Finite-Time Stabilization for Stochastic Inertial Neural Networks with Time-Delay via Nonlinear Delay Controller

This paper pays close attention to the problem of finite-time stabilization related to stochastic inertial neural networks with or without time-delay. By establishing proper Lyapunov-Krasovskii functional and making use of matrix inequalities, some sufficient conditions on finite-time stabilization are obtained and the stochastic settling-time function is also estimated. Furthermore, in order to achieve the finite-time stabilization, both delayed and nondelayed nonlinear feedback controllers are designed, respectively, in terms of solutions to a set of linear matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the correction of the theoretical results and the effectiveness of the proposed control design method.

On the other hand, many researchers have studied Hopfield neural networks [18], cell neural networks, recurrent neural networks [9,19], Cohen-Grossberg neural networks, bidirectional associative memory neural networks, and Lotka-Volterra neural networks, as well as inertial neural networks [12,14,15,20], which are more intricate than all kinds of prementioned neural networks with the standard resistor-capacitor variety [21].The inertial term is taken as a critical tool to bring complex bifurcation behavior and chaos.
It has been confirmed that stochastic disturbances, which are unavoidable in actual applications of artificial neural networks, are probably one of the main sources leading to undesirable behaviors of dynamical systems, especially when a neural network is implemented for applications.Therefore, it is of great significance to study the stability and stabilization problems of neural networks with stochastic disturbances [22][23][24].However, to the best of authors' knowledge, most of the researchers have either investigated the stability for stochastic neural networks with time-delay [25][26][27][28] or studied the stability for inertial neural networks with time-delay [20].There are rare literatures that considered the finitetime stabilization for stochastic inertial neural networks with time-delay.
Inspired by the above comprehensive analysis, in this paper, we are devoted to investigating the finite-time stabilization for stochastic inertial neural networks with timedelay.First, by utilizing an appropriate variable substitution, a stochastic inertial neural network can be transformed into a first-order stochastic differential system.Then, some sufficient conditions on finite-time stability in probability are derived by means of establishing an appropriate Lyapunov function and applying inequality techniques.Moreover, the stochastic settling-time function is also given.

Problem Formulation and Preliminaries
. .Systems Description.Firstly, the inertial neural networks (INNs) without time-delay are considered, which is described as follows: where   () is the state of the -th neuron; the second derivative d 2   ()/d 2 is the inertial term of INNs (1).  > 0,   > 0 are constants.  denotes the connection weight between the -th neuron and the -th neuron.  (⋅) stands for activation function of the -th neuron with   (0) = 0( = 1, 2, . . ., ).  () is the external input on the -th neuron.
The initial conditions of INNs (1) are where   (0) and   (0) are real-valued continuous functions.Suppose that the external input   () is subject to the environmental noise and is described by   () =   () +   (,   ()) ω  (), where   () is known as the control input and   () is a one-dimensional white noise, which is also called Brown motion defined on a complete probability space (Ω, F, ) and satisfied with and   (,   ()) is the intensity function of the noise.Then INNs (1) can be written the following stochastic inertial neural networks (SINNs): . .Problems Formulation.In general, making use of the variable transformation, then the SINNs (4) can be rewritten as and the initial conditions are given as where ã =   −   , b =   +   (  −   ).
is a positive constant with 0 <  < 1.
Remark .There are three cases for the value of .If 0 <  < 1, the controllers (), ]() are continuous functions with respect to  and , respectively, which bring about the continuity of SINNs (10) with respect to the systems state [29,30], but the local Lipschitz condition is dissatisfied.If  = 0, (), ]() turn to be discontinuous ones, which have been studied in [31,32].If  = 1, then they become the typical stabilization issues which only can realize an asymptotical stabilization in infinite time [33,34] due to the local Lipschitz conditions.
Remark .In fact, the control gain matrices  1 ,  2 ,  3 ,  4 in the controllers ]() and () play different roles in ensuring the finite-time stability of the SINNs ( 10) with (11), where  1 and  3 are used to guarantee the Lyapunov stability of the SINNs (10).And the convergence to zero of the SINNs ( 10) is determined by  2 and  4 .
To achieve our main results, some assumptions, lemmas, and definitions are necessary to introduce firstly.
Assumption .The intensity function (, ()) is a continuous function and is supposed to satisfy that where  2 is a known matrix with appropriate dimensions.

