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 2 3 4 . 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 6 7 8 9 10 11 12 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 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 , 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 reaction-diffusion NNs with the Dirichlet boundary conditions in terms of p-norm. In 25 , the synchronization scheme is discussed for a class of delayed NNs with reaction-diffusion terms. In 26 , an adaptive synchronization controller is derived to achieve the exponential synchronization of the drive-response structure of NNs with reaction-diffusion 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 reaction-diffusion 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 time-varying and distributed delays and reaction-diffusion 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.

In this article, a class of delayed BAM NNs with reaction-diffusion terms is described as follows

∂ u i ∂ t = ∑ k = 1 l ∂ ∂ x k D i k ∂ u i ∂ x k - p i u i t , x + ∑ j = 1 n b j i f j v j t , x + ∑ j = 1 n b ̃ j i f j v j t - θ j i t , x + ∑ j = 1 n b ̄ j i ∫ - ∞ t k j i t - s f j v j s , x d s + I i t , ∂ v j ∂ t = ∑ k = 1 l ∂ ∂ x k D j k * ∂ v j ∂ x k - q j v j t , x + ∑ i = 1 m d i j g i u i t , x + ∑ i = 1 m d ̃ i j g i u i t - τ i j t , x + ∑ i = 1 m d ̄ i j ∫ - ∞ t k ̄ i j t - s g i u i s , x d s + J j t ,

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. b j i , b ̃ j i , b ̄ j i , d i j , d ̄ i j , and d ̃ i j 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 time-varying discrete delays, respectively; D_ik ≥ 0 and D j k * ≥ 0 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

∂ u i ∂ n ̄ : = ∂ u i ∂ x 1 , ∂ u i ∂ x 2 , . . . , ∂ u i ∂ x l T = 0 ,   ∂ v j ∂ n ̄ : = ∂ v j ∂ x 1 , ∂ v j ∂ x 2 , . . . , ∂ v j ∂ x l T = 0 , t ≥ 0 , x ∈ ∂ Ω ,

u i s , x = φ u i s , x , v j s , x = φ v j s , x ,   s , x ∈ - ∞ , 0 × Ω .

for any i = 1,2,..., m and j = 1,2,..., n where n ̄ is the outer normal vector of ∂Ω, φ = φ u φ v = φ u 1 , . . . , φ u m , φ v 1 , . . . , φ v n T ∈ C are bounded and continuous, where C = φ | φ = φ u φ v , φ : - ∞ , 0 × ℝ m - ∞ , 0 × ℝ n → ℝ m + n . It is the Banach space of continuous functions which map - ∞ , 0 - ∞ , 0 into ℝ^m+n with the topology of uniform converge for the norm

φ = φ u φ v = sup - ∞ ≤ s ≤ 0 ∫ Ω ∑ i = 1 m φ u i r d x + sup - ∞ ≤ s ≤ 0 ∫ Ω ∑ j = 1 n φ v j r d x , r ≥ 2 .

Throughout this article, we assume that the following conditions are made.

(A1) The functions τ_ij (t), θ_ji (t) are piecewise-continuous of class C ¹ on the closure of each continuity subinterval and satisfy

0 ≤ τ i j t ≤ τ i j , 0 ≤ θ j i t ≤ θ j i , τ ˙ i j t ≤ μ τ < 1 , θ ˙ j i t ≤ μ θ < 1 , τ = max 1 ≤ i ≤ m , 1 ≤ j ≤ n τ i j , θ = max 1 ≤ i ≤ m , 1 ≤ j ≤ n θ j i ,

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 piecewise-continuous of class C ¹ on the closure of each continuity subinterval and satisfy

a i = inf ζ ∈ ℝ p i ′ ζ > 0 , p i 0 = 0 , c j = inf ζ ∈ ℝ q j ′ ζ > 0 , q j 0 = 0 .

(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist positive constants L j f and L i g such that for all η ₁, η ₂ ∈ ℝ

f j η 1 - f j η 2 ≤ L j f η 1 - η 2 ,   g i η 1 - g i η 2 ≤ L i g η 1 - η 2 .

