Abstract and Applied Analysis
Volume 2010 (2010), Article ID 405321, 18 pages
doi:10.1155/2010/405321
Research Article

A Class of Impulsive Pulse-Width Sampler Systems and Its Steady-State Control in Infinite Dimensional Spaces

JinRong Wang, W. Wei, and YanLong Yang

College of Science, Guizhou University, Guiyang, Guizhou 550025, China
College of Technology of Guizhou University, Guiyang, Guizhou 550004, China

Received 11 November 2009; Accepted 25 January 2010

Academic Editor: Paul Eloe

Copyright © 2010 JinRong Wang 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.

Abstract

This paper investigates a class of impulsive pulse-width sampler systems and its steadystate control in the infinite dimensional spaces. Firstly, some definitions of pulse-width sampler systems with impulses are introduced. Then applying impulsive evolution operator and fixed point theorem, some existent results of steady-state of infinite dimensional linear and semilinear pulse-width sampler systems with impulses are obtained. An example to illustrate the theory is presented in the end.

1. Introduction

In the design of distributed parameter control systems, one of the important problems is to choose controller and actuator. As the dimension of an industrial controller in actual applications is finite, it restricts us to consider the distributed parameter system with a finite dimensional output. In industrial process control systems, on-off actuators have been in engineer's good graces because of the cheap price and the high reliability.

The interest in the pulse-width sampler control systems was aroused as early as 1960s. It was motivated by applications to engineering problems and neural nets modeling. In modern times, the development of neurocomputers promises a rebirth of interest in this field. The theory of pulse-width sampler control systems is treated as a very important branch of engineering and mathematics. Nevertheless, it can supply a technical-minded mathematician with a number of new and interesting problems of mathematical nature. There are some results such as steady-state control, stability analysis, robust control of pulse-width sampler systems [1–7], integral control by variable sampling based on steady-state data, and adaptive sampled-data integral control [8–11].

On the other hand, in order to describe dynamics of population, subject to abrupt changes as well as other phenomena, such as harvesting, diseases and so forth, some authors have used impulsive differential equations to describe the model since the last century. The reader can refer the basic theory of impulsive differential equations in finite dimensional spaces to Lakshmikantham's book [12]. Meanwhile, the impulsive evolution equations and its optimal control problems on infinite dimensional Banach spaces have been investigated by many authors including Ahmed, Liu, Nieto, and us (see for instance [13–25] and references therein).

However, to our knowledge, the pulse-width sampler systems with impulse on infinite dimensional spaces have not been investigated extensively. In this paper, we first study the following steady-state control of infinite dimensional linear system with impulses ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝑓 ( 𝑡 ) + 𝐶 𝑢 ( 𝑡 ) , 𝑡 ≠ 𝜏 𝑘 , 𝜏 Δ 𝑥 𝑘 = 𝐵 𝑘 𝑥 𝜏 𝑘 + 𝑐 𝑘 , 𝑘 = 1 , 2 , … , 𝑧 ( 𝑡 ) = 𝐾 1 𝑥 ( 𝑡 ) , ( 1 . 1 ) where the state variable 𝑥 ( 𝑡 ) takes values in a reflexive Banach space 𝑋 , 𝐴 is the infinitesimal generator of a 𝐶 0 -semigroup { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } on the state space 𝑋 , 𝑓 ( 𝑡 ) = 𝑓 ⋅ 1 ( 𝑡 ) is 𝑇 0 -periodic step disturbance of the system and 𝑓 ∈ 𝑋 . Control variable 𝑢 ( 𝑡 ) ∈ ℝ 𝑞 , the input 𝐶 ∶ ℝ 𝑞 → 𝑋 is a bounded linear operator. There is only one time sequences { 𝜏 𝑘 ∣ 𝑘 ∈ ℤ + 0 } satisfing 0 < 𝜏 1 < 𝜏 2 < ⋯ < 𝜏 𝑘 ⋯ and l i m 𝑘 → ∞ 𝜏 𝑘 = ∞ , 𝐵 𝑘 ∶ 𝑋 → 𝑋 , 0 < 𝜏 1 < 𝜏 2 < ⋯ < 𝜏 𝛿 < 𝑇 0 , 𝜏 𝑘 + 𝛿 = 𝜏 𝑘 + 𝑇 0 , Δ 𝑥 ( 𝜏 𝑘 ) = 𝑥 ( 𝜏 + 𝑘 ) − 𝑥 ( 𝜏 − 𝑘 ) , 𝑥 ( 𝜏 + 𝑘 ) = l i m ℎ → 0 + = 𝑥 ( 𝜏 𝑘 + ℎ ) and 𝑥 ( 𝜏 − 𝑘 ) = 𝑥 ( 𝜏 𝑘 ) represent, respectively the right and left limits of 𝑥 ( 𝑡 ) at 𝑡 = 𝜏 𝑘 . 𝐾 1 ∶ 𝑋 → ℝ 𝑝 is a given bounded linear operator; 𝑧 ( 𝑡 ) is the 𝑝 dimensional output of the system (1.1).

