# Input-to-State Stability of Singularly Perturbed Control Systems with Delays

Authors: Yongxiang Zhao; Aiguo Xiao; Li Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the input-to-state stability of singularly perturbed control systems with delays. By using the generalized Halanay inequality and Lyapunov functions, we derive the input-to-state stability of some classes of linear and nonlinear singularly perturbed control systems with delays.


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## Body

## 1. Introduction

The stability properties of control systems are an important research field. The concept of input-to-state stability (ISS) of the control systems was proposed by Sontag [1]. Since then, the ISS of the control systems has been widely studied (cf. [2–12]), and most of the obtained results are often based on the Lyapunov functions.Singularly perturbed control systems are a special class of control systems which is characterized by small parameters multiplying the highest derivates. Recently, many attentions have been devoted to the study of singularly perturbed systems, in particular, to their stability properties. Saberi and Khalil [13] investigated the asymptotic and exponential stability of nonlinear singularly perturbed systems. They obtained a quadratic-type Lyapunov function as a weighted sum of quadratic-type Lyapunov functions of the reduced and the boundary-layer systems. They used the composite Lyapunov function to estimate the degree of exponential stability and the domain of attraction of stable equilibrium point. Corless and Glielmo [14] obtained some results and properties related to exponential stability of singularly perturbed systems. They pointed out that, if both the reduced and the boundary-layer systems are exponentially stable, then, provided that some further regularity conditions are satisfied, the full-order system is exponentially stable for sufficiently small value ϵ. Liu et al. [15] derived the exponential stability criteria of singularly perturbed systems with time delay. Christofides and Teel [11] obtained a type of total stability for the input-to-state stability property with respect to singular perturbations under the assumptions that the reduced system is ISS and the boundary-layer system is uniformly globally asymptotically stable. Tian [16, 17] discussed the analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations. There are some results about the stability of numerical methods for control systems (cf. [18, 19]).The previous studies have mainly focused on the exponential stability of singularly perturbed systems with or without delays and the ISS of singularly perturbed control systems without delay. There are no results about the ISS of delay singulary perturbed control systems. In this paper, we study the ISS of some classes of delay singularly perturbed control systems. By using the generalized Halanay inequality and the Lyapunov functions, we obtain the sufficient conditions under which these delay singularly perturbed control systems are input-to-state stable.


