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

Authors: Deyi Li; Yuanyuan Wang; Guici Chen; Shasha Zhu
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

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.


---

## Body

## 1. Introduction

In recent years, more and more scholars have been attracted by neural networks due to their successful applications in associative memory [1, 2], pattern recognition [3], signal processing, optimization problems, and so forth [4]. These applications always rely on the dynamic behaviors of neural networks. Therefore, the investigation of dynamic trajectories is necessary for applied designation of neural networks. Hence, a large number of studies on stability [5–8], stabilization [9, 10], passivity [11], dissipativity [12, 13], synchronization [14, 15], and state estimation [16, 17] for neural networks have been reported.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 [22–24]. However, to the best of authors’ knowledge, most of the researchers have either investigated the stability for stochastic neural networks with time-delay [25–28] or studied the stability for inertial neural networks with time-delay [20]. There are rare literatures that considered the finite-time 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 time-delay. 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.


## 2. Problem Formulation and Preliminaries

### 2.1. Systems Description

Firstly, the inertial neural networks (INNs) without time-delay are considered, which is described as follows:(1)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+Iit,i=1,2,…,nwhere xi(t) is the state of the i-th neuron; the second derivative d2xi(t)/dt2 is the inertial term of INNs (1). ai>0, bi>0 are constants. cij denotes the connection weight between the i-th neuron and the j-th neuron. fj(·) stands for activation function of the j-th neuron with fj(0)=0(j=1,2,…,n). Ii(t) is the external input on the i-th neuron.The initial conditions of INNs (1) are(2)xi0=φi0,dxi0dt=ψi0,i=1,2,…,n,where φi(0) and ψi(0) are real-valued continuous functions.Suppose that the external inputIi(t) is subject to the environmental noise and is described by Ii(t)=ui(t)+βi(t,xi(t))ω˙i(t), where ui(t) is known as the control input and ωi(t) is a one-dimensional white noise, which is also called Brown motion defined on a complete probability space (Ω,F,P) and satisfied with(3)Eωit=0,Eωi2t=dt,Eωitωjt=0i≠j,and βi(t,xi(t)) is the intensity function of the noise.Then INNs (1) can be written the following stochastic inertial neural networks (SINNs):(4)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+uit+βit,xitω˙it.


