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

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.

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, ‖|𝑥_𝑡|‖=sup_{𝜎∈[𝑡−𝜏,𝑡]}‖𝑥(𝜎)‖ and ‖𝐴‖=max_{1≤𝑗≤𝑚}(∑^𝑛_𝑖=1𝑎²_𝑖𝑗)^1/2 denotes the norm of an 𝑛×𝑚 matrix 𝐴=(𝑎_𝑖𝑗). A matrix 𝐴(𝑡) is bounded means that ‖𝐴(𝑡)‖<∞.(2)𝐴^𝑇 denotes the transpose of the matrix 𝐴, 𝜆_𝑖(𝐴) denotes the 𝑖th eigenvalue of the matrix 𝐴, and Re𝜆_𝑖(𝐴) denotes the real part of 𝜆_𝑖(𝐴).(3)The matrix 𝐴>0 means that 𝐴 is positive-definite. The vector 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇≥0(>0) means each component 𝑣_𝑖≥0(>0),𝑖=1,2,…,𝑛.(4)A real 𝑛×𝑛 matrix 𝐴=(𝑎_𝑖𝑗) with 𝑎_𝑖𝑗≤0 for all 𝑖≠𝑗 is an 𝑀-matrix if 𝐴 is nonsingular and 𝐴⁻¹>0.(5)For any measurable locally essentially bounded function 𝑢∶𝑅_≥0→𝑅^𝑑, ‖𝑢_[0,𝑡)‖=sup_𝑡≥0{‖𝑢(𝑡)‖}.(6)A function 𝛾: 𝑅_≥0→𝑅_≥0 is a 𝜅-function if it is continuous, strictly increasing, and 𝛾(0)=0.(7)A function 𝛽: 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]). Let 𝐴(𝑡) be an 𝑛×𝑛 matrix whose elements are continuous functions defined on the time interval 𝐽=[0,∞) and the following assumptions hold: (i)Re𝜆(𝐴(𝑡))≤−𝑐₁(𝐴(𝑡<0,∀𝑡∈𝐽,ii)‖)‖≤𝑐₂(𝐴,∀𝑡∈𝐽,iii)^′(𝑡)≤𝑐₂,∀𝑡∈𝐽. (1)
Then there exists a positive-definitive matrix 𝑃(𝑡) such that the following algebraic Lyapunov equation holds: 𝐴^𝑇(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)=−𝐼, (2) 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 𝑤^′(𝑡)≤𝜆(𝑡)−𝛼(𝑡)𝑤(𝑡)+𝛽(𝑡)sup_{𝑡−𝜏≤𝜎≤𝑡}𝑤(𝜎),for𝑡≥𝑡₀. (3) 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 𝑤(𝑡)≤𝜆^∗1−𝑞𝛼₀+𝐺𝑒^{−𝜇⋆(𝑡−𝑡0)},𝑡≥𝑡₀, (4)
where 𝐺=sup_{𝑡0−𝜏≤𝜎≤𝑡0}|𝑤(𝜎)| and 𝜇^⋆>0 is defined as 𝜇^⋆=inf_𝑡≥𝑡0𝜇(𝑡)∶𝜇(𝑡)−𝛼(𝑡)+𝛽(𝑡)𝑒^{𝜇(𝑡)𝜏}=0. (5)

Lemma 3. Let 𝐴(𝑡) and 𝐵(𝑡) be 𝑛×𝑛 matrix-value functions, 𝜈(𝑡)=(𝜈₁(𝑡),𝜈₂(𝑡),…,𝜈_𝑛(𝑡))^𝑇, and let Γ(𝑡)=(Γ₁(𝑡),Γ₂(𝑡),…,Γ_𝑛(𝑡))^𝑇 be vector functions of dimensions 𝑛. Assume that (i)𝜆_𝑖(𝐴(𝑡)+𝐴^𝑇(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≥𝑎^∗₀>0; (iv)𝜈^′(𝑡)≤Γ(𝑡)+𝐴(𝑡)𝜈(𝑡)+𝐵(𝑡)sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈(𝜎),‖Γ(𝑡)‖≤Γ^∗, for 𝑡≥𝑡₀, where sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈(𝜎)=(sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈₁(𝜎), sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈₂(𝜎),…,sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈_𝑛(𝜎))^𝑇. Then the following estimate holds ‖𝜈‖≤Γ^∗1−𝑞𝑎^∗₀+𝜈_𝑡0𝑒^{−𝛾∗(𝑡−𝑡0)},for𝑡≥𝑡₀, (6) where 𝛾^∗ is defined as 2𝛾^∗=inf𝛾(𝑡)∶𝛾(𝑡)−(𝑎(𝑡)−‖𝐵(𝑡)‖−1)+‖𝐵(𝑡)‖𝑒^{𝛾(𝑡)𝜏}. (7)