## 2. Preliminary

We introduce the following symbols (cf. [8, 11, 15]).(1)
∥·∥ denotes the standard Euclidean norm of a vector, ∥|xt|∥=supσ∈[t-τ,t]∥x(σ)∥ and ∥A∥=max1≤j≤m(∑‍i=1naij2)1/2 denotes the norm of an n×m matrix A=(aij). A matrix A(t) is bounded means that ∥A(t)∥<∞.(2)
AT denotes the transpose of the matrix A, λi(A) denotes the ith eigenvalue of the matrix A, and Reλi(A) denotes the real part of λi(A).(3)
The matrixA>0 means that A is positive-definite. The vector v=(v1,v2,…,vn)T≥0(>0) means each component vi≥0(>0),i=1,2,…,n.(4)
A realn×n matrix A=(aij) with aij≤0 for all i≠j is an M-matrix if A is nonsingular and A-1>0.(5)
For any measurable locally essentially bounded functionu:R≥0→Rd, ∥u[0,t)∥=supt≥0{∥u(t)∥}.(6)
A functionγ: R≥0→R≥0 is a κ-function if it is continuous, strictly increasing, and γ(0)=0.(7)
A functionβ: R≥0×R≥0→R≥0 is a κφ-function if, for each fixed t≥0, the function β(·,t)∈κ, and for each fixed s≥0, the function β(s,·) is decreasing and β(s,t)→0 as t→∞.Lemma 1 (see [20]).
LetA(t) be an n×n matrix whose elements are continuous functions defined on the time interval J=[0,∞) and the following assumptions hold:
(1)(i)Reλ(A(t))≤-c1<0,∀t∈J,(ii)∥A(t)∥≤c2,∀t∈J,(iii)∥A′(t)∥≤c2,∀t∈J.
Then there exists a positive-definitive matrixP(t) such that the following algebraic Lyapunov equation holds:
(2)AT(t)P(t)+P(t)A(t)=-I,
where c1,c2 are constants, I is the identity matrix, and P(t) is bounded.The following generalized Halanay inequality will play a key role in studying the ISS for the system (9).Lemma 2 (generalized Halanay inequality (see [16, 17])).
Suppose(3)w′(t)≤λ(t)-α(t)w(t)+β(t)supt-τ≤σ≤tw(σ),fort≥t0.
Here w(t) is a non-negative real-value continuous function, τ≥0, λ(t), α(t), and β(t) are continuous with 0≤λ(t)≤λ*, α(t)≥α0>0, and 0<β(t)≤qα(t) for t≥t0 and 0≤q<1. Then
(4)w(t)≤λ*(1-q)α0+Ge-μ⋆(t-t0),t≥t0,
whereG=supt0-τ≤σ≤t0|w(σ)| and μ⋆>0 is defined as
(5)μ⋆=inft≥t0{μ(t):μ(t)-α(t)+β(t)eμ(t)τ=0}.Lemma 3.
