- Research
- Open access
- Published:
Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions
Boundary Value Problems volume 2013, Article number: 105 (2013)
Abstract
In this paper, both global exponential stability and periodic solutions are investigated for a class of delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. By employing suitable Lyapunov functionals, sufficient conditions of the global exponential stability and the existence of periodic solutions are established for reaction-diffusion BAM neural networks with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature. A numerical example is given to show the effectiveness of the obtained results.
1 Introduction
Neural networks (NNs) have been extensively studied in the past few years and have found many applications in different areas such as pattern recognition, associative memory, combinatorial optimization, etc. Delayed versions of NNs were 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 during the last decade (see, e.g., [1–7]). Recently, the authors in [1] and [2] considered the problem of exponential passivity analysis for uncertain NNs with time-varying delays and passivity-based controller design for Hopfield NNs, respectively.
Since NNs related to bidirectional associative memory (BAM) were proposed by Kosko [8], the BAM NNs have been one of the most interesting research topics and have attracted the attention of researchers. In the design and applications of networks, the stability of the designed BAM NNs is one of the most important issues (see, e.g., [9–12]). Many important results concerning mainly the existence and stability of equilibrium of BAM NNs have been obtained (see, e.g., [9–15]).
However, strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. So, we must consider that the activations vary in space as well as in time. In [16–34], the authors considered the stability of NNs with diffusion terms which were expressed by partial differential equations. In particular, the existence and attractivity of periodic solutions for non-autonomous reaction-diffusion Cohen-Grossberg NNs with discrete time delays were investigated in [20]. The authors derived sufficient conditions on the stability and periodic solutions of delayed reaction-diffusion NNs (RDNNs) with Neumann boundary conditions in [21–25]. In these works, due to the divergence theorem employed, a negative integral term with gradient was removed in their deduction. Therefore, the stability criteria acquired by them do not contain diffusion terms; that is to say, the diffusion terms do not have any effect on their deduction and results. Meanwhile, some conditions dependent on the diffusion coefficients were given in [30, 32–34] to ensure the global exponential stability and periodicity of RDNNs with Dirichlet boundary conditions based on 2-norm.
To the best of our knowledge, there are few reports about global exponential stability and periodicity of RDNNs with mixed time delays and Dirichlet boundary conditions, which are very important in theories and applications and also are a very challenging problem. In this paper, by employing suitable Lyapunov functionals, we shall apply inequality techniques to establish global exponential stability criteria of the equilibrium and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature [22, 29].
Throughout this paper, we need the following notations. denotes the n-dimensional Euclidean space. We denote
and
Let , .
The remainder of this paper is organized as follows. In Section 2, the basic notations, model description and assumptions are introduced. In Sections 3 and 4, criteria are proposed to determine global exponential stability, and periodic solutions are considered for reaction-diffusion recurrent neural networks with mixed time delays, respectively. An illustrative example is given to illustrate the effectiveness of the obtained results in Section 5. We also conclude this paper in Section 6.
2 Model description and preliminaries
In this paper, the RDNNs with mixed time delays are described as follows:
The RDNNs model given in (1) can be regarded as RDNNs with two layers, where m is the number of neurons in the first layer and n is the number of neurons in the second layer. , Ω is a compact set with smooth boundary ∂ Ω and in the space ; , . and represent the state of the i th neuron in the first layer and the j th neuron in the second layer at time t and in the space x, respectively. , , , , and are known constants denoting the synaptic connection strengths between the neurons in the two layers, respectively; , , , , and denote the activation functions of the neurons and the signal propagation functions, respectively. and denote the external inputs on the i th neuron and j th neuron, respectively; and are differentiable real functions with positive derivatives defining the neuron charging time; and represent continuous time-varying delay and discrete delay, respectively; and , , and , stand for the transmission diffusion coefficient along the i th neuron and j th neuron, respectively.
System (1) is supplemented with the following boundary conditions and initial values:
for any and , where is the outer normal vector of ∂ Ω, are bounded and continuous, where . It is the Banach space of continuous functions which maps into with the topology of uniform convergence for the norm
Remark 1 Some famous NN models became a special case of system (1). For example, when and (, ), the special case of model (1) is the model which has been studied in [13–15]. When and , , , system (1) became NNs with distributed delays and reaction-diffusion terms [18, 22, 29].
