Attractor and Boundedness of Switched Stochastic Cohen-Grossberg Neural Networks

We address the problem of stochastic attractor and boundedness of a class of switched Cohen-Grossberg neural networks (CGNN) with discrete and infinitely distributed delays. With the help of stochastic analysis technology, the Lyapunov-Krasovskii functional method, linear matrix inequalities technique (LMI), and the average dwell time approach (ADT), some novel sufficient conditions regarding the issues of mean-square uniformly ultimate boundedness, the existence of a stochastic attractor, and the mean-square exponential stability for the switched Cohen-Grossberg neural networks are established. Finally, illustrative examples and their simulations are provided to illustrate the effectiveness of the proposed results.


Introduction
In the last few decades, theoretical and applied researches of artificial neural networks have been the new worldwide focus.Some of the reasons for this are due to the successful hardware implementations and their various applications, such as classification, associative memories, parallel computation, optimization, and signal processing [1,2].It is recognized that such applications of neural networks depend heavily on some dynamic behaviors, such as stability properties, periodic oscillatory behavior, and attractor and boundedness (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein).
Since the seminal work by Cohen and Grossberg [17], Cohen-Grossberg neural networks have been intensively studied [2,[18][19][20][21][22].During hardware implementation, time delays do exist due to the finite switching speed of the amplifiers and communication time; it is important to incorporate delays into the neural networks.Generally speaking, there are two kinds of delays, discrete delays and distributed delays [2,16,23].The utilization of discrete delays in models of delayed feedback provides a good approximation in simple circuits consisting of a small number of cells.When the neural networks have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, it is necessary to incorporate continuously distributed delays.The distributed delay includes finite delay and infinite delay [2,5,18,24].
In real nervous systems, synaptic transmission is a noisy process brought about by random fluctuations from the release of neurotransmitters and other probabilistic causes [19,[25][26][27].It is well known that for stochastic neural networks, it is rather difficult to analyze their dynamic properties due to the introduction of noise.Such studies are however important for understanding the dynamic characteristics of neuron behavior in stochastic environments.For instance, during the implementation of Kalman filter training, stochastic neural networks characterized as zero-mean white noise have been successfully employed [2].
On the other hand, neural networks are complex and large-scale nonlinear dynamics; during hardware implementation, the connection topology of networks may change very quickly and link failures or new creations in networks often bring about switching connection topology [2].
To obtain a deep and clear understanding of the dynamics of this complex system, one of the usual ways is to investigate the switched neural network.As a special class of hybrid systems, switched neural network systems are composed of a family of continuous-time or discrete time subsystems and a rule that orchestrates the switching among the subsystems [28].In general, the switched rule is a piecewise constant function dependent on the state or time.The logical rule that orchestrates switching between these subsystems generates switching signals [29].Recently, switched systems have numerous applications in the control of mechanical systems, the automotive industry, aircraft and air traffic control, switching power converters, and many other fields [30].In [28], Huang et al. are the first to investigate the robust stability of switched Hopfield neural networks with time-varying delays by an arbitrary switched rule.The average dwell time approach provided an effective tool to study the stability of switched systems.Wu et al. used average dwell time approach to analyze the exponential stability of continuoustime switched delayed neural networks in [31].In [27], the average dwell time and LMI method have been utilized to discuss the exponential synchronization of switched stochastic competitive neural networks with mixed delays.In addition, [32] has focused on the delay-dependent global robust asymptotic stability problem of uncertain switched Hopfield neural networks (USHNNs) with discrete interval and distributed time-varying delays and time delay in the leakage term.Moreover, parametric uncertainty which often breaks the stability of systems can be commonly encountered due to modeling inaccuracies or changes in the environment of the model.To deal with the difficulties brought about by uncertainty, exponential stability analysis and  ∞ control of different uncertain systems have received great research attention [33].Moreover, the parametric uncertainty is assumed to be norm-bounded in [34].Unfortunately, up to now, few researchers have considered the mean-square uniformly ultimate boundedness and stochastic attractor for switched SCGNN with discrete delays and infinite distributed delays.
