# Globally Asymptotic Stability of Stochastic Nonlinear Systems with Time-Varying Delays via Output Feedback Control

Authors: Mingzhu Song; Wenwen Cheng; Quanxin Zhu; Hongwei Zhou; Hui Wang
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

We address the problem of globally asymptotic stability for a class of stochastic nonlinear systems with time-varying delays. By the backstepping method and Lyapunov theory, we design a linear output feedback controller recursively based on the observable linearization for a class of stochastic nonlinear systems with time-varying delays to guarantee that the closed-loop system is globally asymptotically stable in probability. In particular, we extend the deterministic nonlinear system to stochastic nonlinear systems with time-varying delays. Finally, an example and its simulations are given to illustrate the theoretical results.


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## Body

## 1. Introduction

In recent years, the stochastic nonlinear system has received much attention and has enjoyed a good development, which has been widely applied in many fields such as engineering and finance [1]. Particularly, owing to the reason that a system by the output feedback is more flexible to respond to the information of control systems, the stochastic nonlinear system via the output feedback control has been more widely studied. Marino and Tomei [2] and Battilotti [3] designed linear observer to study the output feedback control for nonlinear systems. As a result, a large number of researchers have been focused on the stability analysis of stochastic nonlinear systems and the design of controller.Since the backstepping method has been introduced in the nonlinear system, the theory of stochastic nonlinear systems has achieved a remarkable development. According to the different Lyapunov function, there are two ways to solve the problem. One is that Pan and Basar [4–6] considered a quadratic Lyapunov function. By using the backstepping method to design the controller, they discussed the risk-sensitive optimal control problem. The other is that Deng and Krstić [7–10] used a quartic Lyapunov function to guarantee that the closed-loop system is globally asymptotically stable in probability. Based on their works, Liu et al. [11–13] employed the quartic Lyapunov function to design the output feedback control for stochastic nonlinear systems. M.-L. Liu and Y.-G. Liu [14] and Chen et al. [15] used such Lyapunov function to study the state feedback stability for stochastic nonlinear systems with time-varying delays. In [16], Du et al. discussed the global output feedback stability for a class of uncertain upper-triangular systems with the input delay. Finite-time stability for stochastic nonlinear systems in strict-feedback form was considered in [17]. More results can be found in [18–20].In this paper, we are concerned with the globally asymptotic stability of stochastic nonlinear system with time-varying delays. We extend the results of the deterministic nonlinear systems in [21] to the stochastic nonlinear systems with time-varying delays by output feedback control and design a Lyapunov-Krasovskii functional to prove the globally asymptotic stability of the closed-system via the linear observer. It is obvious that the form of the linear controller is simpler than that used in [9]. And compared with the C1 output feedback controller obtained by the homogeneous observers in [20], the controller designed by the linear observers in this paper is smooth.The rest of this paper is organized as follows. In Section2, we present some definitions and establish a new inequality, which plays an important role in the proof of our main results. In Section 3, we construct a Lyapunov-Krasovskii functional and design a linear output feedback control to prove the globally asymptotic stability based on the linear observers. In Section 4, we use an example to illustrate our theoretical results. Finally, we conclude this paper with some general remarks.


