Abstract
In this article, a delaydifferential equation modeling a bidirectional associative memory (BAM) neural networks (NNs) with reactiondiffusion terms is investigated. A feedback control law is derived to achieve the state global exponential synchronization of two identical BAM NNs with reactiondiffusion terms by constructing a suitable Lyapunov functional, using the driveresponse approach and some inequality technique. A novel global exponential synchronization criterion is given in terms of inequalities, which can be checked easily. A numerical example is provided to demonstrate the effectiveness of the proposed results.
Keywords:
neural networks; reactiondiffusion; delays; global exponential synchronization; Lyapunov functional1. Introduction
Aihara et al. [1] firstly proposed chaotic neural network (NN) models to simulate the chaotic behavior of biological neurons. Consequently, chaotic NNs have drawn considerable attention and have successfully been applied in combinational optimization, secure communication, information science, and so on [24]. Since NNs related to bidirectional associative memory (BAM) have been proposed by Kosko [5], the BAM NNs have been one of the most interesting research topics and extensively studied because of its potential applications in pattern recognition, etc. Hence, the study of the stability and periodic oscillatory solution of BAM with delays has raised considerable interest in recent years, see for example [612] and the references cited therein.
Strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. Therefore, we must consider that the activations vary in space as well as in time. In [1327], the authors have considered various dynamical behaviors such as the stability, periodic oscillation, and synchronization of NNs with diffusion terms, which are expressed by partial differential equations. For instance, the authors of [16] discuss the impulsive control and synchronization for a class of delayed reactiondiffusion NNs with the Dirichlet boundary conditions in terms of pnorm. In [25], the synchronization scheme is discussed for a class of delayed NNs with reactiondiffusion terms. In [26], an adaptive synchronization controller is derived to achieve the exponential synchronization of the driveresponse structure of NNs with reactiondiffusion terms. Meanwhile, although the models of delayed feedback with discrete delays are good approximation in simple circuits consisting of a small number of cells, NNs usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there is a distribution of conduction velocities along these pathways and a distribution of propagation delays. Therefore, the models with discrete and continuously distributed delays are more appropriate.
To the best of the authors' knowledge, global exponential synchronization is seldom reported for the class of delayed BAM NNs with reactiondiffusion terms. In the theory of partial differential equations, Poincaré integral inequality is often utilized in the deduction of diffusion operator [28]. In this article, the problem of global exponential synchronization is investigated for the class of BAM NNs with timevarying and distributed delays and reactiondiffusion terms by using Poincaré integral inequality, Young inequality technique, and Lyapunov method, which are very important in theories and applications and also are a very challenging problem. Several sufficient conditions are in the form of a few algebraic inequalities, which are very convenient to verify.
2. Model description and preliminaries
In this article, a class of delayed BAM NNs with reactiondiffusion terms is described as follows
where x = (x_{1}, x_{2 },..,x_{l})^{T }∈ Ω ⊂ ℝ^{l}, Ω is a compact set with smooth boundary ∂Ω and mesΩ > 0 in space ℝ^{l}; u = (u_{1},u_{2},...,u_{m})^{T }∈ ℝ^{m}, (v_{1},v_{2},...,v_{n})^{T }∈ ℝ^{n}, u_{i}(t,x) and v_{j}(t,x) and represent the states of the ith neurons and the jth neurons at time t and in space x, respectively. , and are known constants denoting the synaptic connection strengths between the neurons, respectively; f_{i }and g_{i }denote the activation functions of the neurons and the signal propagation functions, respectively; I_{i }and J_{i }denote the external inputs on the ith and jth neurons, respectively; p_{i }and q_{j }are differentiable real functions with positive derivatives defining the neuron charging time, respectively; τ_{ij}(t) and θ_{ji}(t) represent continuous timevarying discrete delays, respectively; D_{ik }≥ 0 and stand for the transmission diffusion coefficient along the ith and jth neurons, respectively. i = 1, 2, ..., m, k = 1, 2, l and j = 1, 2,..., n.
