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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)
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.
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  and  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 , 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 . 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
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:
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 
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
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
where , .
Let us consider the functions
where , , .
From (8) and (A4), we derive , ; and are continuous for . Moreover, as and as . Thus there exist constants such that
where , .
By using , obviously, we get
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
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
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
where and , , , , , and are Lipschitz constants in (A3).
Proof For any , we denote the solutions of system (1) through , and , as
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