## 2. Preliminaries

In this section, we will give the following notations, definitions, and some preliminary lemmas.R+ denotes the set of all nonnegative real numbers; Rn denotes the real n-dimensional space; Trace{X} denotes the trace for square matrix X; X denotes the Euclidean norm of a vector X. C([-d,0];Rn) denotes the space of continuous Rn-valued functions on [-d,0] endowed with the norm · defined by f=supx∈[-d,0]fx for f∈C([-d,0];Rn); CF0b([-d,0];Rn) denotes the family of all F0-measurable bounded C([-d,0];Rn)-valued random variables ξ={ξ(θ):-d≤θ≤0}. Ci denotes the set of all functions with continuous ith partial derivatives; C2,1(Rn×[-d,∞];R+) denotes the family of all nonnegative functions V(x,t) on Rn×[-d,∞) which are C2 in x and C1 in t; C2,1 denotes the family of all functions which are C2 in the first argument and C1 in the second argument. K denotes the set of all functions: R+→R+, which are continuous and strictly increasing and vanish at zero; K∞ denotes the set of all functions which are of class K and unbounded. KL is the set of all functions β(s,t):R+×R+→R+, which are of K for each fixed t and decrease to zero as t→∞ for each fixed s.Consider the following stochastic time-varying delay system:(1)dxt=fxt,xt-dt,tdt+gxt,xt-dt,tdωt,∀t≥0,with initial date {x(θ):-d≤θ≤0}=ξ∈CF0b([-d,0];Rn), where d(t):R+→[0,d] is a Borel measurable function and ω(t) is an r-dimensional standard Wiener process defined on the complete probability space (Ω,F,{Ft}t≥0,P) with Ω being a sample space, F being a filtration, and P being a probability measure. f:Rn×Rn×R+→Rn,  g:Rn×Rn×R+→Rn×r are assumed to be locally Lipschitz in (x(t),x(t-d(t))) uniformly in t and satisfy f(0,0,t)≡0,  g(0,0,t)≡0.Definition 1 (see [22]).
For any givenV(x(t),t)∈C2,1 associated with system (1), the differential operator L is defined as(2)LVxt,t=∂V∂t+∂V∂xTf+12TracegT∂2V∂x2g,where (1/2)TracegT(∂2V/∂x2)g is called as the Hessian term of L.Definition 2 (see [11, 15]).
For system (1), the equilibrium x=0 is said to be globally asymptotically stable in probability, if, for any ε>0, there exists a class of K function γ(·) such that (3)Pxt<γξ≥1-ε,∀t≥0,∀ξ∈CF0b-d,0;Rn∖0,and for any ξ∈CF0b([-d,0];Rn), one has (4)Plimt→∞⁡xt=0=1.Definition 3 (see [22]).
For fixed coordinates(x1,…,xn)∈Rn and real numbers ri>0,  i=1,…,n.
(1) The dilationΔϵ(x) is defined by Δϵ(x)=(ϵr1x1,…,ϵrnxn) for any ϵ>0; r1,…,rn are called the weights of the coordinates. For simplicity, we define dilation weight Δ=(r1,…,rn).
(2) A functionV∈C(Rn,R) is said to be homogeneous of degree τ if there is a real number τ∈R such that V(Δϵ(x))=ϵτV(x1,…,xn) for any x∈Rn∖{0},  ϵ>0.
(3) A vector fieldh∈C(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that hi(Δϵ(x))=ϵτ+rihi(x) for any x∈Rn∖{0},  ϵ>0.
(4) A homogeneousq-norm is defined as xΔ,q=(Σi=1nxiq/ri)1/q for any x∈Rn, where q>1 is a constant. For simplicity, in this paper, we choose q=2 and write xΔ for xΔ,2.Lemma 4 (see [11, 15, 22]).
For system (1), if there exist a function V(x(t),t)∈C2,1Rn×-d,∞;R+, two class K∞ functions α1,α2, and a class K function α3 such that(5)α1xt≤Vxt,t≤α2sup-d≤σ≤0xσ+t,(6)LVxt,t≤-α3xt,then there exists a unique solution on -d,∞ for each x0∈Rn and the equilibrium x(t)=0 is globally asymptotically stable in probability.Lemma 5 (see [22]).
Suppose thatV:R→R is a homogeneous function of degree τ with respect to the dilation weight Δ; then(i)
∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi;(ii)
there is a constantγ¯ such that V(x(t))≤γ¯x(t)Δτ. Moreover, if V(x(t)) is positive definite, then V(x(t))≥γ_x(t)Δτ, where γ_ is a positive constant.Lemma 6 (see [1]).
LetV:D→R be a continuous positive definite on a domain D=Rn that contains the origin. Let Br⊂D for some r>0. Then, there exist class K functions α1 and α2, defined on [0,r], such that(7)α1xt≤Vxt≤α2xtfor all x∈Br. If D=Rn, the functions α1 and α2 will be defined on [0,∞) and the foregoing inequality will hold for all x∈Rn. Moreover, if V(x(t)) is radially unbounded, then α1 and α2 can be chosen to belong to class K∞.Lemma 7.
For any constantsa>0 and b∈R, one has that, for any x,y∈R,(8)-ax4+bxy3≤k1a-1/3b4/3y4,(9)-ax4+bx3y≤k2b4a-3y4,where k1=4-1/3-4-4/3>0 and k2=3/43-3/44>0.Proof.
We first prove (8). Let Z1(x)=-ax4+bxy3, where y is a parameter. Then, we have(10)Z1′x=-4ax3+by3=0,x=b4a1/3y.It is clear that, for any x∈(-∞,b/4a1/3y), Z1′(x)>0 and x∈((b/4a)1/3y,+∞), Z1′(x)<0. With the sufficient condition of extreme value, x=(b/4a)1/3y is the maximum point of function Z1(x). Therefore, we get(11)Z1x≤-ab4a1/3y4+bb4a1/3yy3=4-1/3-4-4/3a-1/3b4/3y4.We now prove (9). Similarly, letting Z2(x)=-ax4+bx3y, where y is a parameter, we have(12)Z2′x=-4ax3+3bx2y=0,x=3by4aorx=0.Obviously, for any x∈(-∞,3by/4a), Z2′(x)>0 and x∈(3by/4a,+∞), Z2′(x)<0. With the sufficient condition of extreme value, x=3by/4a is the maximum point of function Z2(x). Therefore, we obtain(13)Z2x≤-a3by4a4+b3by4a3y=343-344b4a-3y4.Lemma 8 (Cauchy-Schwartz’s inequality).
For any vectorx=(x1,x2,…,xn)∈Rn and y=(y1,y2,…,yn)∈Rn, one has(14)x1y1+x2y2+⋯+xnyn2≤x12+⋯+xn2y12+⋯+yn2.Lemma 9.
For a sequence of numbersa1,a2,…,an∈R+, one has(15)a1+a2+⋯+an≥a12+a22+⋯+an2,a12+a22+⋯+an2n≥a1+a2+⋯+ann≥a1a2⋯ann≥n1/a1+1/a2+⋯+1/an.Lemma 10 (see [17]).
For any given real numbers c, d and any real-valued functionsf(x,y)>0, g(x,y,z)≥0, the following inequality holds:(16)gx,y,zxcyd≤cc+dfx,yxc+d+dc+dgx,y,zc+d/dfx,y-c/dyc+d,where x,y,z∈R. Particularly when one takes f(x,y)=g(x,y,z)=1, c=3, and d=4, then the inequality will become(17)x3y≤34x4+14y4.