(A4) The delay kernels K j i s , K ̄ i j s : 0 , ∞ → 0 , ∞ , (i = 1, 2,...,m, j = 1, 2,...,n) are real-valued non-negative continuous functions that satisfy the following conditions

(i) ∫ 0 + ∞ K j i s d s = 1 , ∫ 0 + ∞ K ̄ j i s d s = 1 ,

(ii) ∫ 0 + ∞ s K j i s d s < ∞ , ∫ 0 + ∞ s K ̄ i j s d s < ∞ ,

(iii)There exist a positive μ such that

∫ 0 + ∞ s e μ s K j i s d s < ∞ , ∫ 0 + ∞ s e μ s K ̄ i j s d s < ∞ .

We consider system (1) as the drive system. The response system is described by the following equations

∂ ũ i t , x ∂ t = ∑ k = 1 l ∂ ∂ x k D i k ∂ ũ i t , x ∂ x k - p i ũ i t , x + ∑ j = 1 n b j i f j ṽ j t , x + ∑ j = 1 n b ̃ j i f j ṽ j t - θ j i t , x + ∑ j = 1 n b ̄ j i ∫ - ∞ t k j i t - s f j ṽ j s , x d s + I i t + σ i t , x , ∂ ṽ j t , x ∂ t = ∑ k = 1 l ∂ ∂ x k D j k * ∂ ṽ j t , x ∂ x k - q j ṽ j t , x + ∑ i = 1 m d i j g i ũ i t , x + ∑ i = 1 m d ̃ i j g i ũ i t - τ i j t , x + ∑ i = 1 m d ̄ i j ∫ - ∞ t k ̄ i j t - s g i ũ i s , x d s + J j t + ϑ j t , x ,

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 ũ t , x = ũ 1 t , x , . . . , ũ m t , x T , ṽ t , x = ṽ 1 t , x , . . . , ṽ n t , x T , σ t , x = σ 1 t , x , . . . , σ m t , x T and ϑ(t,x) = (ϑ₁(t,x),..., ϑ_n (t,x)) ^T .

The boundary and initial conditions of system (4) are

∂ ũ i ∂ n ̄ : = ∂ ũ i ∂ x 1 , ∂ ũ i ∂ x 2 , . . . , ∂ ũ i ∂ x l T = 0 ,   ∂ ṽ j ∂ n ̄ : = ∂ ṽ j ∂ x 1 , ∂ ṽ j ∂ x 2 , . . . , ∂ ṽ j ∂ x l T = 0   t ≥ 0 , x ∈ ∂ Ω ,

and

ũ i s , x = ψ u i s , x , ṽ j s , x = ψ v j s , x ,   s , x ∈ - ∞ , 0 × Ω ,

where ψ = ψ ũ ψ ṽ = ψ ũ 1 , . . . , ψ ũ m , ψ ṽ 1 , . . . , ψ ṽ n T ∈ C .

Definition 1. Drive-response 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

u t , x - ũ t , x + v t , x - ṽ t , x

≤ β e - 2 α t φ u s , x - ψ ũ s , x + φ v s , x - ψ ṽ s , x , for all t ≥ 0,

in which u t , x - ũ t , x = ∫ Ω ∑ i = 1 m u i t , x - ũ i t , x r d x , v t , x - ṽ t , x = ∫ Ω ∑ j = 1 n v j t , x - ṽ j t , x r d x , r ≥ 2 , and (u(t,x), v(t,x)) and ũ t , x , ṽ t , x are the solutions of drive-response 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 real-valued function belonging to H 0 1 Ω and ∂ u x ∂ n ̄ | ∂ Ω = 0 . Then

∫ Ω u x 2 d x ≤ 1 λ 1 ∫ Ω ∇ u x 2 d x ,

which λ ₁ is the lowest positive eigenvalue of the Neumann boundary problem

- Δ φ x = λ 1 φ x , x ∈ Ω , ∂ u x ∂ n ̄ | ∂ Ω = 0 , x ∈ ∂ Ω .

From the definition of synchronization, we can define the synchronization error signal e i t , x = u i t , x - ũ i t , x , ω j t , x = v j t , x - ṽ j t , x , 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

