Stability of Almost Periodic Solution for a General Class of Discontinuous Neural Networks with Mixed Time-Varying Delays

The global exponential stability issues are considered for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous neuron activations. Some sufficient conditions for the existence, uniqueness, and global exponential stability of almost periodic solution are achieved in terms of certain linear matrix inequalities (LMIs), by applying differential inclusions theory, matrix inequality analysis technique, and generalized Lyapunov functional approach. In addition, the existence and asymptotically almost periodic behavior of the solution of the neural networks are also investigated under the framework of the solution in the sense of Filippov. Two simulation examples are given to illustrate the validity of the theoretical results.


Introduction
When applying neural networks to solve many practical problems in optimization, control, and signal processing, neural networks are usually designed to be globally asymptotical or exponentially stable to avoid spurious responses or the problems of local minima.Hence exploring the global stability of the neural networks is of primary importance.In recent years, neural networks with discontinuous activations, as a special class of dynamical systems described by differential equations with discontinuous right-hand sides, have been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, switching in electronic circuits, systems oscillating under the effect of an earthquake, and control synthesis of uncertain systems, and therefore have received extensive attention from a lot of scholars so far; see, for example, , and references therein.In [1], Forti and Nistri firstly dealt with the global asymptotic stability (GAS) and global convergence in finite time of a unique equilibrium point for the neural networks modeled by differential equations with discontinuous right-hand sides, and by using Lyapunov diagonally stable (LDS) matrix and constructing suitable Lyapunov function, several stability conditions were derived.In the subsequent literature, considerable efforts have been devoted to investigate the issue of stability for the neural network systems with discontinuous activation functions.In [2,3], by applying generalized Lyapunov approach and -matrix, Forti et al. discussed the global exponential stability (GES) of the neural networks with discontinuous or non-Lipschitz activation functions.Arguing as in [1,4], Lu and Chen dealt with GES and GAS of Cohen-Grossberg neural networks with discontinuous activation functions.In [5][6][7][8][9], under the framework of Filippov solutions, by using differential inclusion and Lyapunov approach, a series of results had been obtained for the global stability of the unique equilibrium point of the neural networks with a single constant time-delay and discontinuous activations.In [10], by using differential inclusion and constructing Lyapunov functional, Liu et al. discussed the global stability of the unique equilibrium point for the neural networks with mixed time-varying delays and discontinuous neuron activations, and some stability results were given in terms of certain LMIs.In [11], Nie and Cao studied the existence and global stability of equilibrium point for time-varying delayed competitive 2 Mathematical Problems in Engineering neural networks with discontinuous activation functions.In [12,13], based on Lyapunov stability theory and matrix inequality analysis techniques, the authors analyzed the global robust stability of a unique equilibrium point for the neural networks with time-varying delays and discontinuous neuron activation functions.In [14], Liu and Cao discussed the robust state estimation problems for time-varying delayed neural networks with discontinuous activation functions via differential inclusions, and some criteria had been established to guarantee the existence of robust state estimator.In [15], under the framework of Filippov solutions, by using matrix measure approach, Liu et al. investigated the global dissipativity and quasisynchronization issues for the time-varying delayed neural networks with discontinuous activations and parameter mismatches.
It is well known that any equilibrium point can be regarded as a special case of periodic solution for a neuron system with arbitrary period or zero amplitude.Through the study on periodic solution, more general results can be obtained than those of the study on equilibrium point for a neuron system.Hence to study the global stability of the equilibrium point of the neural networks with discontinuous activation functions, much attention has been paid to deal with the stability of periodic solution for various neural network systems with discontinuous activations; see, for example, [17][18][19][20][21][22][23][24][25][26][27][28].In [18][19][20][21], under the Filippov inclusion framework, by using Leray-Schauder alternative theorem and Lyapunov approach, the authors presented the conditions of the existence and GES or GAS of a unique periodic solution for Hopfield neural networks or BAM neural networks with discontinuous activation functions.In [22,23], Wu et al. discussed the existence and GES of the unique periodic solution for the neural networks with discontinuous activation functions under impulsive control.In [25,26], under the framework of Filippov solutions, by using Lyapunov approach and -matrix, the authors presented the stability results of periodic solution for delayed Cohen-Grossberg neural networks with a single constant timedelay and discontinuous activation functions.In [27,28], the authors explored the periodic dynamical behavior of the neural networks with time-varying delays and discontinuous activation functions.Some conditions were proposed to ensure the existence and GES of the unique periodic solution.