System (1) is supplemented with the following boundary conditions and initial values
for any i = 1,2,..., m and j = 1,2,..., n where is the outer normal vector of ∂Ω, are bounded and continuous, where It is the Banach space of continuous functions which map into ℝ^{m+n }with the topology of uniform converge for the norm
Throughout this article, we assume that the following conditions are made.
(A1) The functions τ_{ij}(t), θ_{ji}(t) are piecewisecontinuous of class C^{1 }on the closure of each continuity subinterval and satisfy
with some constants τ_{ij }≥ 0, θ_{ji }≥ 0, τ > 0, θ > 0, for all t ≥ 0.
(A2) The functions p_{i }(·)and q_{j}(·) are piecewisecontinuous of class C^{1 }on the closure of each continuity subinterval and satisfy
(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist positive constants and such that for all η_{1}, η_{2 }∈ ℝ
(A4) The delay kernels (i = 1, 2,...,m, j = 1, 2,...,n) are realvalued nonnegative continuous functions that satisfy the following conditions
(iii)There exist a positive μ such that
We consider system (1) as the drive system. The response system is described by the following equations
where σ_{i }(t,x) and ϑ_{j}(t,x) denote the external control inputs that will be appropriately designed for a certain control objective. We denote , and ϑ(t,x) = (ϑ_{1}(t,x),..., ϑ_{n}(t,x))^{T}.
The boundary and initial conditions of system (4) are
and
Definition 1. Driveresponse systems (1) and (4) are said to be globally exponentially synchronized, if there are control inputs σ(t,x), ϑ(t,x), and r ≥ 2, further there exist constants α > 0 and β ≥ 1 such that
in which , , and (u(t,x), v(t,x)) and are the solutions of driveresponse systems (1) and (4) satisfying boundary conditions and initial conditions (2), (3) and (5), (6), respectively.
Lemma 1. [21] (Poincaré integral inequality). Let Ω be a bounded domain of ℝ^{m }with a smooth boundary ∂Ω of class C^{2 }by Ω. u(x) is a realvalued function belonging to and Then
which λ_{1 }is the lowest positive eigenvalue of the Neumann boundary problem
3. Main results
From the definition of synchronization, we can define the synchronization error signal , e(t,x) = (e_{1}(t,x),...,e_{m}(t,x))^{T}, and ω(t,x) = (ω_{1}(t,x),..., ω_{n}(t,x))^{T }. Thus, error dynamics between systems (1) and (4) can be expressed by
The control inputs strategy with state feedback are designed as follows:
that is,
where and are the controller gain matrices.
The global exponential synchronization of systems (1) and (4) can be solved if the controller matrices μ and ρ are suitably designed. We have the following result.
Theorem 1. Under the assumptions (A1)(A4), driveresponse systems (1) and (4) are in global exponential synchronization, if there exist w_{i }> 0(i = 1,2,..., n+m), r ≥ 2, γ_{ij }> 0, β_{ji }> 0 such that the controller gain matrices μ and ρ in (9) satisfy
and
in which i = 1, 2, ..., m, j = 1, 2,..., n, and are Lipschitz constants, λ_{1 }is the lowest positive eigenvalue of problem (7).
Proof. If (10) holds, we can always choose a positive number δ > 0 (may be very small) such that
and
where i = 1, 2,..., m, j = 1, 2,..., n.
Let us consider functions
and
where i = 1, 2, ..., m, j = 1, 2, ..., n.
From (12) and (A4), we derive
F_{i}(0) < δ < 0, G_{j}(0) < δ < 0; and are continuous for Moreover, as and as , thus there exist constants ε_{i},ν_{j }∈ [0, +∞) such that
and
and
Multiplying both sides of the first equation of (8) by e_{i }(t,x) and integrating over Ω yields
It is easy to calculate by the Neumann boundary conditions (2) that
Moreover, from Lemma 1, we can derive
From (13)(17), (A2), and (A3), we obtain that
Multiplying both sides of the second equation of (8) by ω_{j }(t,x), similarly, we also have
Consider the following Lyapunov functional