LetA(t) and B(t) be n×n matrix-value functions, ν(t)=(ν1(t),ν2(t),…,νn(t))T, and let Γ(t)=(Γ1(t),Γ2(t),…,Γn(t))T be vector functions of dimensions n. Assume that (i)
λi(A(t)+AT(t))≤-a(t)<0(i=1,2,…,n),B(t) is bounded;(ii)
-qa(t)+(1+q)∥B(t)∥+q≤0 with 0≤q<1;(iii)
a(t)-∥B(t)∥-1≥a0*>0;(iv)
ν′(t)≤Γ(t)+A(t)ν(t)+B(t)supt-τ≤σ≤tν(σ),∥Γ(t)∥≤Γ*, for t≥t0, where supt-τ≤σ≤tν(σ)=(supt-τ≤σ≤tν1(σ), supt-τ≤σ≤tν2(σ),…,supt-τ≤σ≤tνn(σ))T. Then the following estimate holds
(6)∥ν∥≤Γ*(1-q)a0*+∥|νt0|∥e-γ*(t-t0),fort≥t0,
where γ* is defined as
(7)2γ*=inf{γ(t):γ(t)-(a(t)-∥B(t)∥-1)+∥B(t)∥eγ(t)τ}.Proof.
LetV(t)=∥ν∥2=νTν. Then
(8)V′(t)=ν′Tν+νTν′≤2νTΓ(t)+νT(A(t)T+A(t))ν(t)+2νTB(t)supt-τ≤σ≤tν(σ)≤2∥ν(t)∥∥Γ(t)∥-a(t)∥ν(t)∥2+2∥B(t)∥∥ν(t)∥∥|νt|∥≤∥Γ(t)∥2+∥ν(t)∥2-a(t)∥ν(t)∥2+∥B(t)∥(∥ν(t)∥2+∥|νt|∥2)≤∥Γ(t)∥2-(a(t)-∥B(t)∥-1)∥ν(t)∥2+∥B(t)∥∥|νt|∥2.
Moreover, by the conditions (i)–(iii), the estimate (6) can be derived as a consequence of (3)–(5) and (8).Consider the delay singularly perturbed control systems(9)x′(t)=f(t,x(t),x(t-τ),y(t),y(t-τ),u(t)),t≥0,ϵy′(t)=g(t,x(t),x(t-τ),y(t),u(t)),0<ϵ≪1,x(t)=φ(t),y(t)=ψ(t),t∈[-τ,0],
where t∈R is the “time,” x∈Rm and y∈Rn are the state variables, u(t)∈Rd is the control input which is locally essentially bounded, ϵ is the singular perturbation parameter, and τ is a constant time delay. The sufficiently smooth mapping f:R×Rm×Rm×Rn×Rn×Rd→Rm,g:R×Rm×Rm×Rn×Rn×Rd→Rn has bounded derivatives and f(t,0,0,0,0,0)=g(t,0,0,0,0,0)=0. φ∈Rm and ψ∈Rn are given vector-functions and the derivative of ψ exists.Definition 4.
The delay singularly perturbed control system (9) is ISS if there exist κφ-functions β1,β2: R≥0×R≥0→R≥0 and κ-functions γ1,γ2 such that, for any initial functions φ(t),ψ(t) and each essentially bounded input u(t), the solution of (9) satisfy
(10)∥x(t,φ,ψ,u,ϵ)∥≤β1(ξ,t)+γ1(∥u[0,t)∥),∥y(t,φ,ψ,u,ϵ)∥≤β2(ξ,t)+γ2(∥u[0,t)∥),
where x(t,φ,ψ,u,ϵ),y(t,φ,ψ,u,ϵ) are the solutions of (9), ξ=sup-τ≤t≤0∥φ(t)∥+sup-τ≤t≤0∥ψ(t)∥+sup-τ≤t≤0∥ψ′(t)∥.