However, these available literatures mainly consider the stability property of switching neural networks.In fact, except for the stability property, boundedness and attractor are also the foundational concepts of dynamical neural networks, which play important roles in the investigation of the uniqueness of the equilibrium point (periodic solutions), global asymptotic stability, global exponentially stability, and the synchronization [35].To the best of the author's knowledge, few researchers have considered the uniformly ultimate boundedness and attractors for switched CGNN with discrete delays and distributed delays.
Inspired by the above discussions, the objects of this paper are to study the mean-square uniformly ultimate boundedness and stochastic attractor for switched SCGNN with discrete delays and infinitely distributed delays by employing stochastic analysis technology, the Lyapunov-Krasovskii functional method, the linear matrix inequalities (LMI) technique, and the average dwell time approach (ADT).In addition, the parametric uncertainty is considered and assumed to be norm-bounded.
As is well known, mean-square uniformly ultimate boundedness (MSUUB) conditions are derived in terms of linear matrix inequalities (LMIs), which can be easily calculated by the MATLAB LMI control toolbox.All of the above mentioned reasons motivate us to investigate the problems of the MSUUB and stochastic attractor for switched SCGNN in this paper.Numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed criteria.
The rest of this paper is organized as follows.Some preliminaries are given in Section 2. We present some basic definitions and notations, as well as some lemmas needed in later sections.In Section 3, we present some sufficient conditions of MSUUB and stochastic attractor for switched stochastic CGNN.In Section 4, an example is presented to illustrate the effectiveness of the proposed approach.The conclusions are summarized in Section 5.
Notations.The superscript "" stands for matrix transposition;   denotes the -dimensional Euclidean space; the notation  > 0 means that  is real symmetric and positivedefinite;  and  represent the identity matrix and a zero matrix, respectively; diag{⋅ ⋅ ⋅ } stands for a block-diagonal matrix; and  min () ( max ()) denotes the minimum (maximum) eigenvalue of symmetric matrix .In symmetric block matrices or long matrix expressions, a ( * ) is used to represent a term that is induced by symmetry.
In practical systems, the neural network models are disturbed by environmental noises.Therefore, in this paper, we will consider the stochastic Cohen-Grossberg neural networks with mixed time delay described by the following stochastic nonlinear integrodifferential equations: System (2) for convenience can be rewritten as the following vector form: and the delay kernel   is a real valued continuous function defined on [0, +∞] and satisfies; for each , Moreover, there exist  > 0 and matrix  = diag{ 1 (),  2 (), . . .,   ()} > 0, As usual, the initial conditions associated with system (3) are given in the form where the initial value function  ∈   F 0 ([−∞, 0],   ) is the family of all F 0 -measurable ([−∞, 0];   )-valued random variables satisfying sup −∞≤≤0 ‖()‖ 2 < ∞, in which  denotes expectations with respect to P and ([−∞, 0]) denotes the family of all continuous   -valued functions () on [−∞, 0].
To continue our discussion, we give the following basic assumptions.
Remark 2. It is worth mentioning that the structures of the parametric uncertainties with the form ( 13) and ( 14) are more general than those in previous literature in [28,34,37].However,    =    =    has been discussed [28,34,37].Recently, in [35], the attractor and boundedness of stochastic Cohen-Grossberg neural networks without parametric uncertainties were investigated.
Remark 5.It should be pointed out that for the chatter bound  0 , in our work, we take  0 ≥ 1, which is more preferable than those previously reported in [31,37].If   = 0 is equivalent to the existence of a common function for all subsystems, this implies that switching signals can be arbitrary.Hence, the results reported in this paper are more effective than the arbitrary switching signals reported in the previous literatures [28,37].
So, to obtain the main results of this paper, we introduce the following lemmas.
Then, we easily derive Similarly, calculating the operation of L 3 along the trajectory of system (3), one can get ( ())  ( ()) .

Theorem 11. If all of the conditions of Theorem 10 hold, then there exists an attractor A B for the solutions of system (3),
where Theorem 10 shows that for any , there is   > 0, such that ‖(,  0 , )‖ < B for all  ≥   .Let Clearly, A B is closed, bounded, and invariant.Furthermore, lim →∞ sup inf ∈A B ‖(; 0, ) − ‖ = 0. Therefore, A B is a stochastic attractor for the solutions of system (3).