We, then, study the following steady-state control of infinite dimensional semilinear system with impulses ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝐶 𝑢 ( 𝑡 ) , 𝑡 ≠ 𝜏 𝑘 , 𝜏 Δ 𝑥 𝑘 = 𝐵 𝑘 𝑥 𝜏 𝑘 + 𝑐 𝑘 , 𝑘 = 1 , 2 , … , 𝑧 ( 𝑡 ) = 𝐾 1 𝑥 ( 𝑡 ) , ( 1 . 2 ) where 𝑓 ∶ [ 0 , ∞ ) × 𝑋 → 𝑋 is 𝑇 0 -periodic continuous function.

Suppose that control signal 𝑢 ( 𝑡 ) is the output of the 𝑞 dimensional pulse-width sampler controller, and 𝑣 ( 𝑡 ) is the input of the 𝑞 dimensional pulse-width sampler controller, which is the output of some dynamical controller

̇ 𝑣 ( 𝑡 ) = 𝐽 𝑣 ( 𝑡 ) + 𝐾 2 𝑧 ( 𝑡 ) , ( 1 . 3 ) where 𝐽 is a 𝑞 × 𝑞 matrix, 𝐾 2 is a 𝑞 × 𝑝 matrix, 𝐽 is determined by the dynamic characteristics of the controller, and 𝐾 2 is called the feedback matrix which will be chosen in the latter (see Theorem 3.4 and Theorem 3.8). The output signal 𝑢 ( 𝑡 ) = ( 𝑢 1 ( 𝑡 ) , 𝑢 2 ( 𝑡 ) , … , 𝑢 𝑞 ( 𝑡 ) ) 𝑇 and the input signal 𝑣 ( 𝑡 ) = ( 𝑣 1 ( 𝑡 ) , 𝑣 2 ( 𝑡 ) , … , 𝑣 𝑞 ( 𝑡 ) ) 𝑇 of the pulse-width sampler satisfy the following dynamic relation:

𝑢 𝑖 ( 𝑡 ) = s i g n 𝛼 𝑛 𝑖 , 𝑛 𝑇 0 | | 𝛼 ≤ 𝑡 < 𝑛 + 𝑛 𝑖 | | 𝑇 0 | | 𝛼 , 𝑖 = 1 , 2 , … , 𝑞 ; 0 , 𝑛 + 𝑛 𝑖 | | 𝑇 0 ≤ 𝑡 < ( 𝑛 + 1 ) 𝑇 0 𝛼 , 𝑛 = 0 , 1 , … , ( 1 . 4 ) 𝑛 𝑖 = 𝑣 𝑖 𝑛 𝑇 0 , | | 𝑣 𝑖 𝑛 𝑇 0 | | ≤ 1 , 𝑖 = 1 , 2 , … , 𝑞 ; s i g n 𝑣 𝑖 𝑛 𝑇 0 , | | 𝑣 𝑖 𝑛 𝑇 0 | | ≥ 1 , 𝑛 = 0 , 1 , … , ( 1 . 5 ) where 𝑇 0 is called the sampling period of the pulse-width sampler which is the same as the period of 𝑓 and 𝜏 𝑘 , 𝑘 = 1 , 2 , … .

We end this introduction by giving some definitions.

Definition 1.1. The closed-loop system (1.1), (1.3)–(1.5) is called linear pulse-width sampler control system with impulses. The closed-loop system (1.2), (1.3)–(1.5) is called semilinear pulse-width sampler control system with impulses.