### 2.2. Problems Formulation

In general, making use of the variable transformation,(5)yit=dxitdt+ξixit,i=1,2,…,n,then the SINNs (4) can be rewritten as(6)dxit=-ξixit+yitdt,dyit=-a~iyit-b~ixit+∑j=1ncijfjxjt+uitdt+βit,xitdωit,and the initial conditions are given as(7)xi0=φi0,yi0=-ξi0φi0+ψi0,i=1,2,…,n,where a~i=ai-ξi, b~i=bi+ξi(ξi-ai).Moreover, the controllerνi(t) is considered; we have the following SINNs:(8)dxit=-ξixit+yit+νitdt,dyit=-a~iyit-b~ixit+∑j=1ncijfjxjt+uitdt+βit,xitdωit,i=1,2,…,n.Denote(9)xt=x1t,x2t,…,xntT,yt=y1t,y2t,…,yntT,Ξ=diag⁡ξ1,ξ2,…,ξn,νt=ν1t,ν2t,…,νntT,A=diag⁡a~1,a~2,…,a~n,B=diag⁡b~1,b~2,…,b~n,C=cijn×n,βt,xt=β1t,x1t,β2t,x2t,…,βnt,xntT,ωt=ω1t,ω2t,…,ωnt,ut=u1t,u2t,…,untT.Thus, the SINNs (8) can be written in vector form as(10)dxt=-Ξxt+yt+νtdt,dyt=-Ayt-Bxt+Cfxt+utdt+βt,xtdωt.The control inputs to be designed are of the following form:(11)νt=-K1xt-K2sign⁡xtxtμ,ut=-K3yt-K4sign⁡ytytμ,where sign(x(t))=(sign(x1(t)),sign(x2(t)),…, sign(xn(t)))T, sign(y(t))=(sign(y1(t)),sign(y2(t)),…, sign(yn(t)))T, and K1, K2,K3, K4 are the control gain matrices to be determined. μ is a positive constant with 0<μ<1.Remark 1.
There are three cases for the value ofμ. If 0<μ<1, the controllers u(t),ν(t) are continuous functions with respect to x and y, 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 μ=0, u(t),ν(t) turn to be discontinuous ones, which have been studied in [31, 32]. If μ=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 2.
In fact, the control gain matricesK1, K2, K3, K4 in the controllers ν(t) and u(t) play different roles in ensuring the finite-time stability of the SINNs (10) with (11), where K1 and K3 are used to guarantee the Lyapunov stability of the SINNs (10). And the convergence to zero of the SINNs (10) is determined by K2 and K4.To achieve our main results, some assumptions, lemmas, and definitions are necessary to introduce firstly.Assumption 3.
The nonlinear activation functionf satisfies f(0)=0, and there exist some constants m1i-,m1i+(i=1,2,…,n), such that(12)M1-≤fx1-fx2x1-x2≤M1+hold for all x1,x2∈R and x1≠x2, where M1-=diag{m11-,m12-,…,m1n-}, M1+=diag{m11+,m12+,…, m1n+}.Remark 4.
If we choosem1i=max⁡m1i-,m1i+, the inequalities in Assumption 3 can be written as(13)fx1-fx2x1-x2≤M1,where M1=diag{m11,m12,…,m1n}, which has been considerably studied.Assumption 5.
The intensity functionβ(t,x(t)) is a continuous function and is supposed to satisfy that(14)traceβTt,xtβt,xt≤xTtM2TM2xt, where M2 is a known matrix with appropriate dimensions.Definition 6.
The SINNs (10) are said to be finite-time stabilizable by the controller (11); that is, the SINNs (10) are finite-time stable if, for any initial state x(0), y(0), there exists a finite-time function T0 such that (15)Ext=Eyt=0,∀t≥T0, where T0=T0(x(0),y(0),ω)=inf{T≥0:x(t)=y(t)=0,∀t≥T} is called the stochastic settling time function satisfying E[T0]<∞.Lemma 7 (see [35]).
Suppose that SINNs (10) admit a unique solution. If there exist a C2 function V:Rn→R+, K∞ class functions μ1 and μ2, and positive real constant η>0 and 0<γ<1, such that for all x∈Rn and t≥0,(16)μ1x≤Vx≤μ2x,LVx≤-ηVxγ,then the origin of SINNs (10) are stochastically finite-time stable, and E[T0]<E(V(x0))1-γ/η(1-γ).Lemma 8 (see [9]).
Ifa1,a2,…,an are positive number and 0<r<p, then(17)∑i=1naip1/p≤∑i=1nair1/r.


## 2.1. Systems Description

Firstly, the inertial neural networks (INNs) without time-delay are considered, which is described as follows:(1)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+Iit,i=1,2,…,nwhere xi(t) is the state of the i-th neuron; the second derivative d2xi(t)/dt2 is the inertial term of INNs (1). ai>0, bi>0 are constants. cij denotes the connection weight between the i-th neuron and the j-th neuron. fj(·) stands for activation function of the j-th neuron with fj(0)=0(j=1,2,…,n). Ii(t) is the external input on the i-th neuron.The initial conditions of INNs (1) are(2)xi0=φi0,dxi0dt=ψi0,i=1,2,…,n,where φi(0) and ψi(0) are real-valued continuous functions.Suppose that the external inputIi(t) is subject to the environmental noise and is described by Ii(t)=ui(t)+βi(t,xi(t))ω˙i(t), where ui(t) is known as the control input and ωi(t) is a one-dimensional white noise, which is also called Brown motion defined on a complete probability space (Ω,F,P) and satisfied with(3)Eωit=0,Eωi2t=dt,Eωitωjt=0i≠j,and βi(t,xi(t)) is the intensity function of the noise.Then INNs (1) can be written the following stochastic inertial neural networks (SINNs):(4)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+uit+βit,xitω˙it.


