# Stabilization of a Class of Stochastic Systems  with Time Delays

Authors: Jian Wang; Cuixia Li
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The problem of exponential stability is investigated for a class of stochastic time-delay systems. By using the decomposition technique and Lyapunov stability theory, two improved exponential stability criteria are derived. Finally, a numerical example is given to illustrate the effectiveness and the benefit of the proposed method.


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## Body

## 1. Introduction

In fact, time delay constantly occurs in the real world, which results in instability of systems. Thus, the stability problem for time-delay systems has been studied for many years [1–8]. On the other hand, stochastic modelling has come to play an important role in many fields of science or industry. Stability analysis for stochastic systems has become increasingly meaningful. A number of results have appeared in the literature [9–19]. For instance, in [11], the author provided the criteria for the stability of a class of stochastic systems by Lyapunov theory.It’s worth noting that, the grey systems can be established when parameters are evaluated by grey numbers (see [20]). Until now, there have been a few papers tackling the stability of the systems; some important and innovative results are obtained [20–23]. In [23], the authors provided the delay-dependent criteria for exponential robust stability in the forms of nonlinear matrix inequalities and linear matrix inequalities.In this paper, we deal with the exponential stability for the time-delay grey stochastic systems. By using the method of [21–23] and Lyapunov stability theory, two improved criteria of mean-square exponential stability are proposed. At last, a numerical example is given to verify the criteria.Notations. Rn denotes the n dimensional Euclidean space, the superscript “T” denotes matrix transposition, and the notation X≥Y, where X and Y are symmetric matrices, means that X-Y is positive semidefinite. Let (Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). Let w(t)=(w1(t),…,wm(t))T be an m-dimensional Brownian motion defined on the probability space. Let τ>0 and C([-τ,0];Rn) be the family of continuous functions φ from [-τ,0] to Rn. Let LF02([-τ,0];Rn) be the family of all F0-measurable C([-τ,0];Rn)-valued random variables ξ={ξ(θ):-τ≤θ≤0} such that sup⁡-τ≤θ≤0⁡E|ξ(θ)|2<∞.


## 2. Problem Formulation

Consider(1)dx(t)=[Ax(t)+Bx(t-τ)+Du(t)]dt+σ(x(t),x(t-τ),t)dW(t),t≥0,x0=ξ,ξ∈LF02([-τ,0];Rn),-τ≤t≤0.Definition 1.
If there is at least a grey matrix among matricesA,B, and D of system (1), then (1) is called grey stochastic time-delay system.
Hence(2)dx(t)=[A(⊗)x(t)+B(⊗)x(t-τ)+Du(t)]dt+σ(x(t),x(t-τ),t)dW(t),t≥0,x0=ξ,ξ∈LF02([-τ,0];Rn),-τ≤t≤0,
where A(⊗) and B(⊗) are grey matrices and A(⊗)=(⊗ija), B(⊗)=(⊗ijb).
Clearly, if matricesA(⊗) and B(⊗) are replaced by the deterministic matrices A and B, the grey system (2) becomes system (1). The equations
(3)[La,Ua]={A(⊗^)=(aij):aij_≤aij≤aij¯,i,j=1,2,…,n},[Lb,Ub]={B(⊗^)=(bij):bij_≤bij≤bij¯,i,j=1,2,…,n}
are said to be the continuous matrix-covered sets of A(⊗) and B(⊗).Definition 2 (see [20]).
System (2) is said to be robustly exponentially stable in the mean square, if, for all ξ∈LF02([-τ,0];Rn) and arbitrary matrices A(⊗^)∈[La,Ua], B(⊗^)∈[Lb,Ub], there exist scalars r>0 and C>0, such that
(4)E|x(t,ξ)|2≤Ce-rtsup⁡-τ≤θ≤0E|ξ(θ)|2,t≥0.
In addition, the following assumptions are made on the system (2).(H1)
H:Rn×Rn×R+→Rn×n.(H2)
Supposing there exist scalarsα≥0, β≥0, such that, for (x,y,t)∈H:Rn×Rn×R+, the inequality holds, Trace[σT(x,y,t)σ(x,y,t)]≤α|x|2+β|y|2.
Before giving the main results, we first present Lemmas3 and 4, which are important for the proof of main theorems.Lemma 3 (see [20]).
For arbitrary whitened matrixA(⊗^)∈[La,Ua], it follows that(i)
A(⊗^)=La+ΔA(ii)
0≤ΔA≤Ua-La(iii)
∥A(⊗^)∥=∥La∥+∥Ua-La∥,where La=(aij_)m×n, Ua=(aij¯)m×n, ΔA=(δijr^ij)m×n, δij=aij¯-aij_≥0.Lemma 4 (see [18]).
LetN be a real matrix of appropriate dimensions; for any vectors x,y∈Rn, one has 2xTNy≤εxTx+ε-1yTNTNy.