Definition 1.2. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), the 𝑞 dimensional vector 𝛼 𝑛 = ( 𝛼 𝑛 1 , 𝛼 𝑛 2 , … , 𝛼 𝑛 𝑞 ) 𝑇 is called the duration ratio of the pulse-width sampler in the 𝑛 th sampling period, 𝑛 = 0 , 1 , … .
We defined a closed cube Ω in ℝ 𝑞 as 𝛼 Ω = 𝛼 = 1 , 𝛼 2 , … , 𝛼 𝑞 𝑇 ∈ ℝ 𝑞 ∣ | | 𝛼 𝑖 | | ≤ 1 , 𝑖 = 1 , 2 , … , 𝑞 , ( 1 . 6 ) then we have 𝛼 𝑛 ∈ Ω , for 𝑛 = 0 , 1 , … .

Definition 1.3. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), if there exists a 𝑞 dimensional vector 𝛼 𝛼 = 𝑛 1 , 𝛼 𝑛 2 , … , 𝛼 𝑛 𝑞 𝑇 ∈ Ω , ( 1 . 7 ) and a corresponding periodicity rectangular-wave control signal 𝑢 ( 𝑡 ) = 𝑢 ( 𝑡 , 𝛼 ) defined by 𝑢 𝑖 ( 𝑡 ) = 𝑢 𝑖 ( 𝑡 , 𝛼 ) = s i g n 𝛼 𝑖 , 𝑛 𝑇 0 | | 𝛼 ≤ 𝑡 < 𝑛 + 𝑛 𝑖 | | 𝑇 0 | | 𝛼 , 𝑖 = 1 , 2 , … , 𝑞 ; 0 , 𝑛 + 𝑛 𝑖 | | 𝑇 0 ≤ 𝑡 < ( 𝑛 + 1 ) 𝑇 0 , 𝑛 = 0 , 1 , … . ( 1 . 8 ) such that the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), has a corresponding 𝑇 0 -periodic trajectory 𝑥 ( ⋅ ) = 𝑥 ( ⋅ , 𝛼 ) ∶ 𝑥 ( 𝑡 + 𝑇 0 , 𝛼 ) = 𝑥 ( 𝑡 , 𝛼 ) , 𝑡 ≥ 0 , then the control signal (1.8) is called the steady-state control with respect to the disturbance 𝑓 . The 𝑇 0 -periodic trajectory 𝑥 ( ⋅ ) is called steady-state corresponding to steady-state control 𝑢 ( ⋅ ) and the constant vector 𝛼 ∈ Ω of steady-state control (1.8) is called to be a steady-state duration ratio.

Definition 1.4. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), if there exists some 𝛼 ∈ Ω such that l i m 𝑛 → ∞ 𝛼 𝑛 = 𝛼 , w h e r e 𝛼 𝑛 = 𝛼 𝑛 1 , 𝛼 𝑛 2 , … , 𝛼 𝑛 𝑞 𝑇 𝛼 , 𝛼 = 1 , 𝛼 2 , … , 𝛼 𝑞 𝑇 , ( 1 . 9 ) then system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), corresponding to the disturbance 𝑓 is called to be stead-state stable.
Further, system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), corresponding to the perturbation 𝑓 is called stead-state stabilizability if we can choose a suitable 𝑇 0 > 0 and 𝐾 2 such that system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), is stead-state stable.

2. Mathematical Preliminaries

Let L ( 𝑋 , 𝑋 ) denote the space of linear operators from 𝑋 to 𝑋 , L 𝑏 ( 𝑋 , 𝑋 ) denote the space of bounded linear operators from 𝑋 to 𝑋 , L 𝑏 ( ℝ 𝑞 , 𝑋 ) denote the space of bounded linear operators from ℝ 𝑞 to 𝑋 , and L 𝑏 ( 𝑋 , ℝ 𝑝 ) denote the space of bounded linear operators from 𝑋 to ℝ 𝑝 . It is obvious that L 𝑏 ( 𝑋 , 𝑋 ) , L 𝑏 ( ℝ 𝑞 , 𝑋 ) , and L 𝑏 ( 𝑋 , ℝ 𝑝 ) is the Banach space with the usual supremum norm.