## 2.2. Problems Formulation

In general, making use of the variable transformation,(5)yit=dxitdt+ξixit,i=1,2,…,n,then the SINNs (4) can be rewritten as(6)dxit=-ξixit+yitdt,dyit=-a~iyit-b~ixit+∑j=1ncijfjxjt+uitdt+βit,xitdωit,and the initial conditions are given as(7)xi0=φi0,yi0=-ξi0φi0+ψi0,i=1,2,…,n,where a~i=ai-ξi, b~i=bi+ξi(ξi-ai).Moreover, the controllerνi(t) is considered; we have the following SINNs:(8)dxit=-ξixit+yit+νitdt,dyit=-a~iyit-b~ixit+∑j=1ncijfjxjt+uitdt+βit,xitdωit,i=1,2,…,n.Denote(9)xt=x1t,x2t,…,xntT,yt=y1t,y2t,…,yntT,Ξ=diag⁡ξ1,ξ2,…,ξn,νt=ν1t,ν2t,…,νntT,A=diag⁡a~1,a~2,…,a~n,B=diag⁡b~1,b~2,…,b~n,C=cijn×n,βt,xt=β1t,x1t,β2t,x2t,…,βnt,xntT,ωt=ω1t,ω2t,…,ωnt,ut=u1t,u2t,…,untT.Thus, the SINNs (8) can be written in vector form as(10)dxt=-Ξxt+yt+νtdt,dyt=-Ayt-Bxt+Cfxt+utdt+βt,xtdωt.The control inputs to be designed are of the following form:(11)νt=-K1xt-K2sign⁡xtxtμ,ut=-K3yt-K4sign⁡ytytμ,where sign(x(t))=(sign(x1(t)),sign(x2(t)),…, sign(xn(t)))T, sign(y(t))=(sign(y1(t)),sign(y2(t)),…, sign(yn(t)))T, and K1, K2,K3, K4 are the control gain matrices to be determined. μ is a positive constant with 0<μ<1.Remark 1.
There are three cases for the value ofμ. If 0<μ<1, the controllers u(t),ν(t) are continuous functions with respect to x and y, 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 μ=0, u(t),ν(t) turn to be discontinuous ones, which have been studied in [31, 32]. If μ=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 2.
In fact, the control gain matricesK1, K2, K3, K4 in the controllers ν(t) and u(t) play different roles in ensuring the finite-time stability of the SINNs (10) with (11), where K1 and K3 are used to guarantee the Lyapunov stability of the SINNs (10). And the convergence to zero of the SINNs (10) is determined by K2 and K4.To achieve our main results, some assumptions, lemmas, and definitions are necessary to introduce firstly.Assumption 3.
The nonlinear activation functionf satisfies f(0)=0, and there exist some constants m1i-,m1i+(i=1,2,…,n), such that(12)M1-≤fx1-fx2x1-x2≤M1+hold for all x1,x2∈R and x1≠x2, where M1-=diag{m11-,m12-,…,m1n-}, M1+=diag{m11+,m12+,…, m1n+}.Remark 4.
If we choosem1i=max⁡m1i-,m1i+, the inequalities in Assumption 3 can be written as(13)fx1-fx2x1-x2≤M1,where M1=diag{m11,m12,…,m1n}, which has been considerably studied.Assumption 5.
The intensity functionβ(t,x(t)) is a continuous function and is supposed to satisfy that(14)traceβTt,xtβt,xt≤xTtM2TM2xt, where M2 is a known matrix with appropriate dimensions.Definition 6.
The SINNs (10) are said to be finite-time stabilizable by the controller (11); that is, the SINNs (10) are finite-time stable if, for any initial state x(0), y(0), there exists a finite-time function T0 such that (15)Ext=Eyt=0,∀t≥T0, where T0=T0(x(0),y(0),ω)=inf{T≥0:x(t)=y(t)=0,∀t≥T} is called the stochastic settling time function satisfying E[T0]<∞.Lemma 7 (see [35]).
Suppose that SINNs (10) admit a unique solution. If there exist a C2 function V:Rn→R+, K∞ class functions μ1 and μ2, and positive real constant η>0 and 0<γ<1, such that for all x∈Rn and t≥0,(16)μ1x≤Vx≤μ2x,LVx≤-ηVxγ,then the origin of SINNs (10) are stochastically finite-time stable, and E[T0]<E(V(x0))1-γ/η(1-γ).Lemma 8 (see [9]).
Ifa1,a2,…,an are positive number and 0<r<p, then(17)∑i=1naip1/p≤∑i=1nair1/r.