Main Results
Moreover, the upper bound of the stochastic settling time for stabilization can be estimated as , where

Mathematical Problems in Engineering
Proof.Taking controller (11) into SINNs (10) Next, we will prove that system (20) is finite-time stable in the sense of Definition 6.
Summing up the above analysis, some sufficient conditions on finite-time stability for the SINNs (10) with (11) are obtained.In the following, we mainly focus on the design of finite-time stabilizing controllers by transforming the sufficient conditions into solvable linear matrix inequalities.Theorem 10.If there exist some positive define matrices  1 ,  2 ,  3 , matrices  1 , 2 with appropriate dimensions, for fixed control gain matrices  2 and  4 , such that where then the finite-time stabilization problem is solvable for the stochastic inertial neural networks ( ) and the control gain matrices Proof.
can be written as where Then, left-and right-multiplying inequality ( 35) by the blockdiagonal matrix diag{ 1 , }, which follows where and left-and right-multiplying inequality (37) by the blockdiagonal matrix diag{,  1 }, we can obtain where By Schur complement, (33) implies the above inequality (39) holds.This completes the proof.
. .Finite-Time Stabilization Feedback Controller Design with Time-Delay.In the above section, we discussed the finitetime stabilization for stochastic inertial neural networks without time-delay.However, when designing a neural network or implementing it, the occurrence of time-delay is unavoidable.It may cause instability and oscillation [36][37][38].Therefore, in order to reduce the conservatism, in this section, we will study the finite-time stabilization for stochastic inertial neural networks with time-delay.
The nonlinear delay-feedback controller is designed as the following form: where  1 ,  2 ,  3 ,  4 are gain matrices to be determined, and Theorem 11. e SINNs with time-delay ( ) with ( ) are finite-time stable, if there exist some positive-definite matrices where Moreover, the upper bound of the stochastic setting time for stabilization can be estimated as Proof.Construct a Lyapunov function: Calculating the Itô differential of () along with (43), we can obtain We can see that the right of inequality (48) equals (23).Hence, the rest of the proof is the same as that of Theorem 9 and it is omitted here.
Similar to the proof of Theorem 10, we have the following result.) ) where then the finite-time stabilization problem is solvable for the stochastic inertial neural networks ( ) and the control gain matrices

Illustrative Example
Consider the following stochastic inertial neural networks with time-delay: Setting the initial values (0) = (−0.15,0.2)  , (0) = (0.3, 0.3)  , the state trajectories and phrase trajectories of the open-loop system are shown in Figures 1 and 2, respectively.Moreover, take 10 sets of numbers randomly as the initial values of (0) and (0) and satisfy (0) ∈ (−1, 1), (0) ∈ (−3, 3).Then the corresponding state trajectories and phrase trajectories of the open-loop system are shown in Figures 3  and 4, respectively.Obviously, the stochastic inertial neural networks with time-delay (51) are not finite-time stabilization.Hence, we need to design the delay-feedback controller as (44) for system (51), where the parameter  is chosen as 0.6, and the initial values (0) = (−1, 1)  , (0) = (3, −3)  , 2 = ( We can get  1 = (  In order to make the result of the simulation more convincing, we take 100 sets of numbers randomly as the initial values of (0) and (0) and satisfy (0) ∈ (−1, 1), (0) ∈ (−3, 3).Then the corresponding state trajectories are shown in Figure 7 and the corresponding phrase trajectories are shown in Figure 8. Obviously, the stochastic inertial neural networks with time-delay (51) are finite-time stabilization.Moreover, when (0) ∈ (−10, 10), (0) ∈ (−30, 30), we have state trajectories in Figure 9 and phrase trajectories in Figure 10, which also figure out that the stochastic inertial neural networks with time-delay (51) are finite-time stabilization.stochastic inertial neural networks with or without timedelay.Provided that a set of LMIs are feasible, a suitable delayed or nondelayed nonlinear feedback controller can be designed such that finite-time stability in probability can be ensured for the system under study.An example has been given to demonstrate the correctness of the theoretical results and the effectiveness of the proposed methods.