Its upper Diniderivative along the solution to system (8) can be calculated as
From (21) and Young inequality, we can conclude
From (10), we can conclude
Since
Noting that
Let
Clearly, β ≥ 1.
It follows that
for any t ≥ 0 where β ≥ 1 is a constant. This implies that driveresponse systems (1) and (4) are globally exponentially synchronized. This completes the proof of Theorem 1.
Remark 1. In Theorem 1, the Poincaré integral inequality is used firstly. This is a very important step. Thus, the derived sufficient condition includes diffusion terms. We note that, in the proof in the previous articles [2426], a negative integral term with gradient is left out in their deduction. This leads to those criteria that are irrelevant to the diffusion term. Therefore, Theorem 1 is essentially new and more effectiveness than those obtained.
Remark 2. It is noted that we construct a novel Lyapunov functional here as defined in (20) since the considered model contains timevarying and distributed delays and reactiondiffusion terms. We can see that the results and research method obtained in this article can also be extended to many other types of NNs with reactiondiffusion terms, e.g., the cellular NNs, cohengrossberg NNs, etc.
Remark 3. In our result, the effects of the reactiondiffusion terms on the synchronization are considered. Furthermore, we note a very interesting fact, that is, as long as diffusion coefficients in the system are large enough, then condition (10) can always satisfy. This shows that a large enough diffusion coefficient may always make the system globally exponentially synchronous.
Some famous NN models are a special case of model (1). In system (1), ignoring the role of reactiondiffusion, then system (1) will degenerate into the following delayed BAM NNs
and the corresponding response system (4) will become the following form
Define the synchronization error signal , then the error dynamics between systems (27) and (28) can be expressed by
We consider the following control inputs strategy
As a consequence of Theorem 1, we have the following result:
Corollary 1. Under the assumptions (A1)(A4), driveresponse systems (27) and (28) are in global exponential synchronization, if there exist w_{i }> 0 (i = 1, 2,...,n+m), r ≥ 2, γ_{ij }> 0, β_{ji }> 0 such that the controller gain matrices μ and ρ in (9) satisfy
and
in which i = 1, 2,..., m, j = 1, 2,..., n, and are Lipschitz constants.
4. Illustration example
To illustrate the effectiveness of our criterion, we give the following example.
Example 1. Consider the following system on
and
By simple calculation with w_{1 }= w_{2 }= w_{3 }= w_{4 }= 1, β_{11 }= β_{12 }= β_{21 }= β_{22 }= 1,
and γ_{11 }= γ_{12 }= γ_{21 }= γ_{22 }= 1, we get
and
Hence, it follows from Theorem 1 that (32) and (33) are globally exponentially synchronized.
5. Conclusions
In this article, global exponential synchronization has been considered for a class of BAM NNs with timevarying and distributed delays and reactiondiffusion terms. We have established a new sufficient condition which includes the diffusion coefficients by constructing the suitable Lyapunov functional, introducing many real parameters and applying inequality techniques. From condition (10) in Theorem 1, we see that diffusion coefficients directly affect the synchronization behavior of the delayed BAM NNs with reactiondiffusion terms. In comparison with previous literature, diffusion effects are taken into account in our models. A numerical example has been given to show the effectiveness of the obtained results.
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 participated in the design of the study and suggest many good ideas that made this paper possible and helped to draft the first manuscript. All authors read and approved the final manuscript.
Acknowledgements
This study was partially supported by the National Natural Science Foundation of China under Grant No. 60974139 and partially supported by the Fundamental Research Funds for the Central Universities under Grant No. 72103676, the Natural Science Foundation of Shannxi Province, China under Grant No. 2010JQ1013, and the Special research projects in Shannxi Province Department of Education under Grant No. 2010JK896.
References