Throughout this paper, we assume that the following conditions are made.
(A1) The functions , are piecewise-continuous of class on the closure of each continuity subinterval and satisfy
with some constants , , , for all .
(A2) The functions and are piecewise-continuous of class on the closure of each continuity subinterval and satisfy
(A3) The activation functions and the signal propagation functions are bounded and Lipschitz continuous, i.e., there exist positive constants , , , , and such that for all ,
(A4) The delay kernels (, ) are real-valued non-negative continuous functions that satisfy the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
There exist a positive μ such that
Let be the equilibrium point of system (1).
Definition 1 The equilibrium point of system (1) is said to be globally exponentially stable if we can find such that there exist constants and such that
for all .
Remark 2 It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1).
Lemma 1 [33]
Let Ω be a cube (), and let be a real-valued function belonging to which vanishes on the boundary ∂ Ω of Ω, i.e., . Then
3 Global exponential stability
Now we are in a position to investigate the global exponential stability of system (1). By constructing a suitable Lyapunov functional, we arrive at the following conclusion.
Theorem 1 Let (A1)-(A4) be in force. If there exist (), , , such that
and
in which , , , , , , and are Lipschitz constants, , , then the equilibrium point of system (1) is unique and globally exponentially stable.
Proof If (6) holds, we can always choose a positive number (may be very small) such that
and
where , .
Let us consider the functions
and
where , , .
From (8) and (A4), we derive , ; and are continuous for . Moreover, as and as . Thus there exist constants such that
and
where , .
By using , obviously, we get
and
where , .
Suppose is any solution of model (1). Rewrite model (1) as
Multiplying (11) by and integrating over Ω yield
According to Green’s formula and the Dirichlet boundary condition, we get
Moreover from Lemma 1, we have
From (11)-(15), (A2), (A3) and the Holder integral inequality, we obtain that
Multiplying both sides of (12) by , similarly, we also have
Choose a Lyapunov functional as follows:
Its upper Dini-derivative along the solution to system (1) can be calculated as follows:
From (18) and the Young inequality, we can conclude
From (6), we can conclude
Since
Noting that
Let
Clearly, .
It follows that
for any , where is a constant. This implies that the solution of (1) is globally exponentially stable. This completes the proof of Theorem 1. □
Remark 3 In this paper, the derived sufficient condition includes diffusion terms. Unfortunately, in the proof in the previous papers [21–24], a negative integral term with gradient is left out in their deduction. This leads to the fact that those criteria are irrelevant to the diffusion term. Obviously, Lyapunov functional to construct is more general and our results expand the model in [22, 29].
When and (, ), system (1) becomes the following BAM NNs with distributed delays and reaction-diffusion terms:
For (23), we get the following result.
Corollary 1 Let (A1)-(A4) be in force. If there exist (), , , such that
and
where , , , , , , and are Lipschitz constants. Then the equilibrium point of system (1) is unique and globally exponentially stable.
4 Periodic solutions
In this section, we consider the stability criterion for periodic oscillatory solutions of system (1), in which external input , , and , , are continuously periodic functions with period ω, that is,
By constructing a Poincaré mapping, the existence of a unique ω-periodic solution and its stability are readily established.
Theorem 2 Let (A1)-(A4) be in force. There exists only one ω-periodic solution of system (1), and all other solutions converge exponentially to it as if there exist constants (), , , (, ) such that
and
where and , , , , , and are Lipschitz constants in (A3).
Proof For any , we denote the solutions of system (1) through , and , as
and
respectively. Define
Clearly, for any , . Now, we define
Thus, we can obtain from system (1) that
We consider the following Lyapunov functional:
By a minor modification of the proof of Theorem 1, we can easily get
for ,in which is a constant. Now, we can choose a positive integer N such that
Defining a Poincaré mapping by
due to the periodicity of system, we have
Let , then from (26)-(29) we can derive that
which shows that is a contraction mapping. Therefore, there exists a unique fixed point , namely, .
Since , then is also a fixed point of . Because of the uniqueness of a fixed point of , then .
Let be the solution of system (1) through , then is also a solution of system (1). Clearly,
for . Hence for .