## 3. Proof of the Main Theorem

In this section, we discuss the exponential stability for system (2); two improved criteria for robust exponential stability in mean square are proposed.Theorem 5.
System (2) is exponentially robustly stable in mean square. If there exist positive scalars ε1, ε2, and ε3, such that
(5)λmax⁡(La+LaT)+k1+k2<0
here
(6)k1=1+ε1+ε1-1∥Ua-La∥2+ε2+α,k2=ε2-1(1+ε3)λmax⁡(LbTLb)+ε2-1(1+ε3-1)∥Ub-Lb∥2+β.
Then, for all ξ∈LF02([-τ,0];Rn), the following inequality holds:
(7)E|x(t,ξ)|2≤(1+k2τerτ)e-rtsup⁡-τ≤θ≤0E|ξ(θ)|2,t≥0,
where r is the unique positive solution of the following equation:
(8)r+λmax⁡(La+LaT)+k1+k2erτ=0.Proof.
First, define Lyapunov-Krasovskii functional as follows:(9)V(x(t),t)=ertxT(t)x(t)+∫-τ0er(t+θ)xT(t+θ)x(t+θ)dθ.
Then, we have
(10)LV(x(t),t)=(r+1)ertxT(t)x(t)-er(t-τ)xT(t-τ)x(t-τ)+ert{2xT(t)A(⊗^)x(t)hhhhi+2xT(t)B(⊗^)x(t-τ)hhhhi+Trace[σT(x(t),x(t-τ),t)hhhhhhhhhhhi{2xT(t)A(⊗^)x(t)×σ(x(t),x(t-τ),t)σT(x(t),x(t-τ),t)]}.
Using Lemmas 3 and 4, we derive
(11)2xT(t)A(⊗^)x(t)≤λmax⁡(La+LaT)xT(t)x(t)+ε1xT(t)x(t)+ε1-1∥Ua-La∥2xT(t)x(t),2xT(t)B(⊗^)x(t-τ)≤ε2xT(t)x(t)+ε2-1(1+ε3)λmax⁡hh×(LbTLb)xT(t-τ)x(t-τ)+ε2-1(1+ε3-1)∥Ub-Lb∥2xT(t-τ)x(t-τ).
By assumption (H2), we can obtain
(12)Trace[σT(x(t),x(t-τ),t)σ(x(t),x(t-τ),t)]≤αxT(t)x(t)+βxT(t-τ)x(t-τ).
Substituting (11)–(12) into (10), we see that
(13)LV(x(t),t)≤[1+r+λmax⁡(La+LaT)+ε1h+ε1-1∥Ua-La∥2+ε2+α]ertxT(t)x(t)+[ε2-1(1+ε3)λmax⁡(LbTLb)hh+ε2-1(1+ε3-1)∥Ub-Lb∥2+β]×ertxT(t-τ)x(t-τ).
Using Itô’s differential formula and integrating both sides, we obtain
(14)EV(x(t),t)≤sup⁡-τ≤θ≤0E|ξ(θ)|2+[1+r+λmax⁡(La+LaT)hh+ε1+ε1-1∥Ua-La∥2+ε2+α]×∫0tersE|x(s)|2ds+[ε2-1(1+ε3)λmax⁡(LbTLb)hh+ε2-1(1+ε3-1)∥Ub-Lb∥2+β]×∫0tersE|x(s-τ)|2ds.
Moreover, we have
(15)∫0tersE|x(s-τ)|2ds≤erτ∫0teruE|x(u)|2du+τerτsup⁡-τ≤θ≤0E|ξ(θ)|2.
Combining (14) with (15) and noting the definitions of k1, k2, we see that
(16)EV(x(t),t)≤(1+k2τerτ)sup⁡-τ≤θ≤0E|ξ(θ)|2+[r+λmax⁡(La+LaT)+k1+k2erτ]×∫0tersE|x(s)|2ds.
Furthermore,
(17)f(r)=r+λmax⁡(La+LaT)+k1+k2erτ,thenf′(r)=1+k2τerτ.
Since f′(r)>0, f(0)=λmax⁡(La+LaT)+k1+k2 and f(+∞)=+∞.
When (5) holds, (8) must have a unique solution r>0.
Hence, we have(18)E|x(t,ξ)|2≤(1+k2τerτ)e-rtsup⁡-τ≤θ≤0E|ξ(θ)|2,t≥0.
This completes the proof of Theorem 5.By a similar approach in [22], whitened system of (2) can be written as
(19)dx(t)=[(A(⊗^)+Lb)x(t)-Lb(x(t)-x(t-τ))h+ΔBx(t-τ)+Du(t)[(A(⊗^)+Lb)x(t)-Lb(x(t)-x(t-τ))]dt+σ(x(t),x(t-τ),t)dW(t).
Since
(20)x(t)-x(t-τ)=∫t+θt[A(⊗^)x(s)+B(⊗^)x(s-τ)+Du(s)]ds+∫t+θtσ(x(s),x(s-τ),s)dW(s),t≥τ,
then, we can introduce
(21)H(x,t)={∫t+θt[A(⊗^)x(s)+B(⊗^)x(s-τ)]ds+∫t+θtσ(x(s),x(s-τ),s)dW(s),it≥τx(t)-x(t+θ),0≤t≤τ.
Combining (19) and (21) together, whitened system of (2) can be rewritten as
(22)dx(t)=[(A(⊗^)+Lb)x(t)-LbH(x,t)hh+ΔBx(t-τ)+Du(t)(A(⊗^)+Lb)x(t)-LbH(x,t)]dt+σ(x(t),x(t-τ),t)dW(t),t≥0,x0=ξ,ξ∈LF02([-τ,0];Rn),-τ≤t≤0.Lemma 6.