## 3. Linear Systems

In this section, we are concerned with ISS of the following linear delay singularly perturbed control systems as a special class of (9):
(11)x′=A11(t)x+A12(t)xt+B11(t)y+B12(t)yt+C1(t)u(t),t≥0,ϵy′=A21(t)x+A22(t)xt+B21(t)y+C2(t)u(t),0<ϵ≪1,x(t)=φ(t),y(t)=ψ(t),t∈[-τ,0].
Here we let x=x(t), y=y(t),   xt=x(t-τ),  yt=y(t-τ), and  u=u(t) for simplicity; A1j(t)∈Rm×m, A2j(t)∈Rn×m, B1j(t)∈Rm×n(j=1,2), B21(t)∈Rn×n, C1(t)∈Rm×d, and C2(t)∈Rn×d are smooth matrix functions of t, and B21(t) is nonsingular for every t. Now, we introduce some assumptions.Assumption 5.
There exist positive constantsc1 and c2 such that, for for all t∈J=[0,+∞),
(12)Reλ(A11(t))≤-c1,∥A11(t)∥≤c2,∥A11′(t)∥≤c2,Reλ(B21(t))≤-c1,∥B21(t)∥≤c2,∥B21′(t)∥≤c2,∥B21-1(t)∥≤c2,∥B21-1(t)A21(t)∥≤c2,∥B21-1(t)A22(t)∥≤c2,∥B21-1(t)C2(t)∥≤c2.From Assumption5 and Lemma 1, we can easily show that there exist the differentiable positive-definite matrices P1(t) and P2(t) such that (13a)A11T(t)P1(t)+P1(t)A11(t)=-Im,(13b)B21T(t)P2(t)+P2(t)B21(t)=-In,where Im,In are m×m,n×n identity matrices, respectively, [21] shows that Assumption 5 guarantees that Reference, for every t≥t0, (13a), (13b) have unique positive-definite solutions P1(t) and P2(t) given by
(14)P1(t)=∫0∞eA11T(t)θeA11(t)θdθ,P2(t)=∫0∞eB21T(t)θeB21(t)θdθ,
respectively. It follows from the boundness and the positive-definiteness of P1(t) and P2(t) that there exist positive constants Mi,αi,βi(i=1,2) such that
(15)M1≤∥Pi(t)∥≤M2,i=1,2,α1∥x∥2≤xTP1(t)x≤β1∥x∥2,α2∥y∥2≤yTP2(t)y≤β2∥y∥2.Assumption 6.
There exist bounded functionsaij(t),bij(t),andci(t)(i,j=1,2) such that
(16)2xTP1(t)(A12(t)xt+B11(t)y+B12(t)yt+C1(t)u(t))+xTP′1(t)x≤c1(t)∥u(t)∥2+a11(t)∥x∥2+a12(t)∥|xt|∥2+b11(t)∥y-h∥2+b12(t)∥|(y-h)t|∥2,-2(y-h)TP2(t)h′+(y-h)TP2′(t)(y-h)≤c2(t)∥u(t)∥2+a21(t)∥x∥2+a22(t)∥|xt|∥2+b21(t)∥y-h∥2+b22(t)∥|(y-h)t|∥2,
where
(17)h={-B21-1(t)×[A21(t)x+A22(t)xt+C2(t)u(t)],t≥0ψ(t)-ϵB21-1(t)ψ′(t),t∈[-τ,0].Assumption 7.
(1) There exists a positive numberϵ0 such that -A~(t) is an M-matrix;
(2)λi(A~(t)+A~T(t))≤-a(t)<0(i=1,2);
(3)-qa(t)+(1+q)∥B~(t)∥+q≤0 with 0≤q<1;
(4)a(t)-∥B~(t)∥-1≥a0*>0,
where(18)A~(t)=(-1-a11(t)β1b11(t)α2a21(t)α1-1-ϵ0b21(t)ϵ0β2),B~(t)=(a12(t)α1b12(t)α2a22(t)α1b22(t)α2).Theorem 8.
If Assumptions5–7 hold, then the delay singularly perturbed control system (11) is input-to-state stable for ϵ∈(0,ϵ0].Proof.