This shows that is exactly one ω-periodic solution of system (1), and it is easy to see that all other solutions of system (1) converge exponentially to it as . The proof is completed. □
5 Illustration example
In this section, a numerical example is given to illustrate the effectiveness of the obtained results.
Example 1 Consider the following system on :
where , . , , , , , . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . By simple calculation with , and , we have
and
that is, (6) holds.
The simulation results are shown in Figures 1-8. When , the states surfaces of are shown in Figures 1-2, while , the states surfaces of are shown in Figures 3-4. When , the states surfaces of are shown in Figures 5-6, while , the states surfaces of are shown in Figures 7-8, which illustrates that the system states in (30) converge to equilibrium solution. Therefore, it follows from Theorem 1 and the simulation study that (30) has one unique equilibrium solution which is globally exponentially stable.
Remark 4 Since , the conditions of Corollary 3.2 in [22] and , under the conditions of Example 1, the conditions of Theorem 1 in [29] are not satisfied. However, by (31)-(34) and Theorem 1, we can derive that (30) has one unique equilibrium solution which is globally exponentially stable.
6 Conclusions
In this paper, by employing suitable Lyapunov functionals, Young’s inequality and Hölder’s inequality techniques, global exponential stability criteria of the equilibrium point and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions have been derived, respectively. The derived criteria contain and extend some previous NNs in the literature. Hence, our results have an important significance in design as well as in applications of periodic oscillatory NNs with mixed time delays. An example has been given to show the effectiveness of the obtained results.
References
Kwon OM, Park JH, Lee SM, Cha EJ: A new augmented Lyapunov-Krasovskii functional approach to exponential passivity for neural networks with time-varying delays. Appl. Math. Comput. 2011, 217(24):10231-10238. 10.1016/j.amc.2011.05.021
Ji DH, Koo JH, Won SC, Lee SM, Park JH: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput. 2011, 217(13):6168-6175. 10.1016/j.amc.2010.12.100
Lee SM, Kwon OM, Park JH: A novel delay-dependent criterion for delayed neural networks of neutral type. Phys. Lett. A 2010, 374(17-18):1843-1848. 10.1016/j.physleta.2010.02.043
Cao J, Wang J: Global exponential stability and periodicity of recurrent neural networks with time delays. IEEE Trans. Circuits Syst. I, Regul. Pap. 2005, 52(5):920-931.
Huang C, Cao J: Convergence dynamics of stochastic Cohen-Grossberg neural networks with unbounded distributed delays. IEEE Trans. Neural Netw. 2011, 22(4):561-572.
Ensari T, Arik S: Global stability of a class of neural networks with time varying delay. IEEE Trans. Circuits Syst. II, Express Briefs 2005, 52(3):126-130.
Rakkiyappan R, Balasubramaniam P: Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays. Appl. Math. Comput. 2008, 198(2):526-533. 10.1016/j.amc.2007.08.053
Kosko B: Bi-directional associative memories. IEEE Trans. Syst. Man Cybern. 1988, 18(1):49-60. 10.1109/21.87054
Park JH, Kwon OM: Delay-dependent stability criterion for bidirectional associative memory neural networks with interval time-varying delays. Mod. Phys. Lett. B 2009, 23(1):35-46. 10.1142/S0217984909017807
Park JH, Park CH, Kwon OM, Lee SM: New stability criterion for bidirectional associative memory neural networks of neutral-type. Appl. Math. Comput. 2008, 199(2):716-722. 10.1016/j.amc.2007.10.032
Park JH, Kwon OM: On improved delay-dependent criterion for global stability of bidirectional associative memory neural networks with time-varying delays. Appl. Math. Comput. 2008, 199(2):435-446. 10.1016/j.amc.2007.10.001
Cao J, Wang L: Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans. Neural Netw. 2002, 13(2):457-463. 10.1109/72.991431
Park J, Lee SM, Kwon OM: On exponential stability of bidirectional associative memory neural networks with time-varying delays. Chaos Solitons Fractals 2009, 39(3):1083-1091. 10.1016/j.chaos.2007.05.003
Wu R: Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays. Nonlinear Anal., Real World Appl. 2010, 11(1):562-573. 10.1016/j.nonrwa.2009.02.003
Liu X, Martin R, Wu M: Global exponential stability of bidirectional associative memory neural networks with time delays. IEEE Trans. Neural Netw. 2008, 19(2):397-407.
