. In addiample, parameters tion, in assumption we also have not to use the above and , and below bounds of the coefficients of (1). Therefore, we finally have that the main results given in this paper are more general than those in [25]–[28].
. , ,
For (30), we take , ,
,
, . We directly have and . Obviously, – fied. Further choose the parameters ( , 2) and , then we have
, , ,
,
ACKNOWLEDGMENT The authors would like to thank the referees and the Associate Editor for suggesting improvements in presentation.
and
REFERENCES are satis-
and
for all . Hence, also holds. Therefore, by Theorem 1, (30) has a unique almost periodic solution which is globally exponentially stable. But under the conditions above, the condition given in [25]–[28] is not satisfied. V. CONCLUSION In this paper, we study a class of the cellular neural networks with almost periodic coefficients and time-varying delays. By using the existence theorem of almost periodic solutions for general almost periodic differential equations, introducing many real parameters and applying the Lyapunov function method and Young inequality technique, under the assumpand may be tions that nonlinear response functions unbounded on . We establish some sufficient and realistic conditions to ensure the existence and global exponential stability of almost periodic solutions.
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[16] T. Roska, C. W. Wu, and L. O. Chua, “Stability of cellular neural networks with dominant nonlinear and delay-type template,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 40, no. 4, pp. 270–272, 1993. [17] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability and dynamics of delay-type general and cellular neural networks,” IEEE Trans. Circuits Syst., vol. 39, no. 6, pp. 487–490, 1992. [18] V. Singh, “A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Neural Netw., vol. 15, no. 1, pp. 223–225, 2004. [19] C. Sun and C. Feng, “Exponential periodicity and stability of delays neural networks,” Math. Compu. Simul., vol. 66, pp. 469–478, 2004. [20] S. Guo, L. Huang, B. Dai, and Z. Zhang, “Global existence of periodic solutions of BAM neural networks with variable coefficients,” Phys. Lett. A, vol. 317, pp. 97–106, 2003. [21] H. Jiang, Z. Li, and Z. Teng, “Boundedness and stability for nonautonomous cellular neural networks with delay,” Phys. Lett. A, vol. 306, pp. 313–325, 2003. [22] J. Liang and J. Cao, “Boundedness and stability for recurrent neural networks with variable coefficient and time-varying delays,” Phys. Lett. A, vol. 318, pp. 53–64, 2003. [23] Z. Liu and L. Liao, “Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays,” J. Math. Anal. Appl., vol. 290, pp. 247–262, 2004. [24] J. Zhou, Z. Liu, and G. Chen, “Dynamics of periodic delayed neural networks,” Neural Netw., vol. 17, pp. 87–101, 2004. [25] A. Chen, L. Huang, and J. Cao, “Existence and stability of almost periodic solution for BAM neural networks with delays,” Appl. Math. Comput., vol. 137, pp. 177–193, 2003. [26] A. Chen and J. Cao, “Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients,” Appl. Math. Comput., vol. 134, pp. 125–140, 2002. [27] , “Almost periodic solution of shunting inhibitory CNN’s with delays,” Phy. Lett. A, vol. 298, pp. 161–170, 2002. [28] Z. Liu, A. Chen, J. Cao, and L. Huang, “Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays,” Phy. Lett. A, vol. 319, pp. 305–316, 2003. [29] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations. New York: Academic, 1985.
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[30] C. He, Almost Periodic Differential Equations. Beijing, China: Higher Education, 1992. [31] G. Klambauer, Mathematical Analysis. New York: Marcel Dekker, 1975.
Haijun Jiang was born in Hunan, China, in 1968. He received the B.S. degree from the Mathematics Department, Yili Teacher College, Yili, Xinjiang, China, the M.S. degree from the Mathematics Department, East China Normal University, Shanghai, China, and the Ph.D. degree from the College of Mathematics and System Sciences, Xinjiang University, China, in 1990, 1994, and 2004, respectively. He is an Associate Professor with the College of Mathematics and System Sciences, Xinjiang University. His present research interests include nonlinear systems, mathematical biology, empdemiology, and dynamics of neural networks.
Long Zhang was born in Xinjiang, China, in 1978. He received the M.S. degree in mathematics from Xinjiang University in 2004. He is currently working toward the Ph.D. degree. His present research interests include nonlinear systems and dynamics of neural networks.
Zhidong Teng was born in Xinjiang, China, in 1960. He received the B.S. degree in mathematics from Xinjiang University in 1982 and the Ph.D. degree in applied mathematics and mechanics from Kazakhstan State University, Kazakhstan, in 1995. He is a Professor and Doctoral Advisor of Mathematics and System Sciences with Xinjiang University. His current research interests include nonlinear dynamics, delay differential equations, dynamics of neural networks, mathematical biology, and empdemiology.