## 3. Main Results

### 3.1. Finite-Time Stabilization Feedback Controller Design without Time-Delay

Theorem 9.
The controlled systems (10) with (11) are finite-time stable, if there exist some positive-definite matrices P1,P2,M3∈Rn×n and some known constant matrices M1,M2 with compatible dimensions, such that(18)Θ11Θ12∗Θ22<0,Moreover, the upper bound of the stochastic settling time for stabilization can be estimated as E{T0}≤λ2(Ex(0)1-μ+Ey(0)1-μ)/λ1(1-μ),
where(19)Θ11=-P1Ξ-ΞTP1-P1K1-K1TP1+M1TM3-1M1+M2TP2M2,Θ12=P1-BTP2,Θ22=-P2A-ATP2-P2K3-K3TP2+P2CM3CTP2,λ1=min⁡λminP1K2,λminP2K4,λ2=max⁡λmaxP1,λmaxP2.Proof.
Taking controller (11) into SINNs (10), it follows that(20)dxt=-Ξ+K1xt+yt-K2sign⁡xtμdt,dyt=-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμdt+βt,xtdωt.
Next, we will prove that system (20) is finite-time stable in the sense of Definition 6.
Construct a Lyapunov function as(21)Vt=xTtP1xt+yTtP2yt,where P1 and P2 are positive definite matrices. Then, calculate the time derivative of V(t) along the trajectories of systems (20); we get(22)dVt=LVtdt+2yTtP2βt,xtdωt,where(23)LVt=2xTtP1-Ξ+K1xt+yt-K2sign⁡xtxtμ+2yTtP2-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμ+traceβTt,xtP2βt,xt=-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ-2yTtP2A+K3yt+2YTP2Cfxt+traceβTt,xtP2βt,xt.
From (13), we have(24)2yTtP2Cfxt=yTtP2Cfxt+fTxtCTP2yt≤yTtP2CM3CTP2yt+fTxtM3-1fxt≤yTtP2CM3CTP2yt+xTtM1TM3-1M1xt.
From (14), we have(25)traceβTt,xtP2βt,xt≤xTtM2TP2M2xt.Combining (23)-(25) and condition (18), one can follow that(26)LVt≤xTt-2P1Ξ+K1+M1TM3-1M1+M2TP2M2xt+2xTtP1-BTP2yt+yTt-2P2A+K3+P2CM3CTP2yt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤xTtyTtΘ11Θ12∗Θ22xtyt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤-λminP1K2+K2TP1∑i=1nxitμ+1-λminP2K4+K4TP2∑i=1nyitμ+1.
Due to0<μ<1, together with Lemma 8, one has(27)∑i=1nxitμ+11/μ+1≥∑i=1nxit21/2,and then(28)∑i=1nxitμ+1≥∑i=1nxit2μ+1/2=xTtxtμ+1/2.Similarly, we have (29)∑i=1nyitμ+1≥∑i=1nyit2μ+1/2=yTtytμ+1/2.
So, we have(30)LVt≤-λminP1K2+K2TP1xTtxtμ+1/2-λminP2K4+K4TP2yTtytμ+1/2.
Now, taking the expectations on both sides of (22), and letting λ1=min{λmin(P1K2),λmin(P2K4)}, λ2=max{λmax(P1),λmax(P2)}, we can get(31)EdVt≤-2λ1ExTtxtμ+1/2+yTtytμ+1/2≤-2λ1λ2-μ+1/2EVtμ+1/2,EVμ+1/20=EV0μ+1/2.
From Lemma7, we get that the controlled systems (20) are finite-time stable, and the upper bounded stochastic settling time can be estimated by(32)ET0=λ2μ+1/2EV01-μ/22λ11-μ/2≤λ2μ+1/2λ21-μ/2Ex021-μ+Ey021-μλ11-μ=λ2Ex01-μ+Ey01-μλ11-μ.
This completes the proof.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 matricesX1, X2, M3, matrices Y1,Y2 with appropriate dimensions, for fixed control gain matrices K2 and K4, such that(33)Φ11Φ12X1M1TX1M2T0∗Φ2200CM3∗∗-M300∗∗∗-X20∗∗∗∗-M3<0,where (34)Φ11=-ΞX1-X1ΞT-Y1-Y1T,Φ12=X2-X1BT,Φ22=-AX2-X2AT-Y2-Y2T,then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (4) and the control gain matrices K1=Y1X1-1, K3=Y2X2-1.Proof.
SettingP1-1=X1, P2-1=X2, K1X1=Y1, K3X2=Y2, (18) can be written as(35)Θ111Θ112∗Θ122<0,where (36)Θ111=-X1-1Ξ-ΞTX1-1-X1-1K1-K1TX1-1+M1TM3-1M1+M2TX2-1M2,Θ112=X1-1-BTX2-1,Θ122=-X2-1A-ATX2-1-X2-1K3-K3TX2-1+X2-1CM3CTX2-1.Then, left- and right-multiplying inequality (35) by the block-diagonal matrix diag{X1,I}, which follows(37)X100IΘ111Θ112∗Θ122X100I=Θ211Θ212∗Θ222<0,where (38)Θ211=-ΞX1-X1ΞT-K1X1-X1K1T+X1M1TM3-1M1X1+X1M2TX2-1M2X1,Θ212=I-X1BTX2-1,Θ222=-X2-1A-ATX2-1-X2-1K3-K3TX2-1+X2-1CM3CTX2-1,and left- and right-multiplying inequality (37) by the block-diagonal matrix diag{I,X1}, we can obtain(39)I00X1Θ211Θ212∗Θ222I00X1=Θ311Θ312∗Θ322<0,where (40)Θ311=-ΞX1-X1ΞT-K1X1-X1K1T+X1M1TM3-1M1X1+X1M2TX2-1M2X1,Θ312=X2-X1BT,Θ322=-AX2-X2AT-K3X2-X2K3T+CM3CT,By Schur complement, (33) implies the above inequality (39) holds. This completes the proof.