It should be noted that all the results reported in the literature above are concerned with the issue of stability analysis of equilibrium point or periodic solution and neglect the effects of almost periodicity for the neural networks with discontinuous activation functions.As far as we know, the almost periodicity is one of the basic properties for dynamical neural systems and appears to retrace their paths through phase space but not exactly.Almost periodic functions, with a superior spatial structure, can be regarded as a generalization of periodic functions.As pointed out by [29,30], in practice, almost periodic phenomenon was more common than periodic phenomenon, and almost periodic oscillatory behavior was more accordant with reality.In the past few years, a number of results had been obtained for the global stability of the almost periodic solution of various neural networks with continuous activation functions; see, for instance, [31][32][33][34][35][36][37][38][39] and references therein.In [31], Ding and Wang presented the results of the existence and stability of 2  almost periodic attractors for Cohen-Grossberg-type BAM neural networks with variable coefficients.In [37], Wang et al. investigated the multistability of delayed neural networks and described new attraction basins of almost-periodic solutions.Very recently, under the framework of the theory of Filippov differential inclusions, in [29], Allegretto et al. proved the common asymptotic behavior of almost periodic solution for discontinuous, delayed, and impulsive neural networks.In [30,40], Lu and Chen and Qin et al. discussed the existence and uniqueness of an almost periodic solution (as well as its global exponential stability) of delayed neural networks with almost periodic coefficients and discontinuous activations.It should be pointed out that, the network model explored in [29,30,40] was a class of discontinuous neural networks with a single constant time-delay, and the stability conditions were achieved by using LDS matrix or -matrix.Compared with the stability conditions expressed in terms of LMIs, it was obvious that the results obtained in [29,30,40] were very conservative.To the authors' knowledge, very few works have been conducted to deal with the stability analysis issues for almost periodic solution of discontinuous neural networks with time-varying delays, particularly to study the global stability of almost periodic solution by applying LMIs analysis technique, which motivates the work of this paper.
In this paper, our aim is to study the delay-dependent exponential stability problem for almost periodic solution of the neural networks with mixed time-varying delays and discontinuous activation functions.Under the framework of Filippov differential inclusions, by applying the nonsmooth Lyapunov stability theory and highly efficient LMI approach, new delay-dependent sufficient conditions are presented to ensure the existence and GES of almost periodic solution in terms of LMIs, which can be solved efficiently by using recently developed convex optimization algorithms [41].Moreover, the obtained conclusion is applied to prove the existence and stability of periodic solution (or equilibrium point) for the neural networks with mixed time-varying delays and discontinuous activations.
Given a set  ⊂ R  , [] denotes the closure of the convex hull of  and   () denotes the collection of all nonempty, closed, and convex subsets of .
Let  : R  → R be a locally Lipschitz continuous function.Clarke's generalized gradient [42] of  at  is defined by where Ω  ⊂ R  is the set of Lebesgue measure zero where ∇ does not exist and M ⊂ R  is an arbitrary set with measure zero.
Let  ⊂   .A set-valued map  : (graph measurability), where L() is the Lebesgue -field of  and B(R  ) is the Borel -field of R  .
Let ,  be Hausdorff topological spaces and (⋅) :  → 2  \ {0}.We say that the set-valued map (⋅) is upper semicontinuous, if, for all nonempty closed subset  of , The rest of this paper is organized as follows.In Section 2, the model formulation and some preliminaries are given.In Section 3, the existence and asymptotically almost periodic behavior of Filippov solutions are analyzed.Moreover, the proof of the existence of almost periodic solution is given.The global exponential stability is discussed, and a delaydependent criterion is established in terms of LMIs.In Section 4, two numerical examples are presented to demonstrate the validity of the proposed results.Some conclusions are drawn in Section 5.
By the assumption ( 1 )(1), it is easy to check that : is an upper semicontinuous setvalued map with nonempty, compact, and convex values.Hence (, ) is measurable [44].By the measurable selection theorem, if (⋅) is a solution of the system (4), then there exists a measurable function  = ( 1 ,  2 , . . .,   The function () in ( 7) is called an output solution associated with the state variable () and represents the vector of neural network outputs.Definition 3 (see [45]).A continuous function () : R → R  is said to be almost periodic on R, if for any scalar  > 0 there exist scalars  = () > 0 and  = () in any interval with the length of , such that ‖( + ) − ()‖ <  for all  ∈ R.
To obtain the main results of this paper, the following lemmas will be needed.
( ( ( 4 ): There exist a positive definite diagonal matrix  and two positive definite symmetric matrices , , such that

