ASYMPTOTIC BEHAVIOR OF A BAM NEURAL NETWORK WITH DELAYS OF DISTRIBUTED TYPE

Abstract. In this paper, we examine a Bidirectional Associative Memory neural network model with distributed delays. Using a result due to Cid [J. Math. Anal. Appl. 281 (2003) 264–275], we were able to prove an exponential stability result in the case when the standard Lipschitz continuity condition is violated. Indeed, we deal with activation functions which may not be Lipschitz continuous. Therefore, the standard Halanay inequality is not applicable. We will use a nonlinear version of this inequality. At the end, the obtained differential inequality which should imply the exponential stability appears ‘state dependent’. That is the usual constant depends in this case on the state itself. This adds some difficulties which we overcome by a suitable argument.

After a few years, Kosko introduced an expansion of Hopfield neural networks named bidirectional associative memory (BAM) neural network [13][14][15]. It is used in pattern recognition, signal and image processing, optimization problems, and automatic control [21]. Its architecture relies on two interconnected hidden layers of neurons that have no links in a single layer. It is described by the system a ji f j (y j (t)) + I i , i = 1, 2, . . . , n, y j (t) = −c j y j (t) + n i=1ā ij g i (x i (t)) + J j , j = 1, 2, . . . , m.
In applications, it is frequent that, oscillations, divergences, chaos, and bifurcations affect negatively structures. One of the causes may be the occurrence of delays. Owing to the existence of a multitude of parallel paths with axons of different sizes and lengths, neural networks typically have a spatial extent, and therefore a distribution of propagation delays over a time span occurs [19,26,41].
Due to the increasing interest in the asymptotic behavior of solutions for designing neural networks, researchers have recently addressed the stability of delayed neural networks (see for instance [5, 8, 10, 12, 16-18, 20, 23, 24, 38, 39] and references therein). In [18,38,39], a set of sufficient conditions based on the system parameters guaranteeing the exponential stability of various retarded BAM neural network models was derived by analytical techniques and Lyapunov functionals. In addition, the authors in [5,10,17,20,23] obtained some LMI-dependent sufficient conditions ensuring either the exponential or asymptotic stability of BAM neural networks involving delays, via Karasovski Lyapunov functionals and analytical inequalities. In [8,16,24], the asymptotic stability of a class of delayed BAM neural networks was investigated utilizing the LMI approach, Lyapunov functionals, and analytical inequalities. Through a combination of the graph-theoretic approach, degree theory, and Lyapunov functionals, the exponential stability of a BAM neural network with delays was established in [12].
On the other hand, one of the basic components of artificial neural networks involves activation functions, linking the inputs to the outputs of the networks. Generally, the activation function of hidden neurons introduces a degree of nonlinearity that is of significant value in most applications of artificial neural networks. At first, such functions were assumed bounded, smooth, and monotonic [6,36,37]. For example, we can cite the threshold function, the piecewise linear function, and the sigmoid function. Thereafter, these conditions were eased somewhat to be of Lipschitz type, which is commonly considered in the existing literature [12,18,20,23,24,39]. In view of the importance of non-Lipschitz activation functions in implementations [15], a relaxation of the Lipschitz condition is necessary. This has motivated some researchers to consider discontinuous functions and Hölder-type functions, one can refer to [2,3,7,11,[27][28][29][30][31][32][33]35].
As is well known, Cauchy problems described by have local solutions under the continuity of the function f in a neighborhood of (t 0 , x 0 ). This can shown by the theorem of Peano, whilst the uniqueness of the solution is guaranteed under the Lipschitz continuity with respect to the second argument. Weaker conditions have been considered in several papers by Nagumo, Perron, Osgood, Kamke, Tonelli, and many others (see for instance [9]). In particular, Nagumo and Osgood proved the uniqueness of the solution under the condition where φ(u) is a nondecreasing function of 'continuity-modulus' type satisfying the Osgood criterion: φ : [0, ∞) → [0, ∞), with φ(0) = 0 and φ(u) > 0 for u > 0 and In case one of the components of the function f fails to be Lipschitz continuous and f is Lipschitz continuous with respect to all the other remaining components including the first one t, Cid [4] succeeded in proving the uniqueness of the solution for a system of differential equations. Motivated by the discussions above, in this paper, we examine the exponential stability of a BAM neural network model with distributed delays. Our first result is based on a nonlinear Halanay inequality, whilst the second one is proved in a direct fashion. Compared to previous literature, we deal here with less restrictive assumptions on the activation functions, which allows a larger class than the ones considered so far. More precisely, we treat non-linear and non-Lipschitz continuous activation functions, satisfying for some non-decreasing functionφ. This paper is arranged as follows: In Section 2, we introduce some notation, definitions, and technical lemmas, while Section 3 contains our exponential stability results proven with the help of a non-linear Halanay inequality. Numerical illustrations to confirm the obtained results are given in Section 4.