### 3.2. Finite-Time Stabilization Feedback Controller Design with Time-Delay

In the above section, we discussed the finite-time 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 [36–38]. 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.Consider the following SINNs with time-delay,(41)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+∑j=1ncijfjxjt+∑j=1ndijfjxjt-hjt+uit+βit,xitω˙it,where hj(t) is the time-varying delay of j-th neuron with 0≤hj(t)≤h.Denote(42)D=dijn×n,ht=h1t,h2t,…,hntT. Then we have(43)dxt=-Ξxt+yt+νtdt,dyt=-Ayt-Bxt+Cfxt+Dfxt-ht+utdt+βt,xtdωt.The nonlinear delay-feedback controller is designed as the following form:(44)νt=-K1xt-K2signxtxtμ,ut=-K3yt-K4signytytμ-DM1signxtxt-ht,where K1, K2, K3, K4 are gain matrices to be determined, and M1=diagm11,m12,…,m1n, m1i=max⁡m1i-,m1i+.Theorem 11.
The SINNs with time-delay (43) with (44) are finite-time stable, if there exist some positive-definite matrices P1,P2∈Rn×n such that(45)Θ11Θ12∗Θ22<0,where (46)Θ11=-P1Ξ-ΞTP1-P1K1-K1TP1+M1TM3-1M1+M2TP2M2,Θ12=P1-BTP2,Θ22=-P2A-ATP2-P2K3-K3TP2+P2CM3CTP2.Moreover, the upper bound of the stochastic setting time for stabilization can be estimated as E{T0}≤λ2(E{x(0)}1-μ+E{y(0)}1-μ)/λ1(1-μ) with λ1=min{λmin(P1K2),λmin(P2K4)} and λ2=max{λmax(P1),λmax(P2)}.Proof.
Construct a Lyapunov function:(47)Vt=xTtP1xt+yTtP2yt.Calculating the Ito^ differential of V(t) along with (43), we can obtain (48)LVt=2xTtP1-Ξ+K1xt+yt-K2sign⁡xtxtμ+2yTtP2-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμ+traceβTt,xtP2βt,xt=-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4signytytμ-2yTtP2A+K3yt+2YTP2Cfxt+2yTP2D×fxt-ht-M1signxtxt-ht+traceβTt,xtP2βt,xt≤-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2signxtxtμ-2yTtP2K4signytytμ-2yTtP2A+K3yt+2YTP2Cfxt+traceβTt,xtP2βt,xt.
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 Theorem10, we have the following result.Theorem 12.
If there exist some positive define matricesX1, X2, matrices Y1, Y2 with appropriate dimensions, for fixed control gain matrices K2 and K4, such that(49)Φ11Φ12X1M1TX1M2T0∗Φ2200CM3∗∗-M300∗∗∗-X20∗∗∗∗-M3<0,where (50)Φ11=-ΞX1-X1ΞT-Y1-Y1T,Φ12=X2-X1BT,Φ22=-AX2-X2AT-Y2-Y2T, then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (41) and the control gain matrices K1=Y1X1-1, K3=Y2X2-1.