Define 𝐷 = { 𝜏 1 , … , 𝜏 𝛿 } ⊂ [ 0 , 𝑇 0 ] , where 0 < 𝜏 1 < 𝜏 2 < ⋯ < 𝜏 𝛿 < 𝑇 0 . We introduce 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝑋 ) ≡ { 𝑥 ∶ [ 0 , 𝑇 0 ] → 𝑋 | 𝑥 is continuous at 𝑡 ∈ [ 0 , 𝑇 0 𝐷 ] ⧵ , 𝑥 is continuous from left and has right hand limits at 𝑡 ∈ 𝐷 } , and 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝑋 ) ≡ { 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝑋 ) ∣ ̇ 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝑋 ) } . Set

‖ 𝑥 ‖ 𝑃 𝐶 = m a x s u p 𝑡 ∈ 0 , 𝑇 0 ‖ 𝑥 ( 𝑡 + 0 ) ‖ , s u p 𝑡 ∈ 0 , 𝑇 0 ‖ 𝑥 ( 𝑡 − 0 ) ‖ , ‖ 𝑥 ‖ 𝑃 𝐶 1 = ‖ 𝑥 ‖ 𝑃 𝐶 + ‖ ̇ 𝑥 ‖ 𝑃 𝐶 . ( 2 . 1 ) It can be seen that endowed with the norm ‖ ⋅ ‖ 𝑃 𝐶 ( ‖ ⋅ ‖ 𝑃 𝐶 1 ) , 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝑋 ) ( 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝑋 ) ) is a Banach space.

We introduce the following assumption [H1].

(i) [H1.1] 𝐴 is the infinitesimal generator of a 𝐶 0 -semigroup { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } on 𝑋 with domain 𝐷 ( 𝐴 ) . (ii) [H1.2] There exists 𝛿 such that 𝜏 𝑘 + 𝛿 = 𝜏 𝑘 + 𝑇 0 . (iii) [H1.3] For each 𝑘 ∈ ℤ + 0 , 𝐵 𝑘 ∈ L 𝑏 ( 𝑋 , 𝑋 ) and 𝐵 𝑘 + 𝛿 = 𝐵 𝑘 .

We first recall the homogeneous linear impulsive periodic system ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) , 𝑡 ≠ 𝜏 𝑘 , Δ 𝑥 ( 𝑡 ) = 𝐵 𝑘 𝑥 ( 𝑡 ) , 𝑡 = 𝜏 𝑘 , ( 2 . 2 ) and the associated Cauchy problem ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝜏 𝐷 , Δ 𝑥 𝑘 = 𝐵 𝑘 𝑥 𝜏 𝑘 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = 𝑥 . ( 2 . 3 )

If 𝑥 ∈ 𝐷 ( 𝐴 ) and 𝐷 ( 𝐴 ) is an invariant subspace of 𝐵 𝑘 , using [18, Theorem 5.2.2, page 144], step by step, one can verify that the Cauchy problem (2.3) has a unique classical solution 𝑥 ∈ 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝑋 ) represented by 𝑥 ( 𝑡 ) = 𝑆 ( 𝑡 , 0 ) 𝑥 , where

𝑆 ( ⋅ , ⋅ ) ∶ Δ = ( 𝑡 , 𝜃 ) ∈ 0 , 𝑇 0 × 0 , 𝑇 0 ∣ 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 ⟶ L 𝑏 ( 𝑋 , 𝑋 ) ( 2 . 4 )

given by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑆 ( 𝑡 , 𝜃 ) = 𝑇 ( 𝑡 − 𝜃 ) , 𝜏 𝑘 − 1 ≤ 𝜃 ≤ 𝑡 ≤ 𝜏 𝑘 , 𝑇 𝑡 − 𝜏 + 𝑘 𝐼 + 𝐵 𝑘 𝑇 𝜏 𝑘 − 𝜃 , 𝜏 𝑘 − 1 ≤ 𝜃 < 𝜏 𝑘 < 𝑡 ≤ 𝜏 𝑘 + 1 , 𝑇 𝑡 − 𝜏 + 𝑘 ∐ 𝜃 < 𝜏 𝑗 < 𝑡 𝐼 + 𝐵 𝑗 𝑇 𝜏 𝑗 − 𝜏 + 𝑗 − 1 𝐼 + 𝐵 𝑖 𝑇 𝜏 𝑖 , 𝜏 − 𝜃 𝑖 − 1 ≤ 𝜃 < 𝜏 𝑖 ≤ ⋯ < 𝜏 𝑘 < 𝑡 ≤ 𝜏 𝑘 + 1 . ( 2 . 5 ) The operator { 𝑆 ( 𝑡 , 𝜃 ) , ( 𝑡 , 𝜃 ) ∈ Δ } is called impulsive evolution operator associated with { 𝐵 𝑘 ; 𝜏 𝑘 } ∞ 𝑘 = 1 .