Zhang W, Li J: Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions. Bound. Value Probl. 2012., 2012: Article ID 2. doi:10.1186/1687-2770-2012-2
Zhang W, Li J, Shi N: Stability analysis for stochastic Markovian jump reaction-diffusion neural networks with partially known transition probabilities and mixed time delays. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 524187. doi:10.1155/2012/524187
Song Q, Zhao Z, Li YM: Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms. Phys. Lett. A 2005, 335(2-3):213-225. 10.1016/j.physleta.2004.12.007
Zhang W, Li J: Global exponential stability of reaction-diffusion neural networks with discrete and distributed time-varying delays. Chin. Phys. B 2011., 20(3): Article ID 030701
Pan J, Zhan Y: On periodic solutions to a class of non-autonomously delayed reaction-diffusion neural networks. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(1):414-422. 10.1016/j.cnsns.2010.02.022
Song Q, Cao J: Global exponential stability and existence of periodic colutions in BAM with delays and reaction-diffusion terms. Chaos Solitons Fractals 2005, 23(2):421-430. 10.1016/j.chaos.2004.04.011
Cui B, Lou X: Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solitons Fractals 2006, 27(5):1347-1354. 10.1016/j.chaos.2005.04.112
Zhao H, Wang G: Existence of periodic oscillatory solution of reaction-diffusion neural networks with delays. Phys. Lett. A 2005, 343(5):372-383. 10.1016/j.physleta.2005.05.098
Song Q, Cao J, Zhao Z: Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal., Real World Appl. 2006, 7(1):65-80. 10.1016/j.nonrwa.2005.01.004
Wang Z, Zhang H: Global asymptotic stability of reaction-diffusion Cohen-Grossberg neural network with continuously distributed delays. IEEE Trans. Neural Netw. 2010, 21(1):39-49.
Zhang X, Wu S, Li K: Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1524-1532. 10.1016/j.cnsns.2010.06.023
Wang L, Zhang R, Wang Y: Global exponential stability of reaction-diffusion cellular neural networks with S -type distributed time delays. Nonlinear Anal., Real World Appl. 2009, 10(2):1101-1113. 10.1016/j.nonrwa.2007.12.002
Zhu Q, Li X, Yang X: Exponential stability for stochastic reaction-diffusion BAM neural networks with time-varying and distributed delays. Appl. Math. Comput. 2011, 217(13):6078-6091. 10.1016/j.amc.2010.12.077
Lou X, Cui B, Wu W: On global exponential stability and existence of periodic solutions for BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solitons Fractals 2008, 36(4):1044-1054. 10.1016/j.chaos.2006.08.005
Zhang W, Li J, Chen M: Dynamical behaviors of impulsive stochastic reaction-diffusion neural networks with mixed time delays. Abstr. Appl. Anal. 2012., 2012: Article ID 236562. doi:10.1155/2012/236562
Wang Z, Zhang H, Li P: An LMI approach to stability analysis of reaction-diffusion Cohen-Grossberg neural networks concerning Dirichlet boundary conditions and distributed delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 2010, 40(6):1596-1606.
Lu J: Robust global exponential stability for interval reaction-diffusion Hopfield neural networks with distributed delays. IEEE Trans. Circuits Syst. II, Express Briefs 2007, 54(12):1115-1119.
Lu J, Lu L: Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos Solitons Fractals 2009, 39(4):1538-1549. 10.1016/j.chaos.2007.06.040
Wang J, Lu J: Global exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms. Chaos Solitons Fractals 2008, 38(3):878-885. 10.1016/j.chaos.2007.01.032
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work is partially supported by the National Natural Science Foundation of China under Grant No. 60974139, the Special Research Project in Shaanxi Province Department of Education (2013JK0578) and Doctor Introduced project of Xianyang Normal University under Grant No. 12XSYK008.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WZ designed and performed all the steps of proof in this research and also wrote the paper. JL and MC participated in the design of the study and suggested many good ideas that made this paper possible and helped to draft the first manuscript. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhang, W., Li, J. & Chen, M. Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. Bound Value Probl 2013, 105 (2013). https://doi.org/10.1186/1687-2770-2013-105
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2013-105