## 3. The Output Feedback Model and Control Design

In this section, we will design a linear observer system for a class of stochastic nonlinear systems with time-varying delays. Using the backstepping method, a simple linear controller will be constructed to guarantee that the closed-loop stochastic system is globally asymptotically stable in probability.Consider the following stochastic nonlinear system with time-varying delay:(18)dx1=x2+f1t,x¯1,x¯1t-dtdt+g1Tt,x¯1,x¯1t-dtdw,dx2=x3+f2t,x¯2,x¯2t-dtdt+g2Tt,x¯2,x¯2t-dtdw,⋮dxn=u+fnt,x¯n,x¯nt-dtdt+gnTt,x¯n,x¯nt-dtdw,y=x1,where the initial data {x(θ):-d≤θ≤0}=ξ∈CF0b([-d,0];Rn), d(t):R+→[0,d] is a Borel measurable function and d′(t)≤d′<1; ω is an r-dimensional standard Wiener process defined on the complete probability space (Ω,F,{Ft}t≥0,P). x¯i=(x1,…,xi) is the system state, x¯i(t-d(t))=(x1(t-d(t)),…,xi(t-d(t))) is the time-delayed state vectors, u∈R is the system input, and y∈R is the system output. fi:R+×Ri×Ri→R and gi:R+×Ri×Ri→Rr are assumed to be locally Lipschitz in (x(t),x(t-d(t))) uniformly in t with fi(t,0,0)=0, gi(t,0,0)=0, i=1,…,n.Remark 11.
It should be pointed out that Zhai and Zha [23] considered the global adaptive output feedback control for system (18) without the diffusion terms. And Duan and Xie [24] discussed the globally asymptotic stability for system (18) without time-varying delay. Compared with these works, we focus on the output feedback stabilization of stochastic nonlinear systems with time-varying delay (see system (18)).Assumption 12.
There exist nonnegative constantsl1,l2∈R+ such that(19)fit,x¯i,x¯it-θt≤l1x1+⋯+xi+x1t-θt+⋯+xit-θt,git,x¯i,x¯it-θt≤l2x1+⋯+xi+x1t-θt+⋯+xit-θt.The linear observer system is designed as(20)x^˙1=x^2t+La1x1-x^1,x^˙2=x^3t+L2a2x1-x^1,⋮x^˙n-1=x^nt+Ln-1an-1x1-x^1,x^˙n=u+Lnanx1-x^1,where L≥1 is a constant to be determined and ai>0,  i=1,…,n, are coefficients of the Hurwitz polynomial(21)pt=tn+a1tn-1+⋯+an-1t+an.The observation error εi=(xi-x^i)/Li-1 satisfies(22)dε=L-a110⋯0-a201⋯0⋮⋮⋮⋱⋮-an-100⋯0-an00⋯0εdt+f1f2L⋮fn-1Ln-2fnLn-1dt+g1Tg2TL⋮gn-1TLn-2gnTLn-1dw=LAεdt+Fdt+Gdw,where A is a Hurwitz matrix. Therefore, there is a positive-definite matrix P such that (23)ATP+PA=-I.Theorem 13.
Suppose Assumption12 holds. Then, the equilibrium at origin of closed-loop stochastic nonlinear system (18) and (20) with the linear controller (50) below is globally asymptotically stable in probability and P{limt→∞xt=0}=1. Furthermore, the closed-loop system has a unique solution on [-d,∞) for each ξ∈Rn.Proof.