∂ e i t , x ∂ t = ∑ k = 1 l ∂ ∂ x k D i k ∂ e i t , x ∂ x k - p ̃ i e i t , x + ∑ j = 1 n b j i f ̃ j ω j t , x + ∑ j = 1 n b ̃ j i f ̃ j ω j t - θ j i t , x + ∑ j = 1 n b ̄ j i ∫ - ∞ t k j i t - s f ̃ j ω j s , x d s - σ i t , x , ∂ ω j t , x ∂ t = ∑ k = 1 l ∂ ∂ x k D j k * ∂ ω j t , x ∂ x k - q ̃ j ω j t , x + ∑ i = 1 m d i j g ̃ i e i t , x + ∑ i = 1 m d ̃ i j g ̃ i e i t - τ i j t , x + ∑ i = 1 m d ̄ i j ∫ - ∞ t k ̄ i j t - s g ̃ i e i s , x d s - ϑ j t , x ,

where f ̃ j ω j t , x = f j v j t , x - f j ṽ j t , x , g ̃ i e i t , x = g i u i t , x - g i ũ i t , x , p ̃ i e i t , x = p i u i t , x - p i ũ i t , x , q ̃ j ω j t , x = q j v j t , x - q j ṽ j t , x .

The control inputs strategy with state feedback are designed as follows:

σ i t , x = ∑ k = 1 m μ i k e k t , x , ϑ j t , x = ∑ k = 1 n ρ j k ω k t , x ,   i = 1 , 2 , . . . , m , j = 1 , 2 , . . . , n .

that is,

σ t , x = μ e t , x , ϑ t , x = ρ ω t , x ,

where μ = μ i k m × m and ρ = ρ j k n × n 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), drive-response 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

w i - r n a i r - 1 D i λ 1 - r n a i r - r n μ i i a i r - 1 + 2 r - 1 ∑ j = 1 n a i r + r - 1 ∑ j = 1 n a i r β j i - r r - 1 + r - 1 ∑ k = 1 , i ≠ k m a i r + ∑ j = 1 n w m + j m r d i j r L i g r + d ̃ i j r e τ 1 - μ τ L i g r + d ̄ i j r γ i j r L i g r + n r ∑ k = 1 , i ≠ k m μ k i r w k < 0

and

w m + j - r m c j r - 1 D j * λ 1 - r m c j r - r m ρ j j c j r - 1 + 2 r - 1 ∑ i = 1 m c j r + r - 1 ∑ k = 1 , j ≠ k n c j r + r - 1 ∑ i = 1 m c j r γ i j - r r - 1 + ∑ i = 1 m w i n r b j i r L j f r + b ̃ j i r e θ 1 - μ θ L j f r + b ̄ j i r β j i r L j f r + m r ∑ k = 1 , j ≠ k n ρ k j r w k + m < 0 ,

in which i = 1, 2, ..., m, j = 1, 2,..., n, L j f and L i g are Lipschitz constants, D i = min 1 ≤ k ≤ l D i k , D j * = min 1 ≤ k ≤ l D j k * , 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

w i - r n a i r - 1 D i λ 1 - r n a i r - r n μ i i a i r - 1 + 2 r - 1 ∑ j = 1 n a i r + r - 1 ∑ j = 1 n a i r β j i - r r - 1 + r - 1 ∑ k = 1 , i ≠ k m a i r + ∑ j = 1 n w m + j m r d i j r L i g r + d ̃ i j r e τ 1 - μ τ L i g r + d ̄ i j r γ i j r L i g r + n r ∑ k = 1 , i ≠ k m μ k i r w k + δ < 0

and

w m + j - r m c j r - 1 D j * λ 1 - r m c j r - r m ρ j j c j r - 1 + 2 r - 1 ∑ i = 1 m c j r + r - 1 ∑ k = 1 , j ≠ k n c j r + r - 1 ∑ i = 1 m c j r γ i j - r r - 1 + ∑ i = 1 m w i n r b j i r L j f r + b ̃ j i r e θ 1 - μ θ L j f r + b ̄ j i r β j i r L j f r + m r ∑ k = 1 , j ≠ k n ρ k j r w k + m + δ < 0 ,

where i = 1, 2,..., m, j = 1, 2,..., n.

Let us consider functions

F i x i * = w i - r n a i r - 1 D i λ 1 - r n a i r - r n μ i i a i r - 1 + 2 r - 1 ∑ j = 1 n a i r + r - 1 ∑ k = 1 , i ≠ k m a i r + r - 1 ∑ j = 1 n a i r β j i - r r - 1 ∫ 0 + ∞ k j i s d s + 2 x i * n a i r - 1 + ∑ j = 1 n w m + j m r d i j r L i g r + d ̃ i j r e τ 1 - μ τ L i g r + d ̄ i j r γ i j r L i g r ∫ 0 + ∞ e 2 x i * s k ̄ i j s d s + n r ∑ k = 1 , i ≠ k m μ k i r w k