Proof. Let 𝑉(𝑡)=‖𝜈‖²=𝜈^𝑇𝜈. Then 𝑉^′(𝑡)=𝜈^′𝑇𝜈+𝜈^𝑇𝜈^′≤2𝜈^𝑇Γ(𝑡)+𝜈^𝑇𝐴(𝑡)^𝑇+𝐴(𝑡)𝜈(𝑡)+2𝜈^𝑇𝐵(𝑡)sup_{𝑡−𝜏≤𝜎≤𝑡}𝜈(𝜎)≤2‖𝜈(𝑡)‖‖Γ(𝑡)‖−𝑎(𝑡)‖𝜈(𝑡)‖²+2‖𝐵(𝑡)‖‖𝜈(𝑡)‖𝜈_𝑡≤‖Γ(𝑡)‖²+‖𝜈(𝑡)‖²−𝑎(𝑡)‖𝜈(𝑡)‖²+‖𝐵(𝑡)‖‖𝜈(𝑡)‖²+𝜈_𝑡²≤‖Γ(𝑡)‖²−(𝑎(𝑡)−‖𝐵(𝑡)‖−1)‖𝜈(𝑡)‖²+‖𝐵(𝑡)‖𝜈_𝑡². (8)
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 𝑥^′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡),𝑦(𝑡−𝜏),𝑢(𝑡),𝑡≥0,𝜖𝑦^′(𝑡)(𝑡)()(𝑡)(𝑡)𝑥(𝑡)(𝑡)(𝑡)(𝑡)[],=𝑔𝑡,𝑥,𝑥𝑡−𝜏,𝑦,𝑢,0<𝜖≪1,=𝜑,𝑦=𝜓,𝑡∈−𝜏,0 (9) where 𝑡∈𝑅 is the “time,” 𝑥∈𝑅^𝑚 and 𝑦∈𝑅^𝑛 are the state variables, 𝑢(𝑡)∈𝑅^𝑑 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×𝑅^𝑚×𝑅^𝑚×𝑅^𝑛×𝑅^𝑛×𝑅^𝑑→𝑅^𝑚,𝑔∶𝑅×𝑅^𝑚×𝑅^𝑚×𝑅^𝑛×𝑅^𝑛×𝑅^𝑑→𝑅^𝑛 has bounded derivatives and 𝑓(𝑡,0,0,0,0,0)=𝑔(𝑡,0,0,0,0,0)=0. 𝜑∈𝑅^𝑚 and 𝜓∈𝑅^𝑛 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 𝑥𝑡,𝜑,𝜓,𝑢,𝜖≤𝛽₁(𝜉,𝑡)+𝛾₁𝑢_[0,𝑡),𝑦𝑡,𝜑,𝜓,𝑢,𝜖≤𝛽₂(𝜉,𝑡)+𝛾₂𝑢_[0,𝑡), (10) 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): 𝑥^′=𝐴₁₁(𝑡)𝑥+𝐴₁₂(𝑡)𝑥_𝑡+𝐵₁₁(𝑡)𝑦+𝐵₁₂(𝑡)𝑦_𝑡+𝐶₁(𝑡)𝑢(𝑡),𝑡≥0,𝜖𝑦^′=𝐴₂₁(𝑡)𝑥+𝐴₂₂(𝑡)𝑥_𝑡+𝐵₂₁(𝑡)𝑦+𝐶₂(𝑡)𝑢(𝑡)𝑥(𝑡)(𝑡)(𝑡)(𝑡)[].,0<𝜖≪1,=𝜑,𝑦=𝜓,𝑡∈−𝜏,0 (11) Here we let 𝑥=𝑥(𝑡), 𝑦=𝑦(𝑡),   𝑥_𝑡=𝑥(𝑡−𝜏),  𝑦_𝑡=𝑦(𝑡−𝜏), and  𝑢=𝑢(𝑡) for simplicity; 𝐴_1𝑗(𝑡)∈𝑅^𝑚×𝑚, 𝐴_2𝑗(𝑡)∈𝑅^𝑛×𝑚, 𝐵_1𝑗(𝑡)∈𝑅^𝑚×𝑛(𝑗=1,2), 𝐵₂₁(𝑡)∈𝑅^𝑛×𝑛, 𝐶₁(𝑡)∈𝑅^𝑚×𝑑, and 𝐶₂(𝑡)∈𝑅^𝑛×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 constants 𝑐₁ and 𝑐₂ such that, for for all 𝑡∈𝐽=[0,+∞), Re𝜆𝐴₁₁(𝑡)≤−𝑐₁,𝐴₁₁(𝑡)≤𝑐₂,𝐴^′₁₁(𝑡)≤𝑐₂,Re𝜆𝐵₂₁(𝑡)≤−𝑐₁,𝐵₂₁(𝑡)≤𝑐₂,𝐵^′₂₁(𝑡)≤𝑐₂,𝐵⁻¹₂₁(𝑡)≤c₂,𝐵⁻¹₂₁(𝑡)𝐴₂₁(𝑡)≤𝑐₂,𝐵⁻¹₂₁(𝑡)𝐴₂₂(𝑡)≤𝑐₂,𝐵⁻¹₂₁(𝑡)𝐶₂(𝑡)≤𝑐₂. (12)