LetV(t,x)=xTP1(t)x, W(t,x,y)=(y-h)TP2(t)(y-h). For the derivative of V(t,x) along the trajectory of (11), we have
(19)V′(t,x)=[A11(t)x+A12(t)xt+B11(t)y+B12(t)yt+C1(t)u]TP1(t)x+xTP1(t)[A11(t)x+A12(t)xt+B11(t)y+B12(t)yt+C1(t)u]+xTP1′(t)x=xT(A11T(t)P1(t)+P1(t)A11(t))x+2xTP1(t)[A12(t)xt+B11(t)y+B12(t)yt+C1(t)u]+xTP1′(t)x≤-(1-a11(t))∥x∥2+a12(t)∥|xt|∥2+b11(t)∥y-h∥2+b12(t)∥|(y-h)t|∥2+c1(t)∥u∥2.
For the derivative of W(t,x,y) along the trajectory of (11), we have
(20)W′(t,x,y)=[1ϵ(A21(t)x+A22(t)xt+B21(t)y+C2(t)u)-h′]T×P2(t)(y-h)+(y-h)TP2(t)×[1ϵ(A21(t)x+A22(t)xt+B21(t)y+C2(t)u)-h′]+(y-h)TP2′(t)(y-h)=1ϵ(y-h)T[B21T(t)P2(t)+P2(t)B21(t)]T-2(y-h)T(t)h′+(y-h)TP2′(t)(y-h)≤-1ϵ(1-ϵb21(t))∥y-h∥2+a21(t)∥x∥2+a22(t)∥|xt|∥2+b22(t)∥|(y-h)t|∥2+c2(t)∥u∥2.
From(1) of Assumption 7, we can derive (1-ϵ0b21(t))/ϵ0β2>0 and the following inequalities for ϵ∈(0,ϵ0] and t≥0:
(21)V′≤-1-a11(t)β1V+b11(t)α2W+a12(t)α1∥|Vt|∥+b12(t)α2∥|Wt|∥+c1(t)∥u∥2,W′≤a21(t)α1V-1-ϵb21(t)ϵβ2W+a22(t)α1∥|Vt|∥+b22(t)α2∥|Wt|∥+c2(t)∥u∥2≤a21(t)α1V-1-ϵ0b21(t)ϵ0β2W+a22(t)α1∥|Vt|∥+b22(t)α2∥|Wt|∥+c2(t)∥u∥2.
It follows from Lemma 3 that there exist positive constants λ,a0*,andc0* such that
(22)V≤(∥|Vt0|∥+∥|Wt0|∥)e-2λt+c0*∥u[0,t)∥2(1-q)a0*,W≤(∥|Vt0|∥+∥|Wt0|∥)e-2λt+c0*∥u[0,t)∥2(1-q)a0*,
where ∥(c1(t),c2(t))T∥≤c0* and λ is defined by
(23)4λ=inft≥0{λ(t):λ(t)-(a(t)-∥B~(t)∥-1)+∥B~(t)∥eλ(t)τ}.
By the definition of V(t,x) and the positive-definiteness of P1(t), we have(24)α1∥x∥2≤V≤(∥|Vt0|∥+∥|Wt0|∥)e-2λt+c0*∥u[0,t)∥2(1-q)a0*≤(β1∥|x0|∥2+β2∥|(y-h)0|∥2)e-2λt+c0*∥u[0,t)∥2(1-q)a0*≤(β1∥|x0|∥2+β2∥|y0|∥2+β2∥|h0|∥2)e-2λt+c0*∥u[0,t)∥2(1-q)a0*≤(β1∥|x0|∥2+2β2∥|y0|∥2+β2ϵ02c22∥|ψ0′|∥2)e-2λt+c0*∥u[0,t)∥2(1-q)a0*,∥x∥2≤K12(∥|φ0|∥2+∥|ψ0|∥2+∥|ψ0′|∥2)e-2λt+K22∥u[0,t)∥2,
where K12=max{β1/α1,2β2/α1,β2ϵ02c22/α1} and K22=c0*/(1-q)a0*α1. Moreover,
(25)∥x∥≤K1(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λt+K2∥u[0,t)∥.
Thus, (25) and the inequality
(26)∥y∥-∥h∥≤∥(y-h)(t)∥≤(α2)-1/2W1/2
imply that
(27)∥y∥≤∥h∥+(α2)-1/2W1/2≤2c2(K1(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λ(t-τ)+K2∥u[0,t)∥)+c2∥u[0,t)∥+(α2)-1/2W1/2≤(2c2+α1α2)K1(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λ(t-τ)+((2c2+α1α2)K2+c2)∥u[0,t)∥=K1*(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λ(t-τ)+K2*∥u[0,t)∥,
where K1*=(2c2+α1/α2)K1, K2*=(2c2+α1/α2)K2+c2. The proof is complete.