Notez=(x,x^), xi(t)=xi, and x^i(t)=x^i for simplicity. Consider the following Lyapunov function: V0=((n+1)/2)εTPε2. Then by Lemma 10, a direct computation yields(24)LV0=n+1εTPεLεTATP+PAε+2εTPF+12tr⁡GTn+14PεεTP+2εTPεPG≤-n+1λmin⁡Lε4+2n+1λmax⁡2ε3F+3n+1rrλmax⁡2ε2G2≤-n+1λmin⁡Lε4+3n+121+rrλmax⁡2ε4+n+12λmax⁡2F4+3n+12rrλmax⁡2G4,where λmin⁡ denotes the minimum eigenvalue of the matrix P and λmax⁡ denotes the maximum eigenvalue of the matrix P.
By Lemmas8 and 9 and Assumption 12, one gets(25)F4≤l14n5x14+1L4x24+⋯+1L4n-4xn4+l14n5x14t-θt+1L4x24t-θt+⋯+1L4n-4xn4t-θt,G4≤l24n5x14+1L4x24+⋯+1L4n-4xn4+l24n5x14t-θt+1L4x24t-θt+⋯+1L4n-4xn4t-θt.Substituting (25) into (24), we have(26)LV0≤-n+1λmin⁡Lε4+3n+121+rrλmax⁡2ε4+n+1n52λmax⁡2l14+3rrl24∑i=1nxi4t-θtL4i-4+n+1n52λmax⁡2l14+3rrl24∑i=1nxi4L4i-4=-n+1λmin⁡Lε4+3n+121+rrλmax⁡2ε4+α1∑i=1nxi4L4i-4+α1∑i=1nxi4t-dtL4i-4,where(27)α1=n+1n52λmax⁡2l14+3rrl24.Take(28)W=α11-d′∑i=1n∫t-dttxi4sL4i-4ds;then(29)LV0+W≤-n+1λmin⁡L-3n+121+rrλmax⁡2ε4+α11+11-d′∑i=1nxi4tL4i-4.Substituting xi=x^i+Li-1εi,  i=1,…,n, into (29), one gets(30)LV0+W≤-n+1λminL-c0ε4+c1∑i=1nx^i4L4i-4,where(31)c0=3n+121+rrλmax⁡2+8nα11+11-d′,c1=8α11+11-d′.Next, we will combine the backstepping design method with the mathematical induction to design the linear output feedback control.
Step  1. Construct the Lyapunov function V1=V0+W+(1/4)x^14. A direct calculation gives(32)LV1=LV0+LW+x^13x^2+La1ε1≤-n+1λmin⁡L-c0ε4+c1∑i=1nx^i4L4i-4+x^13x^2+La1x^13ε1≤-nλmin⁡L-c0ε4+c1∑i=1nx^i4L4i-4+k1λmin⁡-1/3a14/3Lx^14+x^13x^2,where k1 is defined in Lemma 7.
Defineξ2=x^2-x^2∗ with x^2∗ being a virtual control. Then, we have(33)LV1≤-nλmin⁡L-c0ε4+c1x^34L8+⋯+x^n4L4n-4+c1+k1λmin⁡-1/3a14/3Lx^14+c1L4ξ2+x^2∗4+x^13ξ2+x^2∗≤-nλmin⁡L-c0ε4+c1x^34L8+⋯+x^n4L4n-4+c1+k1λmin⁡-1/3a14/3Lx^14+8c1L4ξ24+8c1L4x^2∗4+x^13ξ2+x^13x^2∗.Choose the virtual controller(34)x^2∗=-Ln+c1+k1λ1-1/3a14/3x^1=-Lb1x^1,where b1>0 is independent of L. Hence,(35)LV1≤-nλminL-c0ε4+c1x^34L8+⋯+x^n4L4n-4+8c1b14-nLx^14+8c1L4ξ24+x^13ξ2.Stepk. Suppose that, at Step k, there exists a smooth Lyapunov function Vk which is positive definite, radially unbounded, and twice continuously differentiable, satisfying(36)LVk≤-n+1-kλminL-c0ε4-∑j=1k1L4j-4n+1-kL-8c1bj4ξj4+c11L4k+4x^k+24+⋯+1L4n-4x^n4+8c1L4kξk+14+1L4k-4ξk3ξk+1,with a set of virtual controllers x^1∗,x^2∗,…,x^k∗, defined by x^1∗=0,  x^i∗=-Lbi-1ξi-1 with bi-1>0 being independent of the gain constant L, where ξi=x^i-x^i∗,  i=1,…,n.
From the above definition, we have(37)ξk+1=x^k+1+Lbkξk=x^k+1+Lbkx^k+L2bkbk-1x^k-1+⋯+Lkbkbk-1⋯b1x^1.A direct computation yields(38)L14L4kξk+14=1L4kξk+13x^k+2+Lk+1ak+1ε1+Lbk∑i=1k∂ξk∂xix^i+1+Liaiε1=1L4kξk+13x^k+2+Lk+1ak+1ε1+∑i=1kLk-i+1bk⋯biξi+1-Lbiξi+Liaiε1=1L4kξk+13x^k+2+Lk+1d0ε1+Lk+1d1ξ1+Lkd2ξ2+⋯+L2dkξk+Ldk+1ξk+1,where(39)d0=ak+1+bk⋯b1a1+bk⋯b2a2+⋯+bkak,d1=-bk⋯b12,d2=bk⋯b1-bk⋯b22,⋮dk=bkbk-1-bk2,dk+1=bk>0.Obviously, d0,d1,…,dk+1 are suitable real numbers and they are independent of the gain constant L.
