Stability Analysis for Stochastic Neutral-Type Memristive Neural Networks with Time-Varying Delay and S-Type Distributed Delays

In this paper, we consider the input-to-stability for a class of stochastic neutral-type memristive neural networks. Neutral terms and S-type distributed delays are taken into account in our system. Using the stochastic analysis theory and Itô formula, we obtain the conditions of mean-square exponential input-to-stability for system. A numerical example is given to illustrate the correctness of our conclusions.


Introduction
The complex network is considered to be one of the leading research subjects of science and technology in twenty-first Century, which include neural networks, communication networks, power networks, and social networks.In particular, the research on the neural network is very widely including the control theory, stability theory, and bifurcation theory (see [1][2][3][4]).As a special case of complex network, memristive neural networks can better simulate the human brain, so it has also become a focus for the majority of scholars.The memristor is a kind of nonlinear resistor which has memory function to simulate the mechanism of human neuron and synapse.Recently, memristive neural networks systems have been successfully applied in associative memory, chaos synchronization, image processing, and so on.Although a lot of achievements have been made in the field of application, most of the current research efforts on memristive neural networks are mainly focused on deterministic models (see [5][6][7]).However, in reality, noise disturbance always exists, which may cause instability and other poor performances.So it is necessary to study the memristive neural networks with random disturbance due to its theoretical and practical significance.
In addition, in order to deal with the dynamic image, we need to introduce delays between the signal transmissions of neurons, which have formed the memristive neural networks with delays.In practical application, it is more common for a dynamic system with time-varying delay, because the constant delay is only an ideal approximation of the timevarying delay.Many scholars have made great achievements in this respect (see [8][9][10][11]).Here, we have to point out that neural networks are composed of a large number of neurons, many of which are clustered into spherical or layered structures and interact with each other and are connected to a variety of complex neural pathways through the axon.Thus, there exists distributed delay in the transmission of signals.Usually, the discrete delays and distributed delays cannot contain each other in the same system; however, in [12] we can see that discrete delays and distributed delays can be written in a unified form under Stieltjes-Lebesgue integral, that is, S-type distributed delays (see [13,14]).In fact, the differential expression of the systems not only is related to the derivative of the current state but also has a great relationship with the derivative of the past state.It is called neutral delay neural network.Therefore, it is very significant to study the stochastic neutral-type memristive neural network with time-varying delays and S-type distributed delays.

Mathematical Problems in Engineering
The control input has a great influence on the dynamic behavior of the neural network.The input-to-state stability (ISS) was first proposed by Sontag to check robust stability, which is more general than the traditional exponential stability.The traditional exponential stability includes asymptotical stability, exponential stability, and almost sure stability.In [15,16], global asymptotical stability analysis for a kind of discrete-time recurrent neural network has been studied.In [17][18][19][20][21][22], exponential stability and almost sure stability of the neural network have been investigated.As far as we know, the traditional stability is that the state of the neural network is close to the equilibrium point when the time approaches infinity.But this does not always happen in our reality.ISS control analysis opens up a new dynamic neural network application in nonlinear system.In [23], input-tostate stability for a class of stochastic memristive neural networks with time-varying delay has been studied.Motivated by the above discussion, even though the stability problem of stochastic neural networks has been studied, there are few studies on the stability of stochastic neutraltype memristive neural network.In this paper, we consider the stochastic neutral-type memristive neural networks to end the gap.Using the stochastic analysis theory and Itô formula, we obtain the sufficient conditions of mean-square exponential input-to-stability and some corollaries for system (1).
The rest of the paper is organized as follows.In Section 2, we present a model and give some hypotheses.The main conclusions are proved in Section 3. In Section 4, a numerical example is given to illustrate the correctness of our conclusions.Finally, the further discussion is drawn in Section 5.
Throughout this paper, solutions of all the systems considered are intended in the Filippovs sense.Let   represent -dimensional Euclidean space.The superscript "" denotes the transpose of a matrix or vector. ∞ denotes the class of essentially bounded functions  from [0, +∞] to   with ‖‖ ∞ = esssup ≥0 || < ∞.Let  > 0 and ([−, 0],   ) denote the family of continuous functions  from [−, 0] to where [⋅] stands for the correspondent expectation operator with respect to the given probability measure .
To obtain the main results, we need the following hypotheses.

Main Results
In this section, the mean-square exponential input-to-state stability of the trivial solution for system (1) is addressed.
Moreover, when we remove the S-type distributed delay system (1)  Remark 6.In particular, when we remove the neutral terms, system (29) becomes the system in [23]; from [23] we can see that the trivial solution of system is mean-square exponentially input-to-state stable and the trivial solution of system with   () = 0 is mean-square exponentially stable in the certain condition.So we can say that our model is the extension of model in [23].
Remark 8. On the achievements of [23,25], this paper discusses a class of more general neural network systems through introducing many factors such as neutral terms, Stype distributed delays, and stochastic perturbations and analyzes the mean-square exponential input-to-state stability of the given neutral stochastic system by utilizing the Lyapunov-Krasovskii functional method, stochastic analysis techniques, and Itô formula.The considered Lyapunov-Krasovskii functional in our paper is more complex comparing with those in [23,25] since it covers neutral terms and double integrals.Therefore, our theoretical results can be seen as an extension in [23,25].In addition, our results are computationally efficient as the sufficient conditions can be easily checked without using linear matrix inequality toolbox.

Conclusions and Discussion
By stochastic analysis theory and Itô formula, mean-square exponential input-to-stability of a class of stochastic neutraltype memristive neural networks is studied.The correctness of our conclusions has been illustrated by a numerical example.In the current papers, there are few studies on stochastic neutral-type memristive neural networks with time-varying delay and S-type distributed delays.Furthermore, in this paper we discuss mean-square exponential input-to-stability of system (1); one may continue to discuss synchronization and passivity as well as some other complex dynamical behaviors on (1).
the conditions of Theorem 2 are satisfied, and we can obtain that the trivial solution of system (1) is mean-square exponentially input-to-state stable with initial values () = (0); see Figure1.When  1 () =  2 () = 0 the trivial solution of system(1) is mean-square exponentially stable; see Figure 2.