The properties of the impulsive evolution operator, { 𝑆 ( 𝑡 , 𝜃 ) , ( 𝑡 , 𝜃 ) ∈ Δ } associated with { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } and { 𝐵 𝑘 ; 𝜏 𝑘 } ∞ 𝑘 = 1 , are collected here.

Lemma 2.1 (see [26, Lemma 2.1] [27]). Let assumption [H1] hold. The impulsive evolution operator { 𝑆 ( 𝑡 , 𝜃 ) , ( 𝑡 , 𝜃 ) ∈ Δ } has the following properties. (1)For 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 , 𝑆 ( 𝑡 , 𝜃 ) ∈ L 𝑏 ( 𝑋 , 𝑋 ) , there exists a 𝑀 𝑇 0 > 0 such that s u p 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 ‖ 𝑆 ( 𝑡 , 𝜃 ) ‖ ≤ 𝑀 𝑇 0 . (2)For 0 ≤ 𝜃 < 𝑟 < 𝑡 ≤ 𝑇 0 , 𝑟 ≠ 𝜏 𝑘 , 𝑆 ( 𝑡 , 𝜃 ) = 𝑆 ( 𝑡 , 𝑟 ) 𝑆 ( 𝑟 , 𝜃 ) .(3)For 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 , 𝑛 ∈ 𝑍 + , 𝑆 ( 𝑡 + 𝑛 𝑇 0 , 𝜃 + 𝑛 𝑇 0 ) = 𝑆 ( 𝑡 , 𝜃 ) .(4)For 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 , 𝑛 ∈ 𝑍 + , 𝑆 ( 𝑡 + 𝑛 𝑇 0 , 0 ) = 𝑆 ( 𝑡 , 0 ) [ 𝑆 ( 𝑇 0 , 0 ) ] 𝑛 .(5)For 0 ≤ 𝜃 < 𝑡 , there exists an 𝑀 ≥ 1 , 𝜔 ∈ ℝ such that ( ⎧ ⎪ ⎨ ⎪ ⎩ ‖ 𝑆 𝑡 , 𝜃 ) ‖ ≤ 𝑀 e x p 𝜔 ( 𝑡 − 𝜃 ) + 𝜃 ≤ 𝜏 𝑛 < 𝑡 𝑀 ‖ ‖ l n 𝐼 + 𝐵 𝑛 ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ . ( 2 . 6 )

The exponential stability of the impulsive evolution operator { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } will be used throughout the paper; we recall them as the following definitions and lemmas.

Definition 2.2. { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is called exponentially stable if there exist 𝐾 ≥ 0 and 𝜈 > 0 such that ‖ 𝑆 ( 𝑡 , 𝜃 ) ‖ ≤ 𝐾 𝑒 − 𝜈 ( 𝑡 − 𝜃 ) , 𝑡 > 𝜃 ≥ 0 . ( 2 . 7 )  Assumption [H2]: { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } is exponentially stable, that is, there exist 𝐾 0 > 0 and 𝜈 0 > 0 such that ‖ 𝑇 ( 𝑡 ) ‖ ≤ 𝐾 0 𝑒 − 𝜈 0 𝑡 , 𝑡 > 0 . ( 2 . 8 )  Two important criteria for exponential stability of a 𝐶 0 -semigroup are collected here.

Lemma 2.3 (see [26, Lemma 2.4]). Assumptions [H1] and [H2] hold. There exists 0 < 𝜆 < 𝜈 0 such that 𝛿 𝑘 = 1 𝐾 0 ‖ ‖ 𝐼 + 𝐵 𝑘 ‖ ‖ 𝑒 − 𝜆 𝑇 0 < 1 . ( 2 . 9 ) Then { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is exponentially stable.