Let us consider the following Lyapunov function:(40)Vk+1=Vk+14L4kξk+14.Then, combining (36) and (38), we obtain(41)LVk+1≤-n+1-kλminL-c0ε4-∑j=1k1L4j-4n+1-kL-8c1bj4ξj4+c11L4k+4x^k+24+⋯+1L4n-4x^n4+8c1L4kξk+14+1L4k-4ξk3ξk+1+1L4kξk+13x^k+2+Lk+1d0ε1+Lk+1d1ξ1+⋯+L2dkξk+Ldk+1ξk+1≤-n-kλminL-c0ε4-∑j=1k1L4j-4n-kL-8c1bj4ξj4+c11L4k+8x^k+34+⋯+1L4n-4x^n4+8c1L4k+4ξk+24+8c1L4k+4x^k+2∗4+1L4kξk+13x^k+2∗+1L4kξk+13ξk+2+dk+1+8c1/LL4k-1ξk+14+-λminLε14+1L3k-1d0ξk+13ε1+-Lξ14+d1L3k-1ξk+13ξ1+-1L3ξ24+d2L3kξk+13ξ2+⋯+-1L4k-9ξk-14+dk-1L4k-3ξk+13ξk-1+-1L4k-5ξk4+dkL4k-2ξk+13ξk+1L4k-4ξk3ξk+1.It follows from Lemma 7 that(42)-λminLε14+1L3k-1d0ξk+13ε1≤k1d04/3λmin-1/3ξk+14L4k-1,-Lξ14+d1L3k-1ξk+13ξ1≤k1d14/3ξk+14L4k-1,-1L3ξ24+d2L3kξk+13ξ2≤k1d24/3ξk+14L4k-1,⋮-1L4k-9ξk-14+dk-1L4k-3ξk+13ξk-1≤k1dk-14/3ξk+14L4k-1.In particular, the last term of (41) yields(43)-1L4k-5ξk4+dkL4k-2ξk+13ξk+1L4k-4ξk3ξk+1≤-12L4k-5ξk4+dkL4k-2ξk+13ξk+-12L4k-5ξk4+1L4k-4ξk3ξk+1≤k112-1/3dk4/3+k212-3ξk+14L4k-1.Hence,(44)LVk+1≤-n-kλminL-c0ε4-∑j=1k1L4j-4n-kL-8c1bj4ξj4+c11L4k+8x^k+34+⋯+1L4n-4x^n4+8c1L4k+4ξk+24+8c1L4k+4x^k+2∗4+1L4kξk+13x^k+2∗+1L4kξk+13ξk+2+1L4k-1d0′+d1′+⋯+dk+1′ξk+14,where(45)d0′=k1d04/3λmin-1/3,d1′=k1d14/3,⋮dk-1′=k1dk-14/3,dk′=k112-1/3dk4/3+k212-3,dk+1′=dk+1+8c1.Now, we choose the following linear controller:(46)x^k+2∗=-Ln-k+d0′+d1′+⋯+dk+1′ξk+1=-Lbk+1ξk+1,with bk+1>0 being independent of L.
So it follows from (44) that(47)LVk+1≤-n-kλmin⁡L-c0ε4-∑j=1k+11L4j-4n-kL-8c1bj4ξj4+c11L4k+8x^k+34+⋯+1L4n-4x^n4+8c1L4k+4ξk+24+1L4kξk+13ξk+2.Stepn. Using the inductive argument step by step, at the (n-1)th step, we get(48)LVn-1≤-2λminL-c0ε4-∑j=1n-11L4j-42L-8c1bj4ξj4+8c1L4n-4ξn4+1L4n-8ξn-13ξn.Noting the function Vn=Vn-1+(1/4L4n-4)ξn4, we have(49)LVn≤-2λminL-c0ε4-∑j=1n-11L4j-42L-8c1bj4ξj4+8c1L4n-4ξn4+1L4n-8ξn-13ξn+1L4n-4ξn3x^n+1+1L4n-4Lnd0ε1+Lnd1ξ1+⋯+L2dn-1ξn-1+Ldnξnξn3≤-λminL-c0ε4-∑j=1n-11L4j-4L-8c1bj4ξj4+1L4n-4ξn3x^n+1+8c1L4n-4+1L4n-5dnξn4+1L4n-5d0′+d1′+⋯+dn-1′ξn4≤-λminL-c0ε4-∑j=1n-11L4j-4L-8c1bj4ξj4+1L4n-4ξn3x^n+1+1L4n-5d0′+d1′+⋯+dn-1′+8c1+dnξn4.We now design the linear controller(50)u=x^n+1=-Lbnξn=-Lbnx^n+Lbn-1x^n-1+L2bn-1bn-2x^n-2+⋯+Ln-1bn-1bn-2⋯b1x^1,where bn=1+d0′+d1′+⋯+dn-1′+8c1+dn>0 is a real number independent of the gain parameter L. Thus,(51)LVn≤-α3ε,ξ1,…,ξn,where α3(ε,ξ1,…,ξn):=(λminL-c0)ε4+∑j=1n-1(1/L4j-4)(L-8c1bj4)ξj4+(1/L4n-5)ξn4.
Choosing the gain constantL=max{1,c0/λmin,8c1b14,…,8c1bn-12}, it is obvious that α3(ε,ξ1,…,ξn) is a positive-definite and proper function. Thus, we see that (6) is true.