From Assumption 5 and Lemma 1, we can easily show that there exist the differentiable positive-definite matrices 𝑃₁(𝑡) and 𝑃₂(𝑡) such that 𝐴^𝑇₁₁(𝑡)𝑃₁(𝑡)+𝑃₁(𝑡)𝐴₁₁(𝑡)=−𝐼_𝑚,𝐵^𝑇₂₁(𝑡)𝑃₂(𝑡)+𝑃₂(𝑡)𝐵₂₁(𝑡)=−𝐼_𝑛, (13a)(13b)where 𝐼_𝑚,𝐼_𝑛 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 𝑃₁(𝑡)=^∞₀𝑒^{𝐴𝑇11(𝑡)𝜃}𝑒^{𝐴11(𝑡)𝜃}𝑃𝑑𝜃,₂(𝑡)=^∞₀𝑒^{𝐵𝑇21(𝑡)𝜃}𝑒^{𝐵21(𝑡)𝜃}𝑑𝜃, (14) respectively. It follows from the boundness and the positive-definiteness of 𝑃₁(𝑡) and 𝑃₂(𝑡) that there exist positive constants 𝑀_𝑖,𝛼_𝑖,𝛽_𝑖(𝑖=1,2) such that 𝑀₁≤𝑃_𝑖(𝑡)≤𝑀₂𝛼,𝑖=1,2,₁‖𝑥‖²≤𝑥^𝑇𝑃₁(𝑡)𝑥≤𝛽₁‖𝑥‖²,𝛼₂𝑦²≤𝑦^𝑇𝑃₂(𝑡)𝑦≤𝛽₂𝑦². (15)