Corollary 12.
In addition to all of the conditions of Theorem 10 that hold, if  = 0 and   (0) = 0, then system (3) has a trivial solution () ≡ 0, and the trivial solution of system ( 3) is meansquare exponentially stable.
In practice, parameter uncertainties in neural networks are always unavoidable, in order to explain such a phenomenon.In this section, we will investigate the mean-square uniform ultimate boundedness of the switching stochastic systems with uncertainties by applying the average dwell time.
Now, we consider the switched stochastic Cohen-Grossberg neural networks with unknown parameters as follows:

Corollary 15.
In addition to all of the conditions of Theorem 13 that hold, if  = 0,   (0) = 0, then system (55) has a trivial solution () ≡ 0, and the trivial solution of system (55) is mean-square exponentially stable.
Remark 16.It is noteworthy that the time-varying delay () restricts the interval [  ,   ] and the lower bound of time delay   may not be equal to 0. In previous work, such as [19,24,26,37], the well-used Lyapunov functional, in which the time delay information is from 0 to an upper bound   , is of the form ∫ Thus the methods in the paper can be adopted to discuss the dynamic behaviors of interval stochastic switched Cohen-Grosberg neural networks with time delays.Therefore, the time-varying from   to   is more general and less conserving of the neural networks models.If the lower bound of time delay   = 0, then our results will turn into the traditional time delay results.
Remark 17.It is known that noise disturbance is a major source of instability and poor performances in neural networks in real neural networks.If (, (), ( − ())) = 0, the switched stochastic Cohen-Grossberg neural networks (8) degenerate into the ordinary switched Cohen-Grossberg neural networks, which have been studied for exponential stability in [22] and robust stability in [37].In addition, when   (  ()) = 1,  = 1, 2, . . ., , the switched Cohen-Grosberg neural networks will turn into the famous switched Hopfield neural networks; this has been investigated in [28] without distributed time delay and for global robust asymptotic stability in [32] with finite distributed time delay.However, the infinite distributed time delay was not taken into account in neural networks.Therefore, our developed results in this paper are more comfortable and general than those reported in [28,32,37].
Remark 18.If  = 1, then the switched stochastic Cohen-Grossberg neural networks (8) degenerate into the ordinary stochastic Cohen-Grossberg neural networks without being switched.The attractor and boundedness for stochastic Cohen-Grossberg neural networks with delays have been discussed in [35] by LaSalle-type theorem and stability has been studied in [2,18,19].So our results generalize these previous results.considered in this paper lead to new dynamic criteria.We make full use of Lemma 6 and do not ignore any terms, which can reduce some conservatism of proposed method.This can be verified from the numerical examples discussed in Section 5.
Remark 20.It should be mentioned that the nonlinear output function in [2,18,24,35,37,38] is required to satisfy   (0) = 0; however, in our paper, the assumption condition was (76) Taking   (0) = 0,  = 0 and using (59), we can obtain the average dwell time   >  *  = max{6.6718,6.8237}.Therefore, one can choose   = 7.The simulations of arbitrary switching signal with the average dwell time   = 7 can be shown in Figure 1.The mean-square exponential stability of system (55) with the initial value as  0 = [−8, 8] can be shown in Figure 2.With the help of MATLAB, the time evolutions of state variables of the system (55) can be shown in Figure 3.In the above conditions, phase portraits of simulations under initial condition of the system (55) are shown in Figure 4.

Conclusion
This paper has studied the problem of boundedness for a class of switched stochastic Cohen-Grossberg neural networks with both average dwell time and norm-bounded parameter uncertainties.By employing multiple Lyapunov-Krasovskii functionals ( 25) and (60), we formulate a method that derives new sufficient conditions guaranteeing the mean-square uniformly ultimate boundedness, the existence of an attractor, and the mean-square exponential stability.A numerical example has been presented to demonstrate the effectiveness and the merits of the proposed method.It is expected that the approach presented in this paper can be easily extended to analyze other neural networks.