## 4. Nonlinear Systems

In this section, we are concerned with ISS of the following nonlinear delay singularly perturbed control systems as a special class of (9):
(28)x′=A(t)x+f(t,x,xt,y,yt,u),t≥0,ϵy′=B(t)y+g(t,x,xt,u),0<ϵ≪1,x(t)=φ(t),y(t)=ψ(t),t∈[-τ,0],
where A(t)∈Rm×m, B(t)∈Rn×n, f(t,0,0,0,0,0)=0, and g(t,0,0,0)=0. Assume that (28) has a unique equilibrium at the origin and the functions f and g are smooth enough and the derivative of ψ exists.Assumption 9.
There exist positive constantsc1,c2 for all t∈J such that
(29)Reλ(A(t))≤-c1,∥A(t)∥≤c2,∥A′(t)∥≤c2,Reλ(B(t))≤-c1,∥B(t)∥≤c2,∥B′(t)∥≤c2.If Assumption9 holds, then there exist the differentiable positive-definite matrices P1(t) and P2(t) such that
(30)AT(t)P1(t)+P1(t)A(t)=-Im,BT(t)P2(t)+P2(t)B(t)=-In,
where Im,In are m×m,n×n identity matrices, respectively. It follows from the boundness and the positive-definiteness of P1(t) and P2(t) that there exist positive constants Mi,αi,andβi(i=1,2) such that
(31)M1≤∥Pi(t)∥≤M2,i=1,2,α1∥x∥2≤xTP1(t)x≤β1∥x∥2,α2∥y∥2≤yTP2(t)y≤β2∥y∥2.Assumption 10.
There exist bounded functionsaij(t),bij(t),andci(t)(i,j=1,2) such that
(32)2xTP1(t)f(t,x,xt,y,yt,u)+xTP1′(t)x≤c1(t)∥u∥2+a11(t)∥x∥2+a12(t)∥|xt|∥2+b11(t)∥y-h∥2+b11(t)∥|(y-h)t|∥2,-2(y-h)TP2(t)h′+(y-h)TP2′(t)(y-h)≤c2(t)∥u∥2+a21(t)∥x∥2+a22(t)∥|xt|∥2+b21(t)∥y-h∥2+b21(t)∥|(y-h)t|∥2,
where
(33)h={-B-1(t)g(t,x,xt,u(t)),t≥0ψ(t)-ϵB-1(t)ψ′(t),t∈[-τ,0].Assumption 11.
(1) There exist a positive numberϵ0 such that -A~(t) is an M-matrix;
(2)λi(A~(t)+A~T(t))≤-a(t)<0(i=1,2);
(3)-qa(t)+(1+q)∥B~(t)∥+q≤0 with 0≤q<1;
(4)a(t)-∥B~(t)∥-1≥a0*>0,
where(34)A~(t)=(-1-a11(t)β1b11(t)α2a21(t)α1-1-ϵ0b21(t)ϵ0β2),B~(t)=(a12(t)α1b12(t)α2a22(t)α1b22(t)α2).Theorem 12.
If Assumptions9–11 hold, then the delay singularly perturbed control system (28) is input-to-state stable for ϵ∈(0,ϵ0].Proof.
LetV(t,x)=xTP1(t)x, W(t,x,y)=(y-h)TP2(t)(y-h). For the derivative of V(t,x) along the trajectory of (28), we have
(35)V′(t,x)=[A(t)x+f(t,x,xt,y,yt,u)]TP1(t)x+xTP1(t)[A(t)x+f(t,x,xt,y,yt,u)]+xTP1′(t)x=xT(AT(t)P1(t)+P1(t)A(t))x+2xTP1(t)f(t,x,xt,y,yt,u)+xTP1′(t)x≤-(1-a11(t))∥x∥2+a12(t)∥|xt|∥2+b11(t)∥y-h∥2+b12(t)∥|(y-h)t|∥2+c1(t)∥u∥2.
For the derivative of W(t,x,y) along the trajectory of (28), we have
(36)W′(t,x,y)=[1ϵ(B(t)y+g(t,x,xt,u))-h′]TP2(t)(y-h)+(y-h)TP2(t)[1ϵ(B(t)y+g(t,x,xt,u))-h′]+(y-h)TP2′(t)(y-h)=1ϵ(y-h)T[BT(t)P2(t)+P2(t)B(t)]T(y-h)-2(y-h)TP2(t)h′+(y-h)TP2′(t)(y-h)≤-1ϵ(1-ϵb21(t))∥y-h∥2+a21(t)∥x∥2+a22(t)∥|xt|∥2+b22(t)∥|(y-h)t|∥2+c2(t)∥u∥2.
From(1) of Assumption 11, we can derive (1-ϵ0b21(t))/ϵ0β2>0 and the following inequalities for ϵ∈(0,ϵ0]:
(37)V′≤-1-a11(t)β1V+b11(t)α2W+a12(t)α1∥|Vt|∥+b12(t)α2∥|Wt|∥+c1(t)∥u∥2,W′≤a21(t)α1V-1-ϵb21(t)ϵβ2W+a22(t)α1∥|Vt|∥+b22(t)α2∥|Wt|∥+c2(t)∥u∥2W′≤a21(t)α1V-1-ϵ0b21(t)ϵ0β2W+a22(t)α1∥|Vt|∥+b22(t)α2∥|Wt|∥+c2(t)∥u∥2.
It follows from Lemma 3 that there exist positive constants λ,a0*,andc0* such that
(38)V≤(∥|Vt0|∥+∥|Wt0|∥)e-2λt+c0*∥u[0,t)∥2(1-q)a0*,W≤(∥|Vt0|∥+∥|Wt0|∥)e-2λt+c0*∥u[0,t)∥2(1-q)a0*,
where ∥(c1(t),c2(t))T∥≤c0* and λ is defined as in (23).
By the definitions ofV(t,x),W(t,x,y), the positive-definiteness of P1(t),P2(t), and the similar proof process to that of Theorem 8, we can obtain
(39)∥x∥≤K1(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λt+K2∥u[0,t)∥,∥y∥≤K1*(∥|φ0|∥+∥|ψ0|∥+∥|ψ0′|∥)e-λ(t-τ)+K2*∥u[0,t)∥,
where K1=max{β1/α1,2β2/α1,β2ϵ02c22/α1}, K2=c0*/(1-q)a0*α1, K1*=(2c2+α1/α2)K1 and K2*=(2c2+α1/α2)K2+c2. The proof is complete.