Model description and preliminaries
In this paper, we consider the following BAM neural network with continuously distributed delays, for i = 1, 2, . . . , n, j = 1, 2, . . . , m where n and m denote the number of neurons in the first layer L x and in the second layer L y , respectively; x i is the state of the ith neuron in L x and y j is the state of the jth neuron in L y ; c i and r j refer to the charging time functions of the ith and the jth neuron, respectively; a ji ,ā ij , d ji andd ij correspond to the constants accounting for the synaptic connection strengths between neurons; f lj and g li for l = 1, 2 represent the activation functions of the ith state and the jth state, respectively; k ji and h ij account for the delay kernel functions; φ i and ϕ j are the history functions of the ith state and the jth state, respectively; I i and J j stand for the external inputs of the ith and the jth neuron.
Throughout this paper, we assume that (A1) The functions c i and r j are monotone increasing continuous functions and there exist constants β i > 0 andβ j > 0 such that x − y ≥ β i , i = 1, 2, ..., n, for all x, y ∈ R with x = y, x − y ≥β j , j = 1, 2, ..., m, for all x, y ∈ R with x = y.
We denote by β = min j . This means that the charging time functions c i and r j are "superlinear". With these assumptions, the first two terms in the right hand sides of (2.1) are indeed dissipative terms (see the first two terms in the right hand sides of (3.4) and (3.5) below).
(A3) The delay kernels k ji and h ij are nonnegative continuous functions such that for i = 1, 2, ..., n, j = 1, 2, ..., m, The significance of these conditions is that the kernels are "subexponential". Roughly, they assert that the densities in the cells are decreasing in time. This is in line with the fading memory principle. Simple examples are the exponentially decaying kernels.
Next, we present some definitions and necessary lemmas for our main results.
Definition 2.2. We say that the system (1) is globally exponentially stable, if for any two solutions (x i (t), y j (t)) and (x i (t),ȳ j (t)) (with (φ i (t), ϕ j (t)) and (φ i (t),φ j (t)) as initial data), there exist two positive constants M and ν such that If there exists a unique equilibrium (x * i , y * j ), i = 1, . . . , n, j = 1, . . . , m, that is a solution of the system then we get the usual exponential stability of this equilibrium.
The above properties are local if they hold only nearby the initial data. ).
The main result of Cid [4] is given below.
where a, b > 0 and k be a nonnegative piecewise continuous function satisfying In order to prove the first exponential stability result, we appeal to a nonlinear Halanay inequality.
This result has been proved in [34] (submitted). For the sake of completeness, we reproduced the proof here.

Exponential stability
In this section, we establish a local exponential stability result of system (2.1). As stated in Theorem 3.1, our result is valid for solutions having close enough initial data. To this end, we need the following assumption on the activation functions.
In the sense of the definition above, the existence of a local solution is guaranteed by Theorem 2.4. We can extend our solution to be global by establishing a priori estimates.
(A5) Assume that the functions f lj0 and g li0 , l = 1, 2 satisfy wheref lj0 andg li0 are nonnegative, non-decreasing and continuous functions such thatf lj0 (0) =g li0 (0) = 0. This assumption avoids and relaxes the classical Lipschitz condition imposed on the activation functions. As a simple example, the pulse-code signal function (average of sampled pulses with an exponential weight) is not Lipschitz continuous.
(A6) The dominance property where This assumption is a natural one, it has been always assumed in such cases. It guarantees the dissipativity of the system as a whole. Indeed, it states that the passive decay rates dominate the synaptic connection strengths.
Theorem 3.1. If (A1)-(A6) hold, then the solutions of (2.1) are exponentially locally stable. That is the difference of any two solutions converges to zero exponentially provided that their initial data are close enough and we do not start from the equilibrium.
For the second result, we need the assumption (A7) We have Theorem 3.2. The conclusion of Theorem 3.1 holds with the same hypotheses except (A6) replaced by (A7).

Proof. According to estimations (3.4) and (3.5), one has
and We introduce the functionals, for i = 1, 2, . . . , n, j = 1, 2, . . . , m for some 0 < β 1 , β 2 < α 0 , and The differentiation of (3.10) and (3.11) for t > 0 yields (3.14) Having in mind that, when the derivative exists, it is equal to all the Dini derivatives, from (3.8)-(3.14), the upper right Dini derivative of L(t) satisfies This is simplified as We may further estimate D + L(t) as follows which we can write in short in the form We shall compare solutions of (3.15) to solutions of the differential problem with In light of our assumptionsf 1j0 (0) =f 2j0 (0) =g 1i0 (0) =g 2i0 (0) = 0, we have p > 0. For q, 0 < q < p, we define Due to the non-decreasingness of the function F and q < p, then Besides, In view of (3.17) and (3.18), then for any positive real number λ that satisfies −λ > −A + F (q), one can find a ξ λ > 0 such that We claim that, if z 0 < q, then We argue by contradiction. Assume that t * > 0 is the first time z(t * ) = qe −ξ λ t * and ψ (t * ) ≥ 0, where ψ(t) := z(t)e ξ λ t . It is easy to see that and therefore, from (3.16) and (3.19), we infer This contradiction confirms the relation (3.20). By comparison, we obtain and thus for some positive constants ξ * and B. This completes the proof.
Remark 3.3. 1. We note that it is not necessary to verify the assumptioñ we need in the nonlinear version of Halanay inequality (Lemma 2.6) the following estimatioñ for Theorem 3.1.

3.
The existence and uniqueness may be deduced from the following argument If k ji and h ij are Lipschitz and bounded, then This, added to our a priori estimates, implies the Lipschitz continuity with respect to t and Cid's Theorem 2.4 [4] implies the well-posedness. 4. The stability of the equilibrium needs additional conditions on the i 0 and j 0 components that ensure the uniqueness of the equilibrium (such as the Osgood condition). However, our argument will then depend on the existence off 1j0 ,f 2j0 ,g 1i0 andg 2i0 (see (1.2)) fulfilling the conditions (see 2. in Rem. 3.3).

Numerical illustration
In this section, we present a numerical example to validate the above results. Observe that, the Hölder continuous functions (having an exponent between 0 and 1) cannot be covered by our results. These were studied in [7,[27][28][29][30][31][32][33]. It is known that when the exponents exceed one, the functions are constant. Neither are the Log-Lipschitz continuous functions, for example, functions with x|ln(x)| as a continuity modulus (because ln(x) is unbounded near 0). These functions do not satisfy the Lipschitz condition, whereas Osgood's condition is verified (which guarantees the uniqueness of the equilibrium). Nevertheless, these results can be applied to locally Lipschitz continuous functions.