Now, we prove (5). From the above process, we have constructed the Lyapunov function(52)Vn=V¯+α11-d′∑i=1n∫t-dttxi4sL4i-4ds,where α1 and d′ are positive parameters and(53)V¯=n+12εTPε2+∑i=1nξi44L4i-4.It is easy to verify that V¯ is C2 on z. Suppose that U(z) is continuous, positive definite, and radially unbounded. Then by Lemmas 5 and 6, there exist positive constants γ_ and γ¯ and two class K∞ functions β1 and α11 such that(54)β1z≤Uz≤α11z,γ_zΔ4≤Uz≤γ¯zΔ4.It follows from d(t):R+→[0,d] that(55)α11-d′∑i=1n∫t-dttxi4sL4i-4ds≤γ∫t-dtt∑i=1nxi4sL4i-4ds≤γ¯∫t-dttzsΔ4ds≤γ~∫t-dttα11zsds=γ~∫-dt0α11zσ+tdσ+t≤γ~∫-d0α11zσ+tdσ+t≤γ^sup-d≤σ≤0⁡α11zσ+t≤α22sup-d≤σ≤0⁡zσ+t,where γ=α1/(1-d′), γ~ and γ^ are positive constants, and α22 is a class K∞ function. Noting that(56)zt≤sup-d≤σ≤0⁡zσ+t,so(57)α11zt≤α11sup-d≤σ≤0⁡zσ+t.Defining(58)β2sup-d≤σ≤0⁡zσ+t≔α11sup-d≤σ≤0⁡zσ+t+α22sup-d≤σ≤0zσ+t,one gets(59)β1zt≤Vnzt≤β2sup-d≤σ≤0zσ+t.Therefore, by Lemma 4, the equilibrium of closed-loop nonlinear stochastic system (18), (20), and (50) is globally asymptotically stable and the closed-loop system has a unique solution on [-d,∞) for each ξ∈Rn.Remark 14.
Lettingfi=0  (i=1…,n) and d(t)≡0 in system (18), then the model will reduce to that introduced in [9]. Therefore, our result extends and improves those given in [9].Remark 15.
It should be pointed out that, compared with theC1 output feedback controller in [20], we see that the linear controller (50) based on the observable linearization is a smooth one.Remark 16.
In this paper, combining the backstepping design with mathematical induction, we extend the method in [21] to the stochastic nonlinear system with time-varying delay. Although the globally asymptotic stability of stochastic nonlinear system has been discussed by Liu and Zhang in [12], they did not consider the term of time-varying delay. Therefore, our conclusion is proposed for the first time.Remark 17.
In [22], Liu and Xie considered the state feedback stability of stochastic feedforward nonlinear systems with time-varying delays. They introduced the homogeneous domination approach to construct a state feedback controller. It is generally known that in real lives the output feedback control is easier to achieve than the state feedback control and has great theory significance and utility value. Therefore, in this paper we combine the backstepping method with the mathematical induction to design a Lyapunov-Krasovskii functional and construct a linear output feedback controller for stochastic nonlinear systems with time-varying delay. Moreover, we impose different restricted conditions on functions fi and gi(i=1,…,n) to ensure the global stability of stochastic nonlinear systems with time-varying delays.