## 5. Examples

Example 1.
Consider the following linear delay system as an application of Theorem8:
(40)x′(t)=-5x(t)+y(t-τ)+u(t),ϵy′(t)=3x(t)-5y(t).
Let V=x2/10, W=(y-h)2/10, h=3x(t)/5. Then
(41)V′(t,x)=15x(-5x+yt+u)≤-375V+35∥|Vt|∥+∥|Wt|∥+15∥u∥2,(42)W′(t,x,y)=15(y-h)(1ϵ(3x-5y)-35(-5x+yt+u))≤3V+(-10ϵ+11425)W+925∥|Vt|∥35∥|Wt|∥+350∥u∥2.
So we obtain the matrices
(43)A~(t)=(-37503-10ϵ+11425),B~(t)=(35192535).
If we require that the constant ϵ0 satisfies -10/ϵ0+114/25≤-37/5; that is, ϵ0≤250/299, then, we can take ϵ0=250/299 such that it is easy to show that the conditions in Assumptions 5–7 will be satisfied for any ϵ∈(0,ϵ0]. Moreover, by Theorem 8, the system (40) is ISS for ϵ∈(0,ϵ0].Example 2.
Consider the following nonlinear delay system as an application of Theorem12:
(44)x′(t)=-15x(t)+5ln(1+x2(t-τ))+sin(y(t))+u(t),ϵy′(t)=2x(t-τ)-6y(t).LetV=x2/30, W=(y-h)2/12, and h=x(t-τ)/3. Then
(45)V′(t,x)=115x(-15x+5ln(1+xt2)+siny+u)≤-683V+25W+163∥|Vt|∥+130∥u∥2,W′(t,x,y)=16(y-h)(1ϵ(2xt-6y)-h′)≤(-12ϵ+679)W+619∥|Vt|∥+13∥|Wt|∥+136∥u∥2.
So we obtain the matrices
(46)A~(t)=(-683250-12ϵ+679),B~(t)=(163061913).
If we require that the constant ϵ0 satisfies -12/ϵ+67/9≤-68/3; that is, ϵ0≤108/271, then, we can take ϵ0=108/271 such that it is easy to show that the conditions in Assumptions 9–11 will be satisfied for any ϵ∈(0,ϵ0]. Moreover, by Theorem 12, the system (44) is ISS for ϵ∈(0,ϵ0].


## 6. Conclusion

In this paper, we have studied the input-to-state stability of two classes of the linear and nonlinear delay singularly perturbed control systems. The generalized Halanay inequality and the Lyapunov function play important roles in obtaining the stability results. The sufficient conditions of input-to-state stability for delay singularly perturbed control systems are given.



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*Source: 401572-2013-06-11.xml*