## 3.1. Finite-Time Stabilization Feedback Controller Design without Time-Delay

Theorem 9.
The controlled systems (10) with (11) are finite-time stable, if there exist some positive-definite matrices P1,P2,M3∈Rn×n and some known constant matrices M1,M2 with compatible dimensions, such that(18)Θ11Θ12∗Θ22<0,Moreover, the upper bound of the stochastic settling time for stabilization can be estimated as E{T0}≤λ2(Ex(0)1-μ+Ey(0)1-μ)/λ1(1-μ),
where(19)Θ11=-P1Ξ-ΞTP1-P1K1-K1TP1+M1TM3-1M1+M2TP2M2,Θ12=P1-BTP2,Θ22=-P2A-ATP2-P2K3-K3TP2+P2CM3CTP2,λ1=min⁡λminP1K2,λminP2K4,λ2=max⁡λmaxP1,λmaxP2.Proof.
Taking controller (11) into SINNs (10), it follows that(20)dxt=-Ξ+K1xt+yt-K2sign⁡xtμdt,dyt=-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμdt+βt,xtdωt.
Next, we will prove that system (20) is finite-time stable in the sense of Definition 6.
Construct a Lyapunov function as(21)Vt=xTtP1xt+yTtP2yt,where P1 and P2 are positive definite matrices. Then, calculate the time derivative of V(t) along the trajectories of systems (20); we get(22)dVt=LVtdt+2yTtP2βt,xtdωt,where(23)LVt=2xTtP1-Ξ+K1xt+yt-K2sign⁡xtxtμ+2yTtP2-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμ+traceβTt,xtP2βt,xt=-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ-2yTtP2A+K3yt+2YTP2Cfxt+traceβTt,xtP2βt,xt.
From (13), we have(24)2yTtP2Cfxt=yTtP2Cfxt+fTxtCTP2yt≤yTtP2CM3CTP2yt+fTxtM3-1fxt≤yTtP2CM3CTP2yt+xTtM1TM3-1M1xt.
From (14), we have(25)traceβTt,xtP2βt,xt≤xTtM2TP2M2xt.Combining (23)-(25) and condition (18), one can follow that(26)LVt≤xTt-2P1Ξ+K1+M1TM3-1M1+M2TP2M2xt+2xTtP1-BTP2yt+yTt-2P2A+K3+P2CM3CTP2yt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤xTtyTtΘ11Θ12∗Θ22xtyt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤-2xTtP1K2sign⁡xtxtμ-2yTtP2K4sign⁡ytytμ≤-λminP1K2+K2TP1∑i=1nxitμ+1-λminP2K4+K4TP2∑i=1nyitμ+1.
Due to0<μ<1, together with Lemma 8, one has(27)∑i=1nxitμ+11/μ+1≥∑i=1nxit21/2,and then(28)∑i=1nxitμ+1≥∑i=1nxit2μ+1/2=xTtxtμ+1/2.Similarly, we have (29)∑i=1nyitμ+1≥∑i=1nyit2μ+1/2=yTtytμ+1/2.
So, we have(30)LVt≤-λminP1K2+K2TP1xTtxtμ+1/2-λminP2K4+K4TP2yTtytμ+1/2.
Now, taking the expectations on both sides of (22), and letting λ1=min{λmin(P1K2),λmin(P2K4)}, λ2=max{λmax(P1),λmax(P2)}, we can get(31)EdVt≤-2λ1ExTtxtμ+1/2+yTtytμ+1/2≤-2λ1λ2-μ+1/2EVtμ+1/2,EVμ+1/20=EV0μ+1/2.
From Lemma7, we get that the controlled systems (20) are finite-time stable, and the upper bounded stochastic settling time can be estimated by(32)ET0=λ2μ+1/2EV01-μ/22λ11-μ/2≤λ2μ+1/2λ21-μ/2Ex021-μ+Ey021-μλ11-μ=λ2Ex01-μ+Ey01-μλ11-μ.
This completes the proof.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 matricesX1, X2, M3, matrices Y1,Y2 with appropriate dimensions, for fixed control gain matrices K2 and K4, such that(33)Φ11Φ12X1M1TX1M2T0∗Φ2200CM3∗∗-M300∗∗∗-X20∗∗∗∗-M3<0,where (34)Φ11=-ΞX1-X1ΞT-Y1-Y1T,Φ12=X2-X1BT,Φ22=-AX2-X2AT-Y2-Y2T,then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (4) and the control gain matrices K1=Y1X1-1, K3=Y2X2-1.Proof.
SettingP1-1=X1, P2-1=X2, K1X1=Y1, K3X2=Y2, (18) can be written as(35)Θ111Θ112∗Θ122<0,where (36)Θ111=-X1-1Ξ-ΞTX1-1-X1-1K1-K1TX1-1+M1TM3-1M1+M2TX2-1M2,Θ112=X1-1-BTX2-1,Θ122=-X2-1A-ATX2-1-X2-1K3-K3TX2-1+X2-1CM3CTX2-1.Then, left- and right-multiplying inequality (35) by the block-diagonal matrix diag{X1,I}, which follows(37)X100IΘ111Θ112∗Θ122X100I=Θ211Θ212∗Θ222<0,where (38)Θ211=-ΞX1-X1ΞT-K1X1-X1K1T+X1M1TM3-1M1X1+X1M2TX2-1M2X1,Θ212=I-X1BTX2-1,Θ222=-X2-1A-ATX2-1-X2-1K3-K3TX2-1+X2-1CM3CTX2-1,and left- and right-multiplying inequality (37) by the block-diagonal matrix diag{I,X1}, we can obtain(39)I00X1Θ211Θ212∗Θ222I00X1=Θ311Θ312∗Θ322<0,where (40)Θ311=-ΞX1-X1ΞT-K1X1-X1K1T+X1M1TM3-1M1X1+X1M2TX2-1M2X1,Θ312=X2-X1BT,Θ322=-AX2-X2AT-K3X2-X2K3T+CM3CT,By Schur complement, (33) implies the above inequality (39) holds. This completes the proof.