Lemma 2.4 (see [26, Lemma 2.5]). Assume that assumption [H1] holds. Suppose 0 < 𝜇 1 = i n f 𝑘 = 1 , 2 , … , 𝛿 𝜏 𝑘 − 𝜏 𝑘 − 1 ≤ s u p 𝑘 = 1 , 2 , … , 𝛿 𝜏 𝑘 − 𝜏 𝑘 − 1 = 𝜇 2 < ∞ . ( 2 . 1 0 ) If there exists 𝛼 > 0 such that 1 𝜔 + 𝜇 𝑀 ‖ ‖ l n 𝐼 + 𝐵 𝑘 ‖ ‖ ≤ − 𝛾 < 0 , 𝑘 = 1 , 2 , … , 𝛿 , ( 2 . 1 1 ) where 𝜇 𝜇 = 1 𝜇 , 𝛾 + 𝜔 < 0 , 2 , 𝛾 + 𝜔 ≥ 0 , ( 2 . 1 2 ) then { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is exponentially stable.

Remark 2.5 (see [26, Theorem 3.2]). If { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is exponentially stable, then [ 𝐼 − 𝑆 ( 𝑇 0 , 0 ) ] is inverse and [ 𝐼 − 𝑆 ( 𝑇 0 , 0 ) ] − 1 ∈ L 𝑏 ( 𝑋 , 𝑋 ) .

3. Steady-State Control

In this section, we study the steady-state control of pulse-width sampler control system with impulses. First we introduce the following assumptions.

[H3]: 𝑓 ( 𝑡 ) , 𝑡 ≥ 0 , is 𝑇 0 -periodic step perturbation. [H4]: Control signal 𝑢 ( 𝑡 ) is 𝑇 0 -periodic, which is defined by the rectangular wave signal 𝑢 ( 𝑡 , 𝛼 ) , 𝛼 ∈ Ω given by (1.8).

 Similar to the proof of Theorem 3.2 [26], one can obtain the following results immediately.

Lemma 3.1. Assumptions [H1], [H3], and [H4] hold. Suppose { 𝑆 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is exponentially stable; for every 𝑢 ( 𝑡 , 𝛼 ) , system (1.1) has a unique 𝑇 0 -periodic 𝑃 𝐶 -mild solution 𝑥 ( 𝑡 , 𝛼 ) = 𝑆 ( 𝑡 , 0 ) 𝑥 0 + 𝑡 0 𝑆 ( 𝑡 , 𝜃 ) ( 𝑓 ( 𝜃 ) + 𝐶 𝑢 ( 𝜃 , 𝛼 ) ) 𝑑 𝜃 + 0 ≤ 𝜏 𝑘 < 𝑡 𝑆 𝑡 , 𝜏 + 𝑘 𝑐 𝑘 , ( 3 . 1 ) where 𝑥 0 = 𝑇 𝐼 − 𝑆 0 , 0 − 1 𝑇 0 0 𝑆 𝑇 0 𝑇 , 𝜃 ( 𝑓 ( 𝜃 ) + 𝐶 𝑢 ( 𝜃 , 𝛼 ) ) 𝑑 𝜃 , 𝐼 − 𝑆 0 , 0 − 1 ∈ L 𝑏 ( 𝑋 , 𝑋 ) , ( 3 . 2 ) which is globally asymptotically stable.

By Lemma 3.1, we have the following results.

Theorem 3.2. Under the assumptions of Lemma 3.1, if the sampler periodic 𝑇 0 has the following properties: 𝑖 𝜔 𝑛 ∈ 𝜌 ( 𝐽 ) , 𝜔 𝑛 = 2 𝑛 𝜋 𝑇 0 , 𝑛 = 0 , ± 1 , ± 2 , … , ( 3 . 3 ) where 𝜌 ( 𝐽 ) is the resolvent set of the matrix 𝐽 , 𝑖 satisfies 𝑖 2 = − 1 , then the following open-loop control system ̇ 𝑥 ( 𝑡 , 𝛼 ) = 𝐴 𝑥 ( 𝑡 , 𝛼 ) + 𝑓 ( 𝑡 ) + 𝐶 𝑢 ( 𝑡 , 𝛼 ) , 𝑡 ≠ 𝜏 𝑘 , Δ 𝑥 ( 𝑡 , 𝛼 ) = 𝐵 𝑘 𝑥 ( 𝑡 , 𝛼 ) + 𝑐 𝑘 , 𝑡 = 𝜏 𝑘 , 𝑧 ( 𝑡 ) = 𝐾 1 ̇ 𝑥 ( 𝑡 ) , 𝑣 ( 𝑡 , 𝛼 ) = 𝐽 𝑣 ( 𝑡 , 𝛼 ) + 𝐾 2 𝑧 ( 𝑡 , 𝛼 ) ( 3 . 4 ) has a unique 𝑇 0 -periodic 𝑃 𝐶 -mild solution 𝑣 ( 𝑡 , 𝛼 ) given by 𝑣 ( 𝑡 , 𝛼 ) = 𝑒 𝐽 𝑡 𝐼 − 𝑒 𝐽 𝑇 0 − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 + 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 𝑡 0 𝑒 𝐽 ( 𝑡 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 , ( 3 . 5 )