Assumption 6. There exist bounded functions 𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),and𝑐_𝑖(𝑡)(𝑖,𝑗=1,2) such that 2𝑥^𝑇𝑃₁(𝑡)𝐴₁₂(𝑡)𝑥_𝑡+𝐵₁₁(𝑡)𝑦+𝐵₁₂(𝑡)𝑦_𝑡+𝐶₁(𝑡)𝑢(𝑡)+𝑥^𝑇𝑃^′₁(𝑡)𝑥≤𝑐₁(𝑡)‖𝑢(𝑡)‖²+𝑎₁₁(𝑡)‖𝑥‖²+𝑎₁₂(𝑡)𝑥_𝑡²+𝑏₁₁(𝑡)𝑦−ℎ²+𝑏₁₂(𝑡)𝑦−ℎ_𝑡²,−2𝑦−ℎ^𝑇𝑃₂(𝑡)ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)𝑦−ℎ≤𝑐₂(𝑡)‖𝑢(𝑡)‖²+𝑎₂₁(𝑡)‖𝑥‖²+𝑎₂₂(𝑡)𝑥_𝑡²+𝑏₂₁(𝑡)𝑦−ℎ²+𝑏₂₂(𝑡)𝑦−ℎ_𝑡², (16) where ℎ=−𝐵⁻¹₂₁(𝑡)×𝐴₂₁(𝑡)𝑥+𝐴₂₂(𝑡)𝑥_𝑡+𝐶₂(𝑡)𝑢(𝑡𝜓(𝑡)),𝑡≥0−𝜖𝐵⁻¹₂₁(𝑡)𝜓^′(𝑡),𝑡∈[−𝜏,0]. (17)

Assumption 7. (1) There exists a positive number 𝜖₀ such that −𝐴(𝑡) is an 𝑀-matrix;
(2) 𝜆_𝑖(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≥𝑎^∗₀>0,
where 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. (18)

Theorem 8. If Assumptions 5–7 hold, then the delay singularly perturbed control system (11) is input-to-state stable for 𝜖∈(0,𝜖₀].