## 3.2. Finite-Time Stabilization Feedback Controller Design with Time-Delay

In the above section, we discussed the finite-time 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 [36–38]. 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.Consider the following SINNs with time-delay,(41)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+∑j=1ncijfjxjt+∑j=1ndijfjxjt-hjt+uit+βit,xitω˙it,where hj(t) is the time-varying delay of j-th neuron with 0≤hj(t)≤h.Denote(42)D=dijn×n,ht=h1t,h2t,…,hntT. Then we have(43)dxt=-Ξxt+yt+νtdt,dyt=-Ayt-Bxt+Cfxt+Dfxt-ht+utdt+βt,xtdωt.The nonlinear delay-feedback controller is designed as the following form:(44)νt=-K1xt-K2signxtxtμ,ut=-K3yt-K4signytytμ-DM1signxtxt-ht,where K1, K2, K3, K4 are gain matrices to be determined, and M1=diagm11,m12,…,m1n, m1i=max⁡m1i-,m1i+.Theorem 11.
The SINNs with time-delay (43) with (44) are finite-time stable, if there exist some positive-definite matrices P1,P2∈Rn×n such that(45)Θ11Θ12∗Θ22<0,where (46)Θ11=-P1Ξ-ΞTP1-P1K1-K1TP1+M1TM3-1M1+M2TP2M2,Θ12=P1-BTP2,Θ22=-P2A-ATP2-P2K3-K3TP2+P2CM3CTP2.Moreover, the upper bound of the stochastic setting time for stabilization can be estimated as E{T0}≤λ2(E{x(0)}1-μ+E{y(0)}1-μ)/λ1(1-μ) with λ1=min{λmin(P1K2),λmin(P2K4)} and λ2=max{λmax(P1),λmax(P2)}.Proof.
Construct a Lyapunov function:(47)Vt=xTtP1xt+yTtP2yt.Calculating the Ito^ differential of V(t) along with (43), we can obtain (48)LVt=2xTtP1-Ξ+K1xt+yt-K2sign⁡xtxtμ+2yTtP2-A+K3yt-Bxt+Cfxt-K4sign⁡ytytμ+traceβTt,xtP2βt,xt=-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2sign⁡xtxtμ-2yTtP2K4signytytμ-2yTtP2A+K3yt+2YTP2Cfxt+2yTP2D×fxt-ht-M1signxtxt-ht+traceβTt,xtP2βt,xt≤-2xTtP1Ξ+K1xt+2xTtP1yt-2yTtP2Bxt-2xTtP1K2signxtxtμ-2yTtP2K4signytytμ-2yTtP2A+K3yt+2YTP2Cfxt+traceβTt,xtP2βt,xt.
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 Theorem10, we have the following result.Theorem 12.
If there exist some positive define matricesX1, X2, matrices Y1, Y2 with appropriate dimensions, for fixed control gain matrices K2 and K4, such that(49)Φ11Φ12X1M1TX1M2T0∗Φ2200CM3∗∗-M300∗∗∗-X20∗∗∗∗-M3<0,where (50)Φ11=-ΞX1-X1ΞT-Y1-Y1T,Φ12=X2-X1BT,Φ22=-AX2-X2AT-Y2-Y2T, then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (41) and the control gain matrices K1=Y1X1-1, K3=Y2X2-1.