## 4. An Example

In this section, we will use a simulation example to illustrate our main results.Consider the following stochastic nonlinear system:(60)dx1=x2+116x1sinx22dt+116cosx2t-dtdw,dx2=udty=x1,where the time-varying delay d(t)=cos(t). Obviously, we have f1x≤(1/16)(x1+x2t-cost), g1x≤(1/16)(x1+x2t-cost), which implies that Assumption 12 holds. Moreover, both of them are locally Lipschitz. The linear observer system is designed as(61)x^˙1=x^2+Lx1-x^1,x^˙2=u+L2x1-x^1with a suitable choice of the parameter L. The observation errors(62)ε1=x1-x^1,ε2=x2-x^2satisfy(63)dε=L-11-10εdt+f1f2Ldt+g1Tg2TLdw=LAεdt+Fdt+Gdw.For the above matrix A, there exists a positive-definite matrix P satisfying ATP+PA=-I, where(64)
P
=
1
-
1
2
-
1
2
3
2
.It is easy to get the corresponding eigenvalues λmin=(5-5)/4 and λmax=(5+5)/4.Consider the Lyapunov-Krasovskii functional(65)V=32εTPε2+α11+sin⁡x∑i=12∫t-cos⁡ttxi4sL4i-4ds+14x^14+14L4x^2+Lb1x^14.Then, we take α1=0.02, b1=2, b2=20, L=160, and the linear control u=-3200(x^2+Lb1x^1). By Theorem 13, we see that closed-loop system (60), (61) and the linear controller u are globally asymptotically stable in probability and there is a unique solution on [-d,∞). (The state response and control input with initial conditions x1(0)=0.5, x2(0)=50, x^1(0)=0.6, and x^2(0)=-100 are presented in Figures 1(a), 1(b), and 2, where hatx1 and hatx2 denote x^1 and x^2.)Figure 1
The state response.
(a)
(b)Figure 2
The control input.


## 5. Conclusion

In this paper, we combine the backstepping method with the mathematical induction to prove that a class of stochastic nonlinear systems with time-varying delays by the output feedback control is globally asymptotically stable in probability. It should be pointed out that we not only extend the results in deterministic nonlinear systems to more complex stochastic case but also add time-varying delays to stochastic nonlinear systems. For the term of time-varying delays, we design a Lyapunov-Krasovskii functional and a linear output feedback controller to render the closed-loop system globally asymptotically stable.



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*Source: 7548156-2016-03-16.xml*