Proof. Let 𝑉(𝑡,𝑥)=𝑥^𝑇𝑃₁(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 𝑉^′(𝑡,𝑥)=𝐴₁₁(𝑡)𝑥+𝐴₁₂(𝑡)𝑥_𝑡+𝐵₁₁(𝑡)𝑦+𝐵₁₂(𝑡)𝑦_𝑡+𝐶₁(𝑡)𝑢^𝑇𝑃₁(𝑡)𝑥+𝑥^𝑇𝑃₁(𝑡)𝐴₁₁(𝑡)𝑥+𝐴₁₂(𝑡)𝑥_𝑡+𝐵₁₁(𝑡)𝑦+𝐵₁₂(𝑡)𝑦_𝑡+𝐶₁(𝑡)𝑢+𝑥^𝑇𝑃^′₁(𝑡)𝑥=𝑥^𝑇𝐴^𝑇₁₁(𝑡)𝑃₁(𝑡)+𝑃₁(𝑡)𝐴₁₁(𝑡)𝑥+2𝑥^𝑇𝑃₁(𝑡)𝐴₁₂(𝑡)𝑥_𝑡+𝐵₁₁(𝑡)𝑦+𝐵₁₂(𝑡)𝑦_𝑡+𝐶₁(𝑡)𝑢+𝑥^𝑇𝑃^′₁(𝑡)𝑥≤−1−𝑎₁₁(𝑡)‖𝑥‖²+𝑎₁₂(𝑡)𝑥_𝑡²+𝑏₁₁(𝑡)𝑦−ℎ²+𝑏₁₂(𝑡)(𝑦−ℎ)_𝑡²+𝑐₁(𝑡)‖𝑢‖². (19) For the derivative of 𝑊(𝑡,𝑥,𝑦) along the trajectory of (11), we have 𝑊^′=1𝑡,𝑥,𝑦𝜖𝐴₂₁(𝑡)𝑥+𝐴₂₂(𝑡)𝑥_𝑡+𝐵₂₁(𝑡)𝑦+𝐶₂(𝑡)𝑢−ℎ^′𝑇×𝑃₂(𝑡)𝑦−ℎ+𝑦−ℎ^𝑇𝑃₂(𝑡)×1𝜖𝐴₂₁(𝑡)𝑥+𝐴₂₂(𝑡)𝑥_𝑡+𝐵₂₁(𝑡)𝑦+𝐶₂(𝑡)𝑢−ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)=1𝑦−ℎ𝜖𝑦−ℎ^𝑇𝐵^𝑇₂₁(𝑡)𝑃₂(𝑡)+𝑃₂(𝑡)𝐵₂₁(𝑡)^𝑇−2𝑦−ℎ^𝑇(𝑡)ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)1𝑦−ℎ≤−𝜖1−𝜖𝑏₂₁(𝑡)𝑦−ℎ²+𝑎₂₁(𝑡)‖𝑥‖²+𝑎₂₂(𝑡)𝑥_𝑡²+𝑏₂₂(𝑡)(𝑦−ℎ)_𝑡²+𝑐₂(𝑡)‖𝑢‖². (20)
From (1) of Assumption 7, we can derive (1−𝜖₀𝑏₂₁(𝑡))/𝜖₀𝛽₂>0 and the following inequalities for 𝜖∈(0,𝜖₀] and 𝑡≥0: 𝑉^′≤−1−𝑎₁₁(𝑡)𝛽₁𝑏𝑉+¹¹(𝑡)𝛼₂𝑊+𝑎₁₂(𝑡)𝛼₁𝑉_𝑡+𝑏₁₂(𝑡)𝛼₂𝑊_𝑡+𝑐₁(𝑡)‖𝑢‖²,𝑊^′≤𝑎₂₁(𝑡)𝛼₁𝑉−1−𝜖𝑏₂₁(𝑡)𝜖𝛽₂𝑎𝑊+²²(𝑡)𝛼₁𝑉_𝑡+𝑏₂₂(𝑡)𝛼₂𝑊_𝑡+𝑐₂(𝑡)‖𝑢‖²≤𝑎₂₁(𝑡)𝛼₁𝑉−1−𝜖₀𝑏₂₁(𝑡)𝜖₀𝛽₂𝑎𝑊+²²(𝑡)𝛼₁𝑉_𝑡+𝑏₂₂(𝑡)𝛼₂𝑊_𝑡+𝑐₂(𝑡)‖𝑢‖². (21) It follows from Lemma 3 that there exist positive constants 𝜆,𝑎^∗₀,and𝑐^∗₀ such that 𝑉≤𝑉_𝑡0+𝑊_𝑡0𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀,𝑊≤𝑉_𝑡0+𝑊_𝑡0𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀, (22) where ‖(𝑐₁(𝑡),𝑐₂(𝑡))^𝑇‖≤𝑐^∗₀ and 𝜆 is defined by 4𝜆=inf_𝑡≥0𝜆(𝑡)∶𝜆(𝑡)−𝑎(𝑡)−𝐵(𝑡)−1+𝐵(𝑡)𝑒^{𝜆(𝑡)𝜏}. (23) By the definition of 𝑉(𝑡,𝑥) and the positive-definiteness of 𝑃₁(𝑡), we have𝛼₁‖𝑥‖²≤𝑉≤𝑉_𝑡0+𝑊_𝑡0𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀≤𝛽₁𝑥₀²+𝛽₂𝑦−ℎ₀²𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀≤𝛽₁𝑥₀²+𝛽₂𝑦₀²+𝛽₂ℎ₀²𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀≤𝛽₁𝑥₀²+2𝛽₂𝑦₀²+𝛽₂𝜖²₀𝑐²₂𝜓^′₀²𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀,‖𝑥‖²≤𝐾²₁𝜑₀²+𝜓₀²+𝜓^′₀²𝑒^−2𝜆𝑡+𝐾²₂𝑢_[0,𝑡)², (24) where 𝐾²₁=max{𝛽₁/𝛼₁,2𝛽₂/𝛼₁,𝛽₂𝜖²₀𝑐²₂/𝛼₁} and 𝐾²₂=𝑐^∗₀/(1−𝑞)𝑎^∗₀𝛼₁. Moreover, ‖𝑥‖≤𝐾₁𝜑₀+𝜓₀+𝜓^′₀𝑒^−𝜆𝑡+𝐾₂𝑢_[0,𝑡). (25) Thus, (25) and the inequality 𝑦−‖ℎ‖≤𝑦−ℎ(𝑡)≤𝛼₂^−1/2𝑊^1/2 (26) imply that 𝑦≤‖ℎ‖+𝛼₂^−1/2𝑊^1/2≤2𝑐₂𝐾₁𝜑₀+𝜓₀+𝜓^′₀𝑒^{−𝜆(𝑡−𝜏)}+𝐾₂𝑢_[0,𝑡)+𝑐₂𝑢_[0,𝑡)+𝛼₂^−1/2𝑊^1/2≤2𝑐₂+𝛼₁𝛼₂𝐾₁𝜑₀+𝜓₀+𝜓^′₀𝑒^{−𝜆(𝑡−𝜏)}+2𝑐₂+𝛼₁𝛼₂𝐾₂+𝑐₂𝑢_[0,𝑡)=𝐾^∗₁𝜑₀+𝜓₀+𝜓^′₀𝑒^{−𝜆(𝑡−𝜏)}+𝐾^∗₂𝑢_[0,𝑡), (27) where 𝐾^∗₁=(2𝑐₂+𝛼₁/𝛼₂)𝐾₁, 𝐾^∗₂=(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): 𝑥^′=𝐴(𝑡)𝑥+𝑓𝑡,𝑥,𝑥_𝑡,𝑦,𝑦_𝑡,𝑢,𝑡≥0,𝜖𝑦^′=𝐵(𝑡)𝑦+𝑔𝑡,𝑥,𝑥_𝑡𝑥(𝑡)(𝑡)(𝑡)(𝑡)[],,𝑢,0<𝜖≪1,=𝜑,𝑦=𝜓,𝑡∈−𝜏,0 (28) where 𝐴(𝑡)∈𝑅^𝑚×𝑚, 𝐵(𝑡)∈𝑅^𝑛×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 constants 𝑐₁,𝑐₂ for all 𝑡∈𝐽 such that Re𝜆(𝐴(𝑡))≤−𝑐₁,‖𝐴(𝑡)‖≤𝑐₂,𝐴^′(𝑡)≤𝑐₂,Re𝜆(𝐵(𝑡))≤−𝑐₁,‖𝐵(𝑡)‖≤𝑐₂,𝐵^′(𝑡)≤𝑐₂. (29)