Aihara, K, Takabe, T, Toyoda, M: Chaotic neural networks. Phys Lett A. 144, 333–340 (1990). Publisher Full Text

Kwok, T, Smith, K: Experimental analysis of chaotic neural network models for combinatorial optimization under a unifying framework. Neural Netw. 13, 731–744 (2000). PubMed Abstract  Publisher Full Text

Yu, W, Cao, J: Cryptography based on delayed chaotic neural networks. Phys Lett A. 356, 333–338 (2006). Publisher Full Text

Cheng, C, Liao, T, Yan, J, Wang, CH: Exponential synchronization of a class of neural networks with timevarying delays. IEEE Trans Syst Man Cybern B. 36, 209–215 (2006)

Kosko, B: Bidirectional associative memories. IEEE Trans Syst Man Cybern. 18(1), 49–60 (1988). Publisher Full Text

Cao, J, Wang, L: Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans Neural Netw. 13(2), 457–463 (2002). PubMed Abstract  Publisher Full Text

Liu, X, Martin, R, Wu, M: Global exponential stability of bidirectional associative memory neural networks with time delays. IEEE Trans Neural Netw. 19(2), 397–407 (2008). PubMed Abstract  Publisher Full Text

Lou, X, Cui, B: Stochastic exponential stability for Markovian jumping BAM neural networks with timevarying delays. IEEE Trans Syst Man Cybern. 37, 713–719 (2007)

Park, JH: A novel criterion for global asymptotic stability of BAM neural networks with time delays. Chaos Solitons Fractals. 29(2), 446–453 (2006). Publisher Full Text

Park, JH, Kwon, OM: Delaydependent stability criterion for bidirectional associative memory neural networks with interval timevarying delays. Mod Phys Lett B. 23(1), 35–46 (2009). Publisher Full Text

Park, JH, Park, CH, Kwon, OM, Lee, SM: A new stability criterion for bidirectional associative memory neural networks of neutraltype. Appl Math Comput. 199(2), 716–722 (2008). Publisher Full Text

Park, JH, Kwon, OM: On improved delaydependent criterion for global stability of bidirectional associative memory neural networks with timevarying delays. Appl Math Comput. 199(2), 435–446 (2008). Publisher Full Text

Zhu, QX, Cao, J: Exponential stability analysis of stochastic reactiondiffusion CohenGrossberg neural networks with mixed delays. Neurocomputing. 74, 3084–3091 (2011). Publisher Full Text

Song, Q, Cao, J: Global exponential stability and existence of periodic solutions in BAM with delays and reactiondiffusion terms. Chaos Solitons Fractals. 23(2), 421–430 (2005). Publisher Full Text

Cui, B, Lou, X: Global asymptotic stability of BAM neural networks with distributed delays and reactiondiffusion terms. Chaos Solitons Fractals. 27(5), 1347–1354 (2006). Publisher Full Text

Hu, C, Jiang, HJ, Teng, ZD: Impulsive control and synchronization for delayed neural networks with reactiondiffusion terms. IEEE Trans Neural Netw. 21(1), 67–81 (2010). PubMed Abstract  Publisher Full Text

Wang, Z, Zhang, H: Global asymptotic stability of reactiondiffusion CohenGrossberg neural network with continuously distributed delays. IEEE Trans Neural Netw. 21(1), 39–49 (2010). PubMed Abstract  Publisher Full Text

Wang, L, Zhang, R, Wang, Y: Global exponential stability of reactiondiffusion cellular neural networks with Stype distributed time delays. Nonlinear Anal Real World Appl. 10(2), 1101–1113 (2009). Publisher Full Text

Balasubramaniam, P, Vidhya, C: Global asymptotic stability of stochastic BAM neural networks with distributed delays and reactiondiffusion terms. J Comput Appl Math. 234, 3458–3466 (2010). Publisher Full Text

Lu, J, Lu, L: Global exponential stability and periodicity of reactiondiffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. Chaos Solitons Fractals. 39(4), 1538–1549 (2009). Publisher Full Text

Song, Q, Zhao, Z, Li, YM: Global exponential stability of BAM neural networks with distributed delays and reactiondiffusion terms. Phys Lett A. 335(23), 213–225 (2005). Publisher Full Text

Zhang, W, Li, J: Global exponential stability of reactiondiffusion neural networks with discrete and distributed timevarying delays. Chin Phys B. 20(3), 030701 (2011). Publisher Full Text

Liao, XX, Yang, SZ, Cheng, SJ, Fu, YL: Stability of generalized networks with reactiondiffusion terms. Sci China (Series F). 44, 87–94 (2001). Publisher Full Text

Lou, X, Cui, B: Asymptotic synchronization of a class of neural networks with reactiondiffusion terms and timevarying delays. Comput Math Appl. 52, 897–904 (2006). Publisher Full Text

Wang, Y, Cao, J: Synchronization of a class of delayed neural networks with reactiondiffusion terms. Phys. Lett A. 369, 201–211 (2007). Publisher Full Text

Sheng, L, Yang, H, Lou, X: Adaptive exponential synchronization of delayed neural networks with reactiondiffusion terms. Chaos Solitons Fractals. 40, 930–939 (2009). Publisher Full Text

Wang, K, Teng, Z, Jiang, H: Global exponential synchronization in delayed reactiondiffusion cellular neural networks with the Dirichlet boundary conditions. Math Comput Model. 52, 12–24 (2010). Publisher Full Text

Temam, R: Infinite Dimensional Dynamical Systems in Mechanics and Physics. SpringerVerlag, New York (1998)