and

G j y j * = w m + j - r m c j r - 1 D j * λ 1 - r m c j r - r m ρ j j c j r - 1 + 2 r - 1 ∑ i = 1 m c j r + r - 1 ∑ k = 1 , j ≠ k n c j r + r - 1 ∑ i = 1 m c j r γ i j - r r - 1 ∫ 0 + ∞ k ̄ i j s d s + 2 y j * m c j r - 1 + ∑ i = 1 m w i n r b j i r L j f r + b ̃ j i r e θ 1 - μ θ L j f r + b ̄ j i r β j i r L j f r ∫ 0 + ∞ e 2 y j * s k j i s d s + m r ∑ k = 1 , j ≠ k n ρ k j r w k + m ,

where x i * , y j * ∈ 0 , + ∞ , i = 1, 2, ..., m, j = 1, 2, ..., n.

From (12) and (A4), we derive

F_i (0) < -δ < 0, G_j (0) < -δ < 0; F i x i * and G j y j * are continuous for x i * , y j * ∈ 0 , + ∞ . Moreover, F i x i * → + ∞ as x i * → + ∞ and G j y j * → + ∞ as y j * → + ∞ , thus there exist constants ε_i,ν_j ∈ [0, +∞) such that

F i ε i = w i - r n a i r - 1 D i λ 1 - r n a i r - r n μ i i a i r - 1 + 2 r - 1 ∑ j = 1 n a i r + r - 1 ∑ k = 1 , i ≠ k m a i r + r - 1 ∑ j = 1 n a i r β j i - r r - 1 ∫ 0 + ∞ k j i s d s + 2 ε i n a i r - 1 + ∑ j = 1 n w m + j m r d i j r L i g r + d ̃ i j r e τ 1 - μ τ L i g r + d ̄ i j r γ i j r L i g r ∫ 0 + ∞ e 2 ε i s k ̄ i j s d s + n r ∑ k = 1 , i ≠ k m μ k i r w k = 0

and

G j ν j = w m + j - r m c j r - 1 D j * λ 1 - r m c j r - r m ρ j j c j r - 1 + 2 r - 1 ∑ i = 1 m c j r + r - 1 ∑ k = 1 , j ≠ k n c j r + r - 1 ∑ i = 1 m c j r γ i j - r r - 1 ∫ 0 + ∞ k ̄ i j s d s + 2 ν j m c j r - 1 + ∑ i = 1 m w i n r b j i r L j f r + b ̃ j i r e θ 1 - μ θ L j f r + b ̄ j i r β j i r L j f r ∫ 0 + ∞ e 2 ν j s k j i s d s + m r ∑ k = 1 , j ≠ k n ρ k j r w k + m = 0 .

By using α = min 1 ≤ i ≤ m , 1 ≤ j ≤ n ε i , ν j , obviously, we get

F i ( α ) = w i ( − r n a i r − 1 D i λ 1 − r n a i r − r n μ i i a i r − 1 + 2 ( r − 1 ) ∑ j = 1 n a i r + ( r − 1 ) ∑ k = 1 , i ≠ k m a i r + ( r − 1 ) ∑ j = 1 n a i r β j i − r r − 1 ∫ 0 + ∞ k j i ( s ) d s + 2 α n a i r − 1 ) + ∑ j = 1 n w m + j m r ( | d i j | r ( L i g ) r + | d ˜ i j | r e τ 1 − μ τ ( L i g ) r + | d ¯ i j | r γ i j r ( L i g ) r ∫ 0 + ∞ e 2 α s k ¯ i j ( s ) d s ) + n r ∑ k = 1 , i ≠ k m | μ k i | r w k ≤ 0

and

G j α = w m + j - r m c j r - 1 D j * λ 1 - r m c j r - r m ρ j j c j r - 1 + 2 r - 1 ∑ i = 1 m c j r + r - 1 ∑ k = 1 , j ≠ k n c j r + r - 1 ∑ i = 1 m c j r γ i j - r r - 1 ∫ 0 + ∞ k ̄ i j s d s + 2 α m c j r - 1 + ∑ i = 1 m w i n r b j i r L j f r + b ̃ j i r e θ 1 - μ θ L j f r + b ̄ j i r β j i r L j f r ∫ 0 + ∞ e 2 α s k j i s d s + m r ∑ k = 1 , j ≠ k n ρ k j r w k + m ≤ 0 .