If Assumption 9 holds, then there exist the differentiable positive-definite matrices 𝑃₁(𝑡) and 𝑃₂(𝑡) such that 𝐴^𝑇(𝑡)𝑃₁(𝑡)+𝑃₁(𝑡)𝐴(𝑡)=−𝐼_𝑚,𝐵^𝑇(𝑡)𝑃₂(𝑡)+𝑃₂(𝑡)𝐵(𝑡)=−𝐼_𝑛, (30) where 𝐼_𝑚,𝐼_𝑛 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 𝑀_𝑖,𝛼_𝑖,and𝛽_𝑖(𝑖=1,2) such that 𝑀₁≤𝑃_𝑖(𝑡)≤𝑀₂𝛼,𝑖=1,2,₁‖𝑥‖²≤𝑥^𝑇𝑃₁(𝑡)𝑥≤𝛽₁‖𝑥‖²,𝛼₂𝑦²≤𝑦^𝑇𝑃₂(𝑡)𝑦≤𝛽₂𝑦². (31)

Assumption 10. There exist bounded functions 𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),and𝑐_𝑖(𝑡)(𝑖,𝑗=1,2) such that 2𝑥^𝑇𝑃₁(𝑡)𝑓𝑡,𝑥,𝑥_𝑡,𝑦,𝑦_𝑡,𝑢+𝑥^𝑇𝑃^′₁(𝑡)𝑥≤𝑐₁(𝑡)‖𝑢‖²+𝑎₁₁(𝑡)‖𝑥‖²+𝑎₁₂(𝑡)𝑥_𝑡²+𝑏₁₁(𝑡)𝑦−ℎ²+𝑏₁₁(𝑡)(𝑦−ℎ)_𝑡²,−2𝑦−ℎ^𝑇𝑃₂(𝑡)ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)𝑦−ℎ≤𝑐₂(𝑡)‖𝑢‖²+𝑎₂₁(𝑡)‖𝑥‖²+𝑎₂₂(𝑡)𝑥_𝑡²+𝑏₂₁(𝑡)𝑦−ℎ²+𝑏₂₁(𝑡)𝑦−ℎ_𝑡², (32) where ℎ=−𝐵⁻¹(𝑡)𝑔𝑡,𝑥,𝑥_𝑡(𝑡𝜓(𝑡),𝑢),𝑡≥0−𝜖𝐵⁻¹(𝑡)𝜓^′(𝑡),𝑡∈[−𝜏,0]. (33)

Assumption 11. (1) There exist a positive number 𝜖₀ such that −𝐴(𝑡) is an 𝑀-matrix;
(2)  𝜆_𝑖(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≥𝑎^∗₀>0,
where 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. (34)

Theorem 12. If Assumptions 9–11 hold, then the delay singularly perturbed control system (28) is input-to-state stable for 𝜖∈(0,𝜖₀].