For allξ∈LF02([-τ,0];Rn) and t>0, the following inequality holds:
(23)∫0tersE|H(x(s),s)|2ds≤(k1τerτ+k2τe2rτ)∫0tersE|x(s)|2ds+k2τ2e2rτsup⁡-τ≤θ≤0E|ξ(θ)|2+sup⁡-τ≤θ≤0ψ(θ),
where
(24)l1=4τ(∥La∥+∥Ua-La∥)2+2α,l2=4τ(∥Lb∥+∥Ub-Lb∥)2+2β,ψ(θ)=∫0τersE|x(s)-x(s-τ)|2ds.Proof.
First, by the definition ofH(x,t), we can derive
(25)E|H(x(s),s)|2≤2E|∫t+θt[A(⊗^)x(s)+B(⊗^)x(s-τ)]ds|2+2E|∫t+θtσ(x(s),x(s-τ),s)dW(s)|2.
Clearly,
(26)E|∫t+θt[A(⊗^)x(s)+B(⊗^)x(s-τ)]ds|2≤2τ(∥La∥+∥Ua-La∥)2∫t+θtE|x(s)|2ds+2τ(∥Lb∥+∥Ub-Lb∥)2∫t+θtE|x(s-τ)|2ds.
By assumption (H2), we see that
(27)E|∫t+θtσ(x(s),x(s-τ),s)dW(s)|2≤α∫t+θtE|x(s)|2ds+β∫t+θtE|x(s-τ)|2ds.
By (25)–(27) and noting the definitions of k1, k2, we can obtain
(28)E|H(t,x(t))|2≤k1∫t+θtE|x(s)|2ds+k2∫t+θtE|x(s-τ)|2ds,t≥τ.
By (28) and integrating both sides, we have
(29)∫0tersE|H(x(s),s)|2ds≤sup⁡-τ≤θ≤0ψ(θ)+k1∫τters∫s+θsE|x(u)|2duds+k2∫τters∫s+θsE|x(u-τ)|2duds.
Moreover, we can obtain
(30)∫τters∫s+θsE|x(u-τ)|2duds≤τe2rτ∫0tersE|x(s)|2ds+τ2e2rτsup⁡-τ≤θ≤0E|ξ(θ)|2,∫τters∫s+θsE|x(u)|2duds≤τerτ∫0tersE|x(s)|2ds.
Substituting (30) into (29) and noting the definitions of l1, l2, and ψ(θ), (23) holds. The proof of Lemma 6 is completed.By (22) and Lemma 6, another criterion for system (2) will be given.Theorem 7.
System (2) is exponentially robustly stable in mean square. If there exist positive scalars ε1, ε2, and ε3, such that
(31)λmax⁡(La+LaT+Lb+LbT)+m1+m2+m3<0,
here
(32)m1=1+ε1+ε1-1∥Ua-La∥2+ε2+ε3+α,m2=ε3-1∥Ub-Lb∥2+β,m3=ε2-1(l1+l2)τλmax⁡(LbTLb).
Then, for all ξ∈LF02([-τ,0];Rn), the following inequality holds:
(33)E|x(t,ξ)|2≤[+ε2-1λmax⁡(LbTLb)sup⁡-τ≤θ≤0ψ(θ)](1+m2τerτ+ε2-1l2τ2e2rτλmax⁡(LbTLb))×sup⁡-τ≤θ≤0E|ξ(θ)|2+ε2-1λmax⁡(LbTLb)sup⁡-τ≤θ≤0ψ(θ)]e-rt,
where r is the unique positive solution of the following equation:
(34)r+λmax⁡(La+LaT+Lb+LbT)+m1+m2erτ+ε2-1(l1τerτ+l2τe2rτ)λmax⁡(LbTLb)=0.Proof.
Similar to the proof process of Theorem5, we can derive
(35)LV(x(t),t)≤[r+λmax⁡(La+LaT+Lb+LbT)+m1]×ertxT(t)x(t)+m2ertxT(t-τ)x(t-τ)+ε2-1λmax⁡(LbTLb)ert|H(x(t),t)|2.
Using Itô’s differential formula and integrating and by the definition of H(x,t), we obtain
(36)EV(x(t),t)≤[r+λmax⁡(La+LaT+Lb+LbT)+m1+m2ert+ε2-1(l1τerτ+l2τe2rτ)×λmax⁡(LbTLb)]×∫0tersE|x(s)|2ds+[1+m2τerτε2-1l2τ2e2rτλmax⁡(LbTLb)hh+ε2-1l2τ2e2rτλmax⁡(LbTLb)]sup⁡-τ≤θ≤0E|ξ(θ)|2+ε2-1λmax⁡(LbTLb)sup⁡-τ≤θ≤0ψ(θ).
If g(r)=r+λmax⁡(La+LaT+Lb+LbT)+m1+m2erτ+ε2-1(l1τerτ+l2τe2rτ)λmax⁡(LbTLb), then g′(r)=1+m2τerτ+ε2-1(l1τ2erτ+2l2τ2e2rτ)λmax⁡(LbTLb).
Becauseg′(r)>0, g(0)=λmax⁡(La+LaT+Lb+LbT)+m1+m2+m3 and g(+∞)=+∞.
If (31) holds, (34) must have a unique solution r>0.
Therefore, we can easily get(37)EV(x(t),t)≤[1+m2τerτ+ε2-1l2τ2e2rτλmax⁡(LbTLb)]×sup⁡-τ≤θ≤0E|ξ(θ)|2+ε2-1λmax⁡(LbTLb)sup⁡-τ≤θ≤0ψ(θ),
or equivalently (33) holds; the proof of Theorem 7 is completed.