## 4. Illustrative Example

Consider the following stochastic inertial neural networks with time-delay:(51)d2xitdt2=-aidxitdt-bixit+∑j=1ncijfjxjt+∑j=1ncijfjxjt+∑j=1ndijfjxjt-hjt+βit,xitω˙it,which are equivalent to the following vector form:(52)dxt=-Ξxt+yt+νtdt,dyt=-Ayt-Bxt+Cfxt+Dfxt-ht+utdt+βt,xtdωt,where (53)Ξ=1003,A=2003,B=4006,C=2-5-1-3,D=1-1-1-2,βt,xt=sin⁡x1+x2002cos⁡2x1-x2,fxt=tanh⁡x1tanh⁡x2,ht=0.25sin⁡t+0.750.25sin⁡t+0.75,M1=1001,M2=112-2.Setting the initial valuesx(0)=(-0.15,0.2)T, y(0)=(0.3,0.3)T, 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 x(0) and y(0) and satisfy x(0)∈(-1,1), y(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.Figure 1
The state trajectories for open-loop SINNs (52) with initial condition x(0)=(-0.15,0.2)T, y(0)=(0.3,0.3)T.Figure 2
The phrase trajectories for open-loop SINNs (52) with initial condition x(0)=(-0.15,0.2)T, y(0)=(0.3,0.3)T.Figure 3
The state trajectories for open-loop SINNs (52) with 10 different initial conditions x(0)∈(-1,1), y(0)∈(-3,3).Figure 4
The phrase trajectories for open-loop SINNs (52) with 10 different initial conditions x(0)∈(-1,1), y(0)∈(-3,3).Hence, we need to design the delay-feedback controller as (44) for system (51), where the parameter μ is chosen as 0.6, and the initial values x(0)=(-1,1)T, y(0)=(3,-3)T,K2=5221, K4=3224. The solution of (49) is derived by resorting to Matlab LMI Control Toolbox: (54)X1=0.07930.02420.02420.0712,X2=0.57050.04920.04920.6263,M3=0.27660.05850.05850.0399,Y1=0.3609-9.91999.87910.2501,Y2=-0.5432-5.94695.4102-1.0974,K1=52.5297152.1518137.7642-43.3383,K3=-0.1341-9.48519.6993-2.5141.We can getP1=14.0697-4.7821-4.782115.6703, P2=1.7648-0.1386-0.13861.6076, T0≤3.2211. The state trajectories and phrase trajectories of close-loop system are shown in Figures 5 and 6, respectively.Figure 5
The state trajectories for closed-loop SINNs (52) with initial condition x(0)=(-1,1)T, y(0)=(-3,3)T.Figure 6
The phrase trajectories for closed-loop SINNs (52) with initial condition x(0)=(-1,1)T, y(0)=(-3,3)T.In order to make the result of the simulation more convincing, we take 100 sets of numbers randomly as the initial values ofx(0) and y(0) and satisfy x(0)∈(-1,1), y(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 x(0)∈(-10,10), y(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.Figure 7
The state trajectories for closed-loop SINNs (52) with 100 different initial conditions x(0)∈(-1,1), y(0)∈(-3,3).Figure 8
The phrase trajectories for closed-loop SINNs (52) with 100 different initial conditions x(0)∈(-1,1), y(0)∈(-3,3).Figure 9
The state trajectories for closed-loop SINNs (52) with 100 different initial conditions x(0)∈(-10,10), y(0)∈(-30,30).Figure 10
The phrase trajectories for closed-loop SINNs (52) with 100 different initial conditions x(0)∈(-10,10), y(0)∈(-30,30).


## 5. Conclusions

In this work, by constructing a proper Lyapunov function, the finite-time stabilization problem has been addressed for stochastic inertial neural networks with or without time-delay. 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.



---

*Source: 2939425-2018-10-09.xml*