Proof. By (3.3), we know that 𝑒 𝑖 𝜔 𝑛 𝑇 0 = 𝑒 𝑖 2 𝑛 𝜋 = 1 , that is 1 ∈ 𝜌 ( 𝑒 𝐽 𝑇 0 ) . Thus ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 exists and is bounded. It is not difficult to see that 𝑣 ( 𝑡 , 𝛼 ) = 𝑒 𝐽 𝑡 𝑣 0 + 𝑡 0 𝑒 𝐽 ( 𝑡 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 , ( 3 . 6 ) where 𝑣 0 = 𝑣 ( 0 , 𝛼 ) .
Consider 𝑦 = ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 , ( 3 . 7 ) which is the unique solution of the following equation: 𝑦 = 𝑒 𝐽 𝑡 𝑦 + 𝑡 0 𝑒 𝐽 ( 𝑡 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 . ( 3 . 8 ) Let 𝑣 0 = 𝑦 = 𝐼 − 𝑒 𝐽 𝑇 0 − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 ; ( 3 . 9 ) it comes from Lemma 3.1 that 𝑧 𝑡 + 𝑇 0 , 𝛼 = 𝑧 ( 𝑡 , 𝛼 ) , 𝑡 ≥ 0 . ( 3 . 1 0 ) It is easy to verify that 𝑣 ( 𝑡 , 𝛼 ) = 𝑒 𝐽 𝑡 ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 + 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 𝑡 0 𝑒 𝐽 ( 𝑡 − 𝑠 ) 𝐾 2 𝑧 ( 𝑠 , 𝛼 ) 𝑑 𝑠 ( 3 . 1 1 ) is just the 𝑇 0 -periodic 𝑃 𝐶 -mild solution 𝑣 ( 𝑡 , 𝛼 ) of open-loop control system (3.4).
In order to discuss the existence of steady-state control of system (1.1), we define a map 𝐺 ∶ Ω ∈ ℝ 𝑞 → ℝ 𝑞 given by 𝐺 ( 𝛼 ) = 𝐼 − 𝑒 𝐽 𝑇 0 − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 𝐾 1 𝑥 ( 𝑠 , 𝛼 ) 𝑑 𝑠 , 𝛼 ∈ Ω , ( 3 . 1 2 ) where 𝑥 ( ⋅ , 𝛼 ) is the 𝑇 0 -periodic 𝑃 𝐶 -mild solution of system (1.1) corresponding to 𝛼 ∈ Ω . Then we have the following result.

Lemma 3.3. Under the assumptions of Theorem 3.2, there exists a constant 𝑀 > 0 such that ‖ ‖ 𝐺 ( 𝛼 ) − 𝐺 𝛼 ‖ ‖ ≤ 𝑀 ‖ ‖ 𝐾 2 ‖ ‖ ‖ ‖ 𝛼 − 𝛼 ‖ ‖ , 𝛼 , 𝛼 ∈ Ω . ( 3 . 1 3 )