Proof. Let 𝑉(𝑡,𝑥)=𝑥^𝑇𝑃₁(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 𝑉^′()=𝐴(𝑡)𝑡,𝑥𝑥+𝑓𝑡,𝑥,𝑥_𝑡,𝑦,𝑦_𝑡,𝑢^𝑇𝑃₁(𝑡)𝑥+𝑥^𝑇𝑃₁(𝑡)𝐴(𝑡)𝑥+𝑓𝑡,𝑥,𝑥_𝑡,𝑦,𝑦_𝑡,𝑢+𝑥^𝑇𝑃^′₁(𝑡)𝑥=𝑥^𝑇𝐴^𝑇(𝑡)𝑃₁(𝑡)+𝑃₁(𝑡)𝐴(𝑡)𝑥+2𝑥^𝑇𝑃₁(𝑡)𝑓𝑡,𝑥,𝑥_𝑡,𝑦,𝑦_𝑡,𝑢+𝑥^𝑇𝑃^′₁(𝑡)𝑥≤−1−𝑎₁₁(𝑡)‖𝑥‖²+𝑎₁₂(𝑡)𝑥_𝑡²+𝑏₁₁(𝑡)𝑦−ℎ²+𝑏₁₂(𝑡)(𝑦−ℎ)_𝑡²+𝑐₁(𝑡)‖𝑢‖². (35) For the derivative of 𝑊(𝑡,𝑥,𝑦) along the trajectory of (28), we have 𝑊^′=1𝑡,𝑥,𝑦𝜖𝐵(𝑡)𝑦+𝑔𝑡,𝑥,𝑥_𝑡,𝑢−ℎ^′𝑇𝑃₂(𝑡)+𝑦−ℎ𝑦−ℎ^𝑇𝑃₂(𝑡1)𝜖𝐵(𝑡)𝑦+𝑔𝑡,𝑥,𝑥_𝑡,𝑢−ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)=1𝑦−ℎ𝜖𝑦−ℎ^𝑇𝐵^𝑇(𝑡)𝑃₂(𝑡)+𝑃₂(𝑡)𝐵(𝑡)^𝑇𝑦−ℎ−2𝑦−ℎ^𝑇𝑃₂(𝑡)ℎ^′+𝑦−ℎ^𝑇𝑃^′₂(𝑡)1𝑦−ℎ≤−𝜖1−𝜖𝑏₂₁(𝑡)𝑦−ℎ²+𝑎₂₁(𝑡)‖𝑥‖²+𝑎₂₂(𝑡)𝑥_𝑡²+𝑏₂₂(𝑡)𝑦−ℎ_𝑡²+𝑐₂(𝑡)‖𝑢‖². (36)
From (1) of Assumption 11, we can derive (1−𝜖₀𝑏₂₁(𝑡))/𝜖₀𝛽₂>0 and the following inequalities for 𝜖∈(0,𝜖₀]: 𝑉^′≤−1−𝑎₁₁(𝑡)𝛽₁𝑏𝑉+¹¹(𝑡)𝛼₂𝑎𝑊+¹²(𝑡)𝛼₁𝑉_𝑡+𝑏₁₂(𝑡)𝛼₂𝑊_𝑡+𝑐₁(𝑡)‖𝑢‖²,𝑊^′≤𝑎₂₁(𝑡)𝛼₁𝑉−1−𝜖𝑏₂₁(𝑡)𝜖𝛽₂𝑎𝑊+²²(𝑡)𝛼₁𝑉_𝑡+𝑏₂₂(𝑡)𝛼₂𝑊_𝑡+𝑐₂(𝑡)‖𝑢‖²≤𝑎₂₁(𝑡)𝛼₁𝑉−1−𝜖₀𝑏₂₁(𝑡)𝜖₀𝛽₂𝑎𝑊+²²(𝑡)𝛼₁𝑉_𝑡+𝑏₂₂(𝑡)𝛼₂𝑊_𝑡+𝑐₂(𝑡)‖𝑢‖². (37) It follows from Lemma 3 that there exist positive constants 𝜆,𝑎^∗₀,and𝑐^∗₀ such that 𝑉≤𝑉_𝑡0+𝑊_𝑡0𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀,𝑊≤𝑉_𝑡0+𝑊_𝑡0𝑒^−2𝜆𝑡+𝑐^∗₀𝑢_[0,𝑡)²1−𝑞𝑎^∗₀, (38) where ‖(𝑐₁(𝑡),𝑐₂(𝑡))^𝑇‖≤𝑐^∗₀ and 𝜆 is defined as in (23).
By the definitions of 𝑉(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 ‖𝑥‖≤𝐾₁𝜑₀+𝜓₀+𝜓^′₀𝑒^−𝜆𝑡+𝐾₂𝑢_[0,𝑡),𝑦≤𝐾^∗₁𝜑₀+𝜓₀+𝜓^′₀𝑒^{−𝜆(𝑡−𝜏)}+𝐾^∗₂𝑢_[0,𝑡), (39) where 𝐾₁=max{𝛽₁/𝛼₁,2𝛽₂/𝛼₁,𝛽₂𝜖²₀𝑐²₂/𝛼₁}, 𝐾₂=𝑐^∗₀/(1−𝑞)𝑎^∗₀𝛼₁, 𝐾^∗₁=(2𝑐₂+𝛼₁/𝛼₂)𝐾₁ and 𝐾^∗₂=(2𝑐₂+𝛼₁/𝛼₂)𝐾₂+𝑐₂. The proof is complete.

5. Examples

Example 1. Consider the following linear delay system as an application of Theorem 8: 𝑥^′(𝑡)=−5𝑥(𝑡)+𝑦(𝑡−𝜏)+𝑢(𝑡),𝜖𝑦^′(𝑡)=3𝑥(𝑡)−5𝑦(𝑡). (40) Let 𝑉=𝑥²/10, 𝑊=(𝑦−ℎ)²/10, ℎ=3𝑥(𝑡)/5. Then 𝑉^′()=1𝑡,𝑥5𝑥−5𝑥+𝑦_𝑡+𝑢≤−3753𝑉+5𝑉_𝑡+𝑊_𝑡+15‖𝑢‖²,𝑊^′=1𝑡,𝑥,𝑦51𝑦−ℎ𝜖−33𝑥−5𝑦5−5𝑥+𝑦_𝑡−+𝑢≤3𝑉+10𝜖+114𝑊+925𝑉25^𝑡35𝑊_𝑡+3‖𝑢‖50². (41)(42) So we obtain the matrices 𝐴(𝑡)=−37503−10𝜖+114,𝐵(𝑡)=3255193255. (43) 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 Theorem 12: 𝑥^′(𝑡)=−15𝑥(𝑡)+5ln1+𝑥²(𝑡−𝜏)+sin𝑦(𝑡)+𝑢(𝑡),𝜖𝑦^′(𝑡)=2𝑥(𝑡−𝜏)−6𝑦(𝑡). (44)

Let 𝑉=𝑥²/30, 𝑊=(𝑦−ℎ)²/12, and ℎ=𝑥(𝑡−𝜏)/3. Then 𝑉^′()=1𝑡,𝑥𝑥15−15𝑥+5ln1+𝑥²_𝑡≤+sin𝑦+𝑢−6832𝑉+5𝑊+163𝑉_𝑡+1‖𝑢‖30²,𝑊^′=1𝑡,𝑥,𝑦61𝑦−ℎ𝜖2𝑥_𝑡−6𝑦−ℎ^′≤−12𝜖+679𝑊+619𝑉_𝑡+13𝑊_𝑡+1‖𝑢‖36². (45) So we obtain the matrices 𝐴(𝑡)=−683250−12𝜖+679,𝐵(𝑡)=163061913. (46) 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.

Acknowledgment

This work is supported by projects NSF of China (11126329, 11271311, 11201510), NSF of Hunan Province (09JJ3002), Projects Board of Education of Chongqing City (KJ121110). The authors express their sincere thanks to the referees for their useful comments and suggestions, which led to improvements of the presentation.