## 4. Examples

Consider a stochastic time-delay system(38)dx(t)=[A(⊗)x(t)+B(⊗)x(t-0.5)]dt+σ(x(t),x(t-0.5),t)dW(t),x0=ξ,ξ∈LF02([-0.5,0];R2),-0.5≤t≤0,
where
(39)La=[-3.350.220.23-3.34];Ua=[-3.150.320.31-3.45]Lb=[-1.150.200.23-1.16];Ub=[-1.120.220.31-1.09].Respectively,La, Ua, Lb, and Ub are the lower bound and upper bound matrices of A(⊗) and B(⊗).In addition,(40)σ(x(t),x(t-0.5),t)=[12x1(t)⁡sin(x2(t-0.5))12x2(t)⁡sin(x1(t-0.5))].
Clearly,
(41)Trace[σT(x(t),x(t-0.5),t)σ(x(t),x(t-0.5),t)]≤0.25x2(t).
By using the method of [21], we can obtain that r=1.7513 or r=1.8162, which indicates that the system (38) is exponentially stable in mean square.


## 5. Conclusion

In this paper, we have investigated a class of grey stochastic systems with time delay; by constructing a suitable Lyapunov-Krasovskii functional combined with Itô’s differential formula, two improved exponential stability criteria are derived. The criteria obtained in this paper are so conveniently verified that the results in this paper should be proved to be very useful in applications.



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