Proof. Suppose 𝑥 1 ( 𝑡 , 𝛼 ) and 𝑥 2 ( 𝑡 , 𝛼 ) are the 𝑇 0 -periodic 𝑃 𝐶 -mild solution of system (1.1) corresponding to 𝛼 and 𝛼 ∈ Ω , respectively, then 𝑥 1 ( 0 ) − 𝑥 2 ( 0 ) = 𝑥 1 𝑇 0 − 𝑥 2 𝑇 0 𝑇 = 𝑆 0 𝑥 , 0 1 ( 0 ) − 𝑥 2 + ( 0 ) 𝑇 0 0 𝑆 𝑇 0 𝐶 , 𝜃 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 𝜃 , 𝛼 𝑑 𝜃 . ( 3 . 1 4 ) Thus, ‖ ‖ 𝑥 1 ( 0 ) − 𝑥 2 ‖ ‖ ≤ ‖ ‖ ( 0 ) [ 𝐼 − 𝑆 ( 𝑇 0 , 0 ) ] − 1 ‖ ‖ ‖ ‖ 𝑆 𝑇 0 ‖ ‖ , 𝜃 ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 . ( 3 . 1 5 ) For 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 , we obtain ‖ ‖ 𝑥 1 ( 𝑡 , 𝛼 ) − 𝑥 2 𝑡 , 𝛼 ‖ ‖ ‖ ‖ 𝑥 ≤ ‖ 𝑆 ( 𝑡 , 0 ) ‖ 1 ( 0 ) − 𝑥 2 ‖ ‖ + ‖ ‖ 𝑆 𝑇 ( 0 ) 0 ‖ ‖ , 𝜃 ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 ≤ 𝐾 ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) ( ‖ 𝑄 ‖ 𝐾 + 1 ) 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 ≤ 𝑑 𝜃 𝑀 1 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 , ( 3 . 1 6 ) where 𝑀 1 = 𝐾 ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) ( ) ‖ 𝑄 ‖ 𝐾 + 1 , 𝑄 = [ 𝐼 − 𝑆 ( 𝑇 0 , 0 ) ] − 1 . ( 3 . 1 7 ) By elementaly computation, ‖ ‖ 𝐺 ( 𝛼 ) − 𝐺 𝛼 ‖ ‖ ≤ ‖ ‖ ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 ‖ ‖ ‖ ‖ 𝑒 𝐽 𝑇 0 ‖ ‖ ‖ ‖ 𝐾 2 ‖ ‖ ‖ ‖ 𝐾 1 ‖ ‖ L 𝑏 ( 𝑋 , ℝ 𝑝 ) 𝑇 0 0 ‖ ‖ 𝑥 1 ( 𝑠 , 𝛼 ) − 𝑥 2 𝑠 , 𝛼 ‖ ‖ ≤ ‖ ‖ 𝑑 𝑠 ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 ‖ ‖ ‖ ‖ 𝑒 𝐽 𝑇 0 ‖ ‖ ‖ ‖ 𝐾 2 ‖ ‖ ‖ ‖ 𝐾 1 ‖ ‖ L 𝑏 ( 𝑋 , ℝ 𝑝 ) 𝑀 1 𝑇 0 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 ≤ 𝑑 𝜃 𝑀 2 ‖ ‖ 𝐾 2 ‖ ‖ 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 , ( 3 . 1 8 ) where 𝑀 2 = ‖ ‖ ( 𝐼 − 𝑒 𝐽 𝑇 0 ) − 1 ‖ ‖ ‖ ‖ 𝑒 𝐽 𝑇 0 ‖ ‖ ‖ ‖ 𝐾 1 ‖ ‖ L 𝑏 ( 𝑋 , ℝ 𝑝 ) 𝑀 1 𝑇 0 . ( 3 . 1 9 ) (i)For 𝛼 𝑙 𝛼 𝑙 > 0 . Without loss of generality, we suppose that 0 < 𝛼 𝑙 < 𝛼 𝑙 , then we have 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 ≤ 𝛼 𝑙 𝑇 0 𝛼 𝑙 𝑇 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 ≤ 𝑇 0 ‖ ‖ 𝛼 − 𝛼 ‖ ‖ . ( 3 . 2 0 ) (ii)For 𝛼 𝑙 𝛼 𝑙 < 0 . For example, 𝛼 𝑙 < 0 < 𝛼 𝑙 , | 𝛼 𝑙 | > 𝛼 𝑙 , we have 𝑇 0 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 ≤ | 𝛼 𝑙 | 𝑇 0 𝛼 𝑙 𝑇 0 ‖ ‖ 𝑢 ( 𝜃 , 𝛼 ) − 𝑢 ( 𝜃 , ‖ ‖ 𝛼 ) ℝ 𝑞 𝑑 𝜃 ≤ 2 𝑇 0 ‖ ‖ 𝛼 − 𝛼 ‖ ‖ . ( 3 . 