Main Results
In this section, the main results concerned with the stability of the almost periodic solution are addressed for a general class of neural networks (4).Firstly, we give the proof of the existence of the solution and discuss the asymptotically almost periodic behavior of the solution for the system (4).Proposition 8.If the assumption ( 4 ) holds, then there exist a positive constant  <  = min{ 1 ,  2 , . . .,   }, a positive diagonal matrix , and two positive definite symmetric matrices  and  such that where Γ =  +    +     +       +   and   is the -dimensional identity matrix.
Using Lemmas 6 and 7, we can obtain that where is defined as in Proposition 8.By the assumption ( 2 ), () is bounded for  ⩾ 0. Hence there exists a constant  > 0 such that This implies that Integrating both sides of ( 27) from 0 to ,  ∈ [0, ), it follows that In view of the definition of () in ( 17) and the fact that all the terms in () are not negative, we have By combining ( 28) and ( 29), it is easy to obtain Therefore, lim  →  − ‖()‖ < +∞.By viability theorem in differential inclusions theory [44], one yields  = +∞.That is, the system (4) has a solution of IVP on [0, +∞) for any initial value.The proof is completed.
Theorem 10.Suppose that the assumptions ( 1 )-( 4 ) are satisfied; then any solution of the neural network system (4) is asymptotically almost periodic.
Next, we will discuss the existence, uniqueness, and global exponential stability of the almost periodic solutions for the system (4).Theorem 12. Suppose that the assumptions ( 1 )-( 4 ) hold; then the neural network system (4) has a unique almost periodic solution which is globally exponentially stable.
Proof.Firstly, we prove the existence of the almost periodic solution for the system (4).
Remark 13.Under the assumptions ( 1 ) and ( 2 ), if the assumption ( 4 ) changes as the assumption (A 4 ), there exists a positive diagonal matrix , such that the coefficient matrices , , and  of the system (4) satisfy the following LMI: then, the conclusion of Theorem 9 is still true.In fact, similar with the proof of Proposition 8, we can choose positive constants  and  < , such that where Σ = ( +    +       +         +   ).Consider the following Lyapunov functional candidate: Under the assumptions ( 1 ) and ( 2 ), calculating the time derivative of V() along the local solution of the system (4), we can obtain where This implies that the conclusion of Theorem 9 is still true.In addition, under the case above, if the assumption ( 3 ) is satisfied, then it is easy to get that the conclusions of Theorems 10 and 12 are also correct.
When the external input of the network () = (1, −3)  is constant, by Corollary 15, this neural network has a unique equilibrium point which is globally exponentially stable.) . (55) The assumption ( 4 ) is also satisfied.When the external input of the network () = (2 sin , − cos , −5 sin 2)  is a periodic function, by Corollary 14, this neural network has a unique periodic solution which is globally exponentially stable.
When the external input of the network () = (−3, −4, −5)  is constant, by Corollary 15, this neural network has a unique equilibrium point which is globally exponentially stable.equilibrium point of the network.This is in accordance with the conclusion of Corollary 15.

Conclusion
In this paper, the almost periodic oscillation issue has been investigated for the neural networks with mixed timevarying delays and discontinuous activation functions.Some sufficient conditions which ensure the existence, uniqueness, and global exponential stability of almost periodic solution have been obtained in terms of LMIs, which is easy to be checked and applied in practice.Two numerical examples have been given to illustrate the effectiveness of the present results.
In [2], Forti et al. conjectured that all solutions of the delayed neural networks with discontinuous neuron activations and periodic inputs converge into an asymptotically stable limit cycle.In this paper, under the assumptions ( 1 )-( 4 ), the obtained results confirm that Forti's conjecture is true for the neural networks with mixed timevarying delays and discontinuous activation functions.In the near future, the stability issues of almost periodic solution for stochastic/impulsive neural networks with discontinuous activations will be expected to be solved.

Figure 3 Example 2 .Figure 3 :
Figure 3 displays the state trajectories of this neural network with state initial value () = (sin 2, cos 2)  ,  ∈ [−1, 0], when () = (1, −3)  .It can be seen that these trajectories converge into the unique equilibrium point of the network.This is in accordance with the conclusion of Corollary 15.Example 2. Consider the third-order neural network (4) with the following system parameters: