Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 401572, 8 pages
http://dx.doi.org/10.1155/2013/401572
Research Article
Input-to-State Stability of Singularly Perturbed
Control Systems with Delays
Yongxiang Zhao,1,2 Aiguo Xiao,2 and Li Li1
1
2
School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404000, China
School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and
Engineering, Xiangtan University, Xiangtan, Hunan 411105, China
Correspondence should be addressed to Yongxiang Zhao; zyxlily80@126.com
Received 16 December 2012; Revised 9 April 2013; Accepted 14 May 2013
Academic Editor: Kang Liu
Copyright © 2013 Yongxiang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 boundarylayer 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-tostate 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
Journal of Applied Mathematics
where 𝐺 = sup𝑡0 −𝜏≤𝜎≤𝑡0 |𝑤(𝜎)| and 𝜇⋆ > 0 is defined as
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 (∑𝑛𝑖=1 𝑎𝑖𝑗2 )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, 𝜆 𝑖 (𝐴)
denotes the 𝑖th eigenvalue of the matrix 𝐴, and
Re 𝜆 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 , . . . , V𝑛 )𝑇 ≥ 0 (> 0) means each
component V𝑖 ≥ 0 (> 0), 𝑖 = 1, 2, . . . , 𝑛.
(4) A real 𝑛 × 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 function 𝑢 : 𝑅≥0 → 𝑅𝑑 , ‖𝑢[0,𝑡) ‖ = sup𝑡≥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) Re 𝜆 (𝐴 (𝑡)) ≤ −𝑐1 < 0,
(ii) ‖𝐴 (𝑡)‖ ≤ 𝑐2 ,
󵄩
󵄩
(iii) 󵄩󵄩󵄩󵄩𝐴󸀠 (𝑡)󵄩󵄩󵄩󵄩 ≤ 𝑐2 ,
∀𝑡 ∈ 𝐽,
∀𝑡 ∈ 𝐽,
(1)
Then there exists a positive-definitive matrix 𝑃(𝑡) such that
the following algebraic Lyapunov equation holds:
𝐴𝑇 (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 𝑤 (𝜎) ,
for 𝑡 ≥ 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
+ 𝐺𝑒−𝜇 (𝑡−𝑡0 ) ,
(1 − 𝑞) 𝛼0
𝑡≥𝑡0
(5)
Lemma 3. Let 𝐴(𝑡) and 𝐵(𝑡) be 𝑛 × 𝑛 matrix-value functions, ](𝑡) = (]1 (𝑡),]2 (𝑡), . . . , ]𝑛 (𝑡))𝑇 , and let Γ(𝑡) =
(Γ1 (𝑡), Γ2 (𝑡), . . . , Γ𝑛 (𝑡))𝑇 be vector functions of dimensions 𝑛.
Assume that
(i) 𝜆 𝑖 (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) ]󸀠 (𝑡) ≤ Γ(𝑡) + 𝐴(𝑡)](𝑡) + 𝐵(𝑡)sup𝑡−𝜏≤𝜎≤𝑡 ](𝜎), ‖Γ(𝑡)‖ ≤
Γ∗ ,
for 𝑡 ≥ 𝑡0 , where sup𝑡−𝜏≤𝜎≤𝑡 ](𝜎) = (sup𝑡−𝜏≤𝜎≤𝑡 ]1 (𝜎),
sup𝑡−𝜏≤𝜎≤𝑡 ]2 (𝜎),. . . , sup𝑡−𝜏≤𝜎≤𝑡 ]𝑛 (𝜎))𝑇 . Then the following estimate holds
‖]‖ ≤
Γ∗
√(1 − 𝑞) 𝑎0∗
󵄩󵄨 󵄨󵄩 ∗
+ 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨]𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 𝑒−𝛾 (𝑡−𝑡0 ) ,
for 𝑡 ≥ 𝑡0 ,
(6)
where 𝛾∗ is defined as
2𝛾∗ = inf {𝛾 (𝑡) : 𝛾 (𝑡) − (𝑎 (𝑡) − ‖𝐵 (𝑡)‖ − 1) + ‖𝐵 (𝑡)‖ 𝑒𝛾(𝑡)𝜏 } .
(7)
Proof. Let 𝑉(𝑡) = ‖]‖2 = ]𝑇 ]. Then
𝑇
𝑉󸀠 (𝑡) = ]󸀠 ] + ]𝑇 ]󸀠
≤ 2]𝑇 Γ (𝑡) + ]𝑇 (𝐴(𝑡)𝑇 + 𝐴 (𝑡)) ] (𝑡)
+ 2]𝑇 𝐵 (𝑡) sup ] (𝜎)
∀𝑡 ∈ 𝐽.
𝑡−𝜏≤𝜎≤𝑡
𝜇⋆ = inf {𝜇 (𝑡) : 𝜇 (𝑡) − 𝛼 (𝑡) + 𝛽 (𝑡) 𝑒𝜇(𝑡)𝜏 = 0} .
𝑡 ≥ 𝑡0 ,
(4)
𝑡−𝜏≤𝜎≤𝑡
≤ 2 ‖] (𝑡)‖ ‖Γ (𝑡)‖ − 𝑎 (𝑡) ‖] (𝑡)‖2
󵄩󵄨 󵄨󵄩
+ 2 ‖𝐵 (𝑡)‖ ‖] (𝑡)‖ 󵄩󵄩󵄩󵄨󵄨󵄨]𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
(8)
≤ ‖Γ (𝑡)‖2 + ‖] (𝑡)‖2 − 𝑎 (𝑡) ‖] (𝑡)‖2
󵄩󵄨 󵄨󵄩2
+ ‖𝐵 (𝑡)‖ (‖] (𝑡)‖2 + 󵄩󵄩󵄩󵄨󵄨󵄨]𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 )
≤ ‖Γ (𝑡)‖2 − (𝑎 (𝑡) − ‖𝐵 (𝑡)‖ − 1) ‖] (𝑡)‖2
󵄩󵄨 󵄨󵄩2
+ ‖𝐵 (𝑡)‖ 󵄩󵄩󵄩󵄨󵄨󵄨]𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 .
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) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑥 (𝑡 − 𝜏) , 𝑦 (𝑡) , 𝑦 (𝑡 − 𝜏) , 𝑢 (𝑡)) ,
𝜖𝑦󸀠 (𝑡) = 𝑔 (𝑡, 𝑥 (𝑡) , 𝑥 (𝑡 − 𝜏) , 𝑦 (𝑡) , 𝑢 (𝑡)) ,
𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑦 (𝑡) = 𝜓 (𝑡) ,
𝑡 ≥ 0,
0 < 𝜖 ≪ 1,
𝑡 ∈ [−𝜏, 0] ,
(9)
Journal of Applied Mathematics
3
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
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩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) 󵄩󵄩󵄩) ,
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
𝐴𝑇11 (𝑡) 𝑃1 (𝑡) + 𝑃1 (𝑡) 𝐴 11 (𝑡) = −𝐼𝑚 ,
(13a)
𝑇
𝐵21
(𝑡) 𝑃2 (𝑡) + 𝑃2 (𝑡) 𝐵21 (𝑡) = −𝐼𝑛 ,
(13b)
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
∞
0
∞
(14)
𝑇
𝐵21
(𝑡)𝜃 𝐵21 (𝑡)𝜃
𝑃2 (𝑡) = ∫ 𝑒
(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)‖.
𝑇
𝑃1 (𝑡) = ∫ 𝑒𝐴 11 (𝑡)𝜃 𝑒𝐴 11 (𝑡)𝜃 𝑑𝜃,
𝑒
0
𝑑𝜃,
respectively. It follows from the boundness and the positivedefiniteness of 𝑃1 (𝑡) and 𝑃2 (𝑡) that there exist positive constants 𝑀𝑖 , 𝛼𝑖 , 𝛽𝑖 (𝑖 = 1, 2) such that
󵄩
󵄩
𝑀1 ≤ 󵄩󵄩󵄩𝑃𝑖 (𝑡)󵄩󵄩󵄩 ≤ 𝑀2 ,
𝑖 = 1, 2,
3. Linear Systems
𝛼1 ‖𝑥‖2 ≤ 𝑥𝑇 𝑃1 (𝑡) 𝑥 ≤ 𝛽1 ‖𝑥‖2 ,
In this section, we are concerned with ISS of the following
linear delay singularly perturbed control systems as a special
class of (9):
󵄩 󵄩2
󵄩 󵄩2
𝛼2 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑦𝑇 𝑃2 (𝑡) 𝑦 ≤ 𝛽2 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 .
𝑥󸀠 = 𝐴 11 (𝑡) 𝑥 + 𝐴 12 (𝑡) 𝑥𝑡 + 𝐵11 (𝑡) 𝑦
+ 𝐵12 (𝑡) 𝑦𝑡 + 𝐶1 (𝑡) 𝑢 (𝑡) ,
+ 𝐶2 (𝑡) 𝑢 (𝑡) ,
𝑥 (𝑡) = 𝜑 (𝑡) ,
󵄩󵄨 󵄨󵄩2
≤ 𝑐1 (𝑡) ‖𝑢 (𝑡)‖2 + 𝑎11 (𝑡) ‖𝑥‖2 + 𝑎12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
󵄩2
󵄩
󵄩󵄨
󵄨 󵄩2
+ 𝑏11 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑏12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)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; 𝐴 1𝑗 (𝑡) ∈ 𝑅𝑚×𝑚 , 𝐴 2𝑗 (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 constants 𝑐1 and 𝑐2 such
that, for for all 𝑡 ∈ 𝐽 = [0, +∞),
󵄩
󵄩󵄩
󵄩󵄩󵄩𝐴󸀠 (𝑡)󵄩󵄩󵄩 ≤ 𝑐 ,
󵄩󵄩𝐴 11 (𝑡)󵄩󵄩󵄩 ≤ 𝑐2 ,
󵄩󵄩 11 󵄩󵄩 2
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝐵󸀠 (𝑡)󵄩󵄩󵄩 ≤ 𝑐 ,
Re 𝜆 (𝐵21 (𝑡)) ≤ −𝑐1 ,
󵄩󵄩𝐵21 (𝑡)󵄩󵄩󵄩 ≤ 𝑐2 ,
󵄩󵄩 21 󵄩󵄩 2
󵄩󵄩
󵄩󵄩 −1 󵄩󵄩
󵄩󵄩 −1
󵄩󵄩𝐵21 (𝑡)󵄩󵄩 ≤ c2 ,
󵄩󵄩𝐵21 (𝑡) 𝐴 21 (𝑡)󵄩󵄩 ≤ 𝑐2 ,
󵄩
󵄩
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩 −1
󵄩󵄩 −1
󵄩󵄩𝐵21 (𝑡) 𝐴 22 (𝑡)󵄩󵄩 ≤ 𝑐2 ,
󵄩󵄩𝐵21 (𝑡) 𝐶2 (𝑡)󵄩󵄩󵄩 ≤ 𝑐2 .
󵄩
󵄩
󵄩
󵄩
(12)
Re 𝜆 (𝐴 11 (𝑡)) ≤ −𝑐1 ,
+ 𝑥𝑇 𝑃󸀠 1 (𝑡) 𝑥
(11)
0 < 𝜖 ≪ 1,
𝑦 (𝑡) = 𝜓 (𝑡) ,
Assumption 6. There exist bounded functions 𝑎𝑖𝑗 (𝑡), 𝑏𝑖𝑗 (𝑡),
and 𝑐𝑖 (𝑡) (𝑖, 𝑗 = 1, 2) such that
2𝑥𝑇 𝑃1 (𝑡) (𝐴 12 (𝑡) 𝑥𝑡 + 𝐵11 (𝑡) 𝑦 + 𝐵12 (𝑡) 𝑦𝑡 + 𝐶1 (𝑡) 𝑢 (𝑡))
𝑡 ≥ 0,
𝜖𝑦󸀠 = 𝐴 21 (𝑡) 𝑥 + 𝐴 22 (𝑡) 𝑥𝑡 + 𝐵21 (𝑡) 𝑦
(15)
𝑇
𝑇
− 2(𝑦 − ℎ) 𝑃2 (𝑡) ℎ󸀠 + (𝑦 − ℎ) 𝑃2󸀠 (𝑡) (𝑦 − ℎ)
󵄩󵄨 󵄨󵄩2
≤ 𝑐2 (𝑡) ‖𝑢 (𝑡)‖2 + 𝑎21 (𝑡) ‖𝑥‖2 + 𝑎22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
󵄩2
󵄩
󵄩󵄨
󵄨 󵄩2
+ 𝑏21 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑏22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 ,
(16)
where
−𝐵−1 (𝑡)
{
{ 21
ℎ = { × [𝐴 21 (𝑡) 𝑥 + 𝐴 22 (𝑡) 𝑥𝑡 + 𝐶2 (𝑡) 𝑢 (𝑡)] , 𝑡 ≥ 0
{
−1
󸀠
𝑡 ∈ [−𝜏, 0] .
{𝜓 (𝑡) − 𝜖𝐵21 (𝑡) 𝜓 (𝑡) ,
(17)
Assumption 7. (1) There exists a positive number 𝜖0 such that
̃ is an 𝑀-matrix;
−𝐴(𝑡)
̃ +𝐴
̃𝑇 (𝑡)) ≤ −𝑎(𝑡) < 0 (𝑖 = 1, 2);
(2) 𝜆 𝑖 (𝐴(𝑡)
̃
(3) −𝑞𝑎(𝑡) + (1 + 𝑞)‖𝐵(𝑡)‖
+ 𝑞 ≤ 0 with 0 ≤ 𝑞 < 1;
∗
̃
(4) 𝑎(𝑡) − ‖𝐵(𝑡)‖ − 1 ≥ 𝑎0 > 0,
4
Journal of Applied Mathematics
𝑇
1
𝑇 𝑇
= (𝑦 − ℎ) [𝐵21
(𝑡) 𝑃2 (𝑡) + 𝑃2 (𝑡) 𝐵21 (𝑡)]
𝜖
where
1 − 𝑎11 (𝑡)
𝑏11 (𝑡)
𝛽1
𝛼2
̃ (𝑡) = (
𝐴
),
1 − 𝜖0 𝑏21 (𝑡)
𝑎21 (𝑡)
−
𝛼1
𝜖0 𝛽2
𝑇
𝑎12 (𝑡) 𝑏12 (𝑡)
𝛼
𝛼2
𝐵̃ (𝑡) = ( 1
).
𝑎22 (𝑡) 𝑏22 (𝑡)
𝛼1
𝛼2
𝑇
− 2(𝑦 − ℎ) (𝑡) ℎ󸀠 + (𝑦 − ℎ) 𝑃2󸀠 (𝑡) (𝑦 − ℎ)
−
(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) = 𝑥𝑇 𝑃1 (𝑡)𝑥, 𝑊(𝑡, 𝑥, 𝑦) = (𝑦 − ℎ)𝑇 𝑃2 (𝑡)(𝑦 −
ℎ). For the derivative of 𝑉(𝑡, 𝑥) along the trajectory of (11), we
have
1
󵄩2
󵄩
≤ − (1 − 𝜖𝑏21 (𝑡)) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑎21 (𝑡) ‖𝑥‖2
𝜖
󵄩󵄨 󵄨󵄩2
󵄩󵄨
󵄨󵄩2
+ 𝑎22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝑏22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝑐2 (𝑡) ‖𝑢‖2 .
(20)
From (1) of Assumption 7, we can derive (1 − 𝜖0 𝑏21 (𝑡))/
𝜖0 𝛽2 > 0 and the following inequalities for 𝜖 ∈ (0, 𝜖0 ] and
𝑡 ≥ 0:
𝑉󸀠 ≤ −
+
𝑉󸀠 (𝑡, 𝑥) = [𝐴 11 (𝑡) 𝑥 + 𝐴 12 (𝑡) 𝑥𝑡 + 𝐵11 (𝑡) 𝑦
𝑇
+𝐵12 (𝑡) 𝑦𝑡 + 𝐶1 (𝑡) 𝑢] 𝑃1 (𝑡) 𝑥
+ 𝑥𝑇 𝑃1 (𝑡) [𝐴 11 (𝑡) 𝑥 + 𝐴 12 (𝑡) 𝑥𝑡
𝑊󸀠 ≤
𝑎12 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 𝑏12 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
󵄩󵄨𝑉 󵄨󵄩 +
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖ ,
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 1
𝑎21 (𝑡)
1 − 𝜖𝑏21 (𝑡)
𝑎 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
𝑉−
𝑊 + 22
󵄩󵄨𝑉 󵄨󵄩
𝛼1
𝜖𝛽2
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
(21)
𝑏 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
+ 22
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 2
+𝐵11 (𝑡) 𝑦 + 𝐵12 (𝑡) 𝑦𝑡 + 𝐶1 (𝑡) 𝑢]
+ 𝑥𝑇 𝑃1󸀠 (𝑡) 𝑥
= 𝑥𝑇 (𝐴𝑇11 (𝑡) 𝑃1 (𝑡) + 𝑃1 (𝑡) 𝐴 11 (𝑡)) 𝑥
1 − 𝑎11 (𝑡)
𝑏 (𝑡)
𝑉 + 11 𝑊
𝛽1
𝛼2
≤
+ 2𝑥𝑇 𝑃1 (𝑡) [𝐴 12 (𝑡) 𝑥𝑡 + 𝐵11 (𝑡) 𝑦
1 − 𝜖0 𝑏21 (𝑡)
𝑎21 (𝑡)
𝑎 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
𝑉−
𝑊 + 22
󵄩󵄨𝑉 󵄨󵄩
𝛼1
𝜖0 𝛽2
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
+
+𝐵12 (𝑡) 𝑦𝑡 + 𝐶1 (𝑡) 𝑢]
𝑏22 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖ .
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 2
+ 𝑥𝑇 𝑃1󸀠 (𝑡) 𝑥
󵄩󵄨 󵄨󵄩2
≤ − (1 − 𝑎11 (𝑡)) ‖𝑥‖2 + 𝑎12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
󵄩2
󵄩
󵄩󵄨
󵄨󵄩2
+ 𝑏11 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 +𝑏12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 +𝑐1 (𝑡) ‖𝑢‖2 .
(19)
For the derivative of 𝑊(𝑡, 𝑥, 𝑦) along the trajectory of (11), we
have
𝑇
1
= [ (𝐴 21 (𝑡) 𝑥 + 𝐴 22 (𝑡) 𝑥𝑡 + 𝐵21 (𝑡) 𝑦 + 𝐶2 (𝑡) 𝑢) − ℎ󸀠 ]
𝜖
𝑇
× 𝑃2 (𝑡) (𝑦 − ℎ) + (𝑦 − ℎ) 𝑃2 (𝑡)
1
× [ (𝐴 21 (𝑡) 𝑥 + 𝐴 22 (𝑡) 𝑥𝑡 + 𝐵21 (𝑡) 𝑦 + 𝐶2 (𝑡) 𝑢) − ℎ󸀠 ]
𝜖
+ (𝑦 −
󵄩
󵄩2
𝑐0∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 −2𝜆𝑡
󵄨
󵄩
󵄨
󵄩
󵄩
󵄩
󵄨
󵄨
𝑉 ≤ (󵄩󵄩󵄨󵄨𝑉𝑡0 󵄨󵄨󵄩󵄩 + 󵄩󵄩󵄨󵄨𝑊𝑡0 󵄨󵄨󵄩󵄩) 𝑒
+
,
√(1 − 𝑞) 𝑎0∗
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
𝑊 ≤ (󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑉𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑊𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−2𝜆𝑡 +
𝑊󸀠 (𝑡, 𝑥, 𝑦)
𝑇
ℎ) 𝑃2󸀠
It follows from Lemma 3 that there exist positive constants
𝜆, 𝑎0∗ , and 𝑐0∗ such that
(𝑡) (𝑦 − ℎ)
󵄩
󵄩2
𝑐0∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
√(1 − 𝑞) 𝑎0∗
(22)
,
where ‖(𝑐1 (𝑡), 𝑐2 (𝑡))𝑇 ‖ ≤ 𝑐0∗ and 𝜆 is defined by
󵄩
󵄩
󵄩
󵄩
4𝜆 = inf {𝜆 (𝑡) : 𝜆 (𝑡) − (𝑎 (𝑡) − 󵄩󵄩󵄩󵄩𝐵̃ (𝑡)󵄩󵄩󵄩󵄩 − 1) + 󵄩󵄩󵄩󵄩𝐵̃ (𝑡)󵄩󵄩󵄩󵄩 𝑒𝜆(𝑡)𝜏 } .
𝑡≥0
(23)
Journal of Applied Mathematics
5
By the definition of 𝑉(𝑡, 𝑥) and the positive-definiteness of
𝑃1 (𝑡), we have
𝛼1 ‖𝑥‖2 ≤ 𝑉
󵄩
󵄩2
𝑐∗ 󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
≤ (󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑉𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑊𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−2𝜆𝑡 + 0 󵄩
√(1 − 𝑞) 𝑎0∗
󵄩
󵄩2
𝑐0∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩
󵄩2
𝑐0∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩󵄨 󵄨󵄩2
󵄩󵄨 󵄨󵄩2
󵄩󵄨 󵄨󵄩2
≤ (𝛽1 󵄩󵄩󵄩󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨󵄩󵄩󵄩 + 2𝛽2 󵄩󵄩󵄩󵄨󵄨󵄨𝑦0 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝛽2 𝜖02 𝑐22 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 ) 𝑒−2𝜆𝑡
+
√(1 − 𝑞) 𝑎0∗
𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [−𝜏, 0] ,
𝑦 (𝑡) = 𝜓 (𝑡) ,
(28)
Re 𝜆 (𝐴 (𝑡)) ≤ −𝑐1 ,
‖𝐴 (𝑡)‖ ≤ 𝑐2 ,
Re 𝜆 (𝐵 (𝑡)) ≤ −𝑐1 ,
‖𝐵 (𝑡)‖ ≤ 𝑐2 ,
=
𝑐0∗ /√(1 − 𝑞)𝑎0∗ 𝛼1 . Moreover,
Thus, (25) and the inequality
(26)
𝐴𝑇 (𝑡) 𝑃1 (𝑡) + 𝑃1 (𝑡) 𝐴 (𝑡) = −𝐼𝑚 ,
−1/2 1/2
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦󵄩󵄩 ≤ ‖ℎ‖ + (𝛼2 ) 𝑊
󵄩
󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
≤ 2𝑐2 (𝐾1 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−𝜆(𝑡−𝜏) + 𝐾2 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩)
−1/2
󵄩
󵄩
+ 𝑐2 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 + (𝛼2 ) 𝑊1/2
𝛼1
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
) 𝐾1 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−𝜆(𝑡−𝜏)
𝛼2
(30)
where 𝐼𝑚 , 𝐼𝑛 are 𝑚 × 𝑚, 𝑛 × 𝑛 identity matrices, respectively. It follows from the boundness and the positivedefiniteness of 𝑃1 (𝑡) and 𝑃2 (𝑡) that there exist positive constants 𝑀𝑖 , 𝛼𝑖 , and 𝛽𝑖 (𝑖 = 1, 2) such that
󵄩
󵄩
𝑀1 ≤ 󵄩󵄩󵄩𝑃𝑖 (𝑡)󵄩󵄩󵄩 ≤ 𝑀2 ,
imply that
𝑖 = 1, 2,
𝛼1 ‖𝑥‖2 ≤ 𝑥𝑇 𝑃1 (𝑡) 𝑥 ≤ 𝛽1 ‖𝑥‖2 ,
(31)
󵄩 󵄩2
󵄩 󵄩2
𝛼2 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑦𝑇 𝑃2 (𝑡) 𝑦 ≤ 𝛽2 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 .
Assumption 10. There exist bounded functions 𝑎𝑖𝑗 (𝑡), 𝑏𝑖𝑗 (𝑡),
and 𝑐𝑖 (𝑡) (𝑖, 𝑗 = 1, 2) such that
2𝑥𝑇 𝑃1 (𝑡) 𝑓 (𝑡, 𝑥, 𝑥𝑡 , 𝑦, 𝑦𝑡 , 𝑢) + 𝑥𝑇 𝑃1󸀠 (𝑡) 𝑥
󵄩󵄨 󵄨󵄩2
≤ 𝑐1 (𝑡) ‖𝑢‖2 + 𝑎11 (𝑡) ‖𝑥‖2 + 𝑎12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
𝛼
󵄩
󵄩
+ ((2𝑐2 + √ 1 ) 𝐾2 + 𝑐2 ) 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
𝛼2
󵄩
󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
= 𝐾1∗ (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−𝜆(𝑡−𝜏) + 𝐾2∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 ,
If Assumption 9 holds, then there exist the differentiable
positive-definite matrices 𝑃1 (𝑡) and 𝑃2 (𝑡) such that
𝐵𝑇 (𝑡) 𝑃2 (𝑡) + 𝑃2 (𝑡) 𝐵 (𝑡) = −𝐼𝑛 ,
󵄩
󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
‖𝑥‖ ≤ 𝐾1 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−𝜆𝑡 + 𝐾2 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 . (25)
−1/2 1/2
󵄩
󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦󵄩󵄩 − ‖ℎ‖ ≤ 󵄩󵄩󵄩(𝑦 − ℎ) (𝑡)󵄩󵄩󵄩 ≤ (𝛼2 ) 𝑊
󵄩󵄩 󸀠 󵄩󵄩
󵄩󵄩𝐴 (𝑡)󵄩󵄩 ≤ 𝑐2 ,
󵄩
󵄩
󵄩󵄩 󸀠 󵄩󵄩
󵄩󵄩𝐵 (𝑡)󵄩󵄩 ≤ 𝑐2 .
󵄩
󵄩
(29)
= max{𝛽1 /𝛼1 , 2𝛽2 /𝛼1 , 𝛽2 𝜖02 𝑐22 /𝛼1 } and 𝐾22
≤ (2𝑐2 + √
0 < 𝜖 ≪ 1,
,
󵄩
󵄩2
󵄩󵄨 󵄨󵄩2 󵄩󵄨 󵄨󵄩2 󵄩󵄨 󵄨󵄩2
‖𝑥‖2 ≤ 𝐾12 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 ) 𝑒−2𝜆𝑡 +𝐾22 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 ,
(24)
where 𝐾12
𝜖𝑦󸀠 = 𝐵 (𝑡) 𝑦 + 𝑔 (𝑡, 𝑥, 𝑥𝑡 , 𝑢) ,
Assumption 9. There exist positive constants 𝑐1 , 𝑐2 for all 𝑡 ∈ 𝐽
such that
√(1 − 𝑞) 𝑎0∗
󵄩
󵄩2
𝑐0∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
𝑡 ≥ 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.
√(1 − 𝑞) 𝑎0∗
󵄩󵄨 󵄨󵄩2
󵄩󵄨 󵄨󵄩2
󵄩󵄨 󵄨󵄩2
≤ (𝛽1 󵄩󵄩󵄩󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝛽2 󵄩󵄩󵄩󵄨󵄨󵄨𝑦0 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝛽2 󵄩󵄩󵄩󵄨󵄨󵄨ℎ0 󵄨󵄨󵄨󵄩󵄩󵄩 ) 𝑒−2𝜆𝑡
+
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 + 𝑓 (𝑡, 𝑥, 𝑥𝑡 , 𝑦, 𝑦𝑡 , 𝑢) ,
󵄩󵄨 󵄨󵄩2
󵄩󵄨
󵄨󵄩2
≤ (𝛽1 󵄩󵄩󵄩󵄨󵄨󵄨𝑥0 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝛽2 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)0 󵄨󵄨󵄨󵄩󵄩󵄩 ) 𝑒−2𝜆𝑡
+
4. Nonlinear Systems
󵄩2
󵄩
󵄩󵄨
󵄨󵄩2
+ 𝑏11 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑏11 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 ,
𝑇
(27)
where 𝐾1∗ = (2𝑐2 + √𝛼1 /𝛼2 )𝐾1 , 𝐾2∗ = (2𝑐2 + √𝛼1 /𝛼2 )𝐾2 + 𝑐2 .
The proof is complete.
𝑇
− 2(𝑦 − ℎ) 𝑃2 (𝑡) ℎ󸀠 + (𝑦 − ℎ) 𝑃2󸀠 (𝑡) (𝑦 − ℎ)
󵄩󵄨 󵄨󵄩2
≤ 𝑐2 (𝑡) ‖𝑢‖2 + 𝑎21 (𝑡) ‖𝑥‖2 + 𝑎22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
󵄩2
󵄩
󵄩󵄨
󵄨󵄩2
+ 𝑏21 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑏21 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 ,
(32)
6
Journal of Applied Mathematics
For the derivative of 𝑊(𝑡, 𝑥, 𝑦) along the trajectory of (28),
we have
where
−𝐵−1 (𝑡) 𝑔 (𝑡, 𝑥, 𝑥𝑡 , 𝑢 (𝑡)) , 𝑡 ≥ 0
ℎ={
𝑡 ∈ [−𝜏, 0] .
𝜓 (𝑡) − 𝜖𝐵−1 (𝑡) 𝜓󸀠 (𝑡) ,
(33)
𝑇
1
= [ (𝐵 (𝑡) 𝑦 + 𝑔 (𝑡, 𝑥, 𝑥𝑡 , 𝑢)) − ℎ󸀠 ] 𝑃2 (𝑡) (𝑦 − ℎ)
𝜖
Assumption 11. (1) There exist a positive number 𝜖0 such that
̃ is an 𝑀-matrix;
−𝐴(𝑡)
̃ +𝐴
̃𝑇 (𝑡)) ≤ −𝑎(𝑡) < 0 (𝑖 = 1, 2);
(2) 𝜆 𝑖 (𝐴(𝑡)
1
𝑇
+ (𝑦 − ℎ) 𝑃2 (𝑡) [ (𝐵 (𝑡) 𝑦 + 𝑔 (𝑡, 𝑥, 𝑥𝑡 , 𝑢)) − ℎ󸀠 ]
𝜖
𝑇
+ (𝑦 − ℎ) 𝑃2󸀠 (𝑡) (𝑦 − ℎ)
𝑇
1
𝑇
= (𝑦 − ℎ) [𝐵𝑇 (𝑡) 𝑃2 (𝑡) + 𝑃2 (𝑡) 𝐵 (𝑡)] (𝑦 − ℎ)
𝜖
̃
(3) −𝑞𝑎(𝑡) + (1 + 𝑞)‖𝐵(𝑡)‖
+ 𝑞 ≤ 0 with 0 ≤ 𝑞 < 1;
̃
(4) 𝑎(𝑡) − ‖𝐵(𝑡)‖
− 1 ≥ 𝑎0∗ > 0,
𝑇
𝑇
− 2(𝑦 − ℎ) 𝑃2 (𝑡) ℎ󸀠 + (𝑦 − ℎ) 𝑃2󸀠 (𝑡) (𝑦 − ℎ)
1
󵄩2
󵄩󵄨 󵄨󵄩2
󵄩
≤ − (1 − 𝜖𝑏21 (𝑡)) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑎21 (𝑡) ‖𝑥‖2 + 𝑎22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
𝜖
󵄩󵄨
󵄨󵄩2
+ 𝑏22 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝑐2 (𝑡) ‖𝑢‖2 .
(36)
where
−
̃
𝐴 (𝑡) = (
𝑊󸀠 (𝑡, 𝑥, 𝑦)
1 − 𝑎11 (𝑡)
𝑏11 (𝑡)
𝛽1
𝛼2
),
1 − 𝜖0 𝑏21 (𝑡)
𝑎21 (𝑡)
−
𝛼1
𝜖0 𝛽2
𝑎12 (𝑡) 𝑏12 (𝑡)
𝛼
𝛼2
𝐵̃ (𝑡) = ( 1
).
𝑎22 (𝑡) 𝑏22 (𝑡)
𝛼1
𝛼2
(34)
From (1) of Assumption 11, we can derive (1 − 𝜖0 𝑏21 (𝑡))/
𝜖0 𝛽2 > 0 and the following inequalities for 𝜖 ∈ (0, 𝜖0 ]:
𝑉󸀠 ≤ −
+
Theorem 12. If Assumptions 9–11 hold, then the delay singularly perturbed control system (28) is input-to-state stable for
𝜖 ∈ (0, 𝜖0 ].
𝑊󸀠 ≤
Proof. Let 𝑉(𝑡, 𝑥) = 𝑥 𝑃1 (𝑡)𝑥, 𝑊(𝑡, 𝑥, 𝑦) = (𝑦 − ℎ) 𝑃2 (𝑡)(𝑦 −
ℎ). For the derivative of 𝑉(𝑡, 𝑥) along the trajectory of (28),
we have
𝑉󸀠 (𝑡, 𝑥)
𝑏12 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖ ,
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 1
𝑎21 (𝑡)
1 − 𝜖𝑏21 (𝑡)
𝑎 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
𝑉−
𝑊 + 22
󵄩󵄨𝑉 󵄨󵄩
𝛼1
𝜖𝛽2
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
𝑏 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
+ 22
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 2
𝑇
𝑇
1 − 𝑎11 (𝑡)
𝑏 (𝑡)
𝑎 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
𝑉 + 11 𝑊 + 12
󵄩󵄨𝑉 󵄨󵄩
𝛽1
𝛼2
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
≤
(37)
1 − 𝜖0 𝑏21 (𝑡)
𝑎21 (𝑡)
𝑎 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
𝑉−
𝑊 + 22
󵄩󵄨𝑉 󵄨󵄩
𝛼1
𝜖0 𝛽2
𝛼1 󵄩󵄨 𝑡 󵄨󵄩
+
𝑏22 (𝑡) 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
2
󵄩󵄨𝑊 󵄨󵄩 + 𝑐 (𝑡) ‖𝑢‖ .
𝛼2 󵄩󵄨 𝑡 󵄨󵄩 2
𝑇
= [𝐴 (𝑡) 𝑥 + 𝑓 (𝑡, 𝑥, 𝑥𝑡 , 𝑦, 𝑦𝑡 , 𝑢)] 𝑃1 (𝑡) 𝑥
𝑇
+ 𝑥 𝑃1 (𝑡) [𝐴 (𝑡) 𝑥 + 𝑓 (𝑡, 𝑥, 𝑥𝑡 , 𝑦, 𝑦𝑡 , 𝑢)] +
𝑥𝑇 𝑃1󸀠
(𝑡) 𝑥
= 𝑥𝑇 (𝐴𝑇 (𝑡) 𝑃1 (𝑡) + 𝑃1 (𝑡) 𝐴 (𝑡)) 𝑥
+ 2𝑥𝑇 𝑃1 (𝑡) 𝑓 (𝑡, 𝑥, 𝑥𝑡 , 𝑦, 𝑦𝑡 , 𝑢) + 𝑥𝑇 𝑃1󸀠 (𝑡) 𝑥
󵄩󵄨 󵄨󵄩2
≤ − (1 − 𝑎11 (𝑡)) ‖𝑥‖2 + 𝑎12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨𝑥𝑡 󵄨󵄨󵄨󵄩󵄩󵄩
󵄩2
󵄩
󵄩󵄨
󵄨󵄩2
+ 𝑏11 (𝑡) 󵄩󵄩󵄩𝑦 − ℎ󵄩󵄩󵄩 + 𝑏12 (𝑡) 󵄩󵄩󵄩󵄨󵄨󵄨(𝑦 − ℎ)𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝑐1 (𝑡) ‖𝑢‖2 .
(35)
It follows from Lemma 3 that there exist positive constants
𝜆, 𝑎0∗ , and 𝑐0∗ such that
󵄩
󵄩2
𝑐∗ 󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
𝑉 ≤ (󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑉𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑊𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−2𝜆𝑡 + 0 󵄩
,
√(1 − 𝑞) 𝑎0∗
󵄩
󵄩2
𝑐∗ 󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
𝑊 ≤ (󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑉𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝑊𝑡0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−2𝜆𝑡 + 0 󵄩
,
√(1 − 𝑞) 𝑎0∗
where ‖(𝑐1 (𝑡), 𝑐2 (𝑡))𝑇 ‖ ≤ 𝑐0∗ and 𝜆 is defined as in (23).
(38)
Journal of Applied Mathematics
7
By the definitions of 𝑉(𝑡, 𝑥), 𝑊(𝑡, 𝑥, 𝑦), the positivedefiniteness of 𝑃1 (𝑡), 𝑃2 (𝑡), and the similar proof process to
that of Theorem 8, we can obtain
󵄩
󵄩
󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩
‖𝑥‖ ≤ 𝐾1 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0󸀠 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒−𝜆𝑡 + 𝐾2 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 ,
󵄩󵄩 󵄩󵄩
󵄩
󵄩
∗ 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 󵄩󵄨 󸀠 󵄨󵄩 −𝜆(𝑡−𝜏)
+ 𝐾2∗ 󵄩󵄩󵄩𝑢[0,𝑡) 󵄩󵄩󵄩 ,
󵄩󵄩𝑦󵄩󵄩 ≤ 𝐾1 (󵄩󵄩󵄩󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜓0 󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩) 𝑒
(39)
where 𝐾1
√max{𝛽1 /𝛼1 , 2𝛽2 /𝛼1 , 𝛽2 𝜖02 𝑐22 /𝛼1 }, 𝐾2
=
=
√ 𝑐0∗ /√(1 − 𝑞)𝑎0∗ 𝛼1 , 𝐾1∗ = (2𝑐2 + √𝛼1 /𝛼2 )𝐾1 and 𝐾2∗ =
(2𝑐2 + √𝛼1 /𝛼2 )𝐾2 + 𝑐2 . The proof is complete.
Example 2. Consider the following nonlinear delay system as
an application of Theorem 12:
𝑥󸀠 (𝑡) = −15𝑥 (𝑡)+5 ln (1+𝑥2 (𝑡 − 𝜏))+sin (𝑦 (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󸀠 (𝑡, 𝑥) =
≤
𝑊󸀠 (𝑡, 𝑥, 𝑦) =
5. Examples
1
𝑥 (−15𝑥 + 5 ln (1 + 𝑥𝑡2 ) + sin 𝑦 + 𝑢)
15
−68
2
16 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 1
2
𝑉+ 𝑊+
󵄩󵄨𝑉 󵄨󵄩 + ‖𝑢‖ ,
3
5
3 󵄩󵄨 𝑡 󵄨󵄩 30
1
1
(𝑦 − ℎ) ( (2𝑥𝑡 − 6𝑦) − ℎ󸀠 )
6
𝜖
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)
1
𝑉󸀠 (𝑡, 𝑥) = 𝑥 (−5𝑥 + 𝑦𝑡 + 𝑢)
5
37
3 󵄩󵄨 󵄨󵄩 󵄩󵄨 󵄨󵄩 1
≤ − 𝑉 + 󵄩󵄩󵄩󵄨󵄨󵄨𝑉𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄨󵄨󵄨𝑊𝑡 󵄨󵄨󵄨󵄩󵄩󵄩 + ‖𝑢‖2 ,
5
5
5
𝑊󸀠 (𝑡, 𝑥, 𝑦) =
+
̃ (𝑡) = (
𝐴
(41)
10 114
+
)𝑊
𝜖
25
9 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 3 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 3
2
󵄩󵄨𝑉 󵄨󵄩 󵄩󵄨𝑊 󵄨󵄩 + ‖𝑢‖ .
25 󵄩󵄨 𝑡 󵄨󵄩 5 󵄩󵄨 𝑡 󵄨󵄩 50
(42)
̃ (𝑡) = (
𝐴
2
5
68
3
0
−
12 67 ) ,
+
𝜖
9
16
0
3
𝐵̃ (𝑡) = ( 61 1 ) .
9 3
(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
So we obtain the matrices
−
1 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩 1
2
󵄩󵄨𝑊 󵄨󵄩 + ‖𝑢‖ .
3 󵄩󵄨 𝑡 󵄨󵄩 36
So we obtain the matrices
1
1
3
(𝑦 − ℎ) ( (3𝑥 − 5𝑦) − (−5𝑥 + 𝑦𝑡 + 𝑢))
5
𝜖
5
≤ 3𝑉 + (−
12 67
61 󵄩󵄩󵄨󵄨 󵄨󵄨󵄩󵄩
+ )𝑊 +
󵄩󵄨𝑉 󵄨󵄩
𝜖
9
9 󵄩󵄨 𝑡 󵄨󵄩
−
Let 𝑉 = 𝑥2 /10, 𝑊 = (𝑦 − ℎ)2 /10, ℎ = 3𝑥(𝑡)/5. Then
(45)
37
5
3
0
−
10 114 ) ,
+
𝜖
25
3
1
5
𝐵̃ (𝑡) = ( 9 3 ) .
25 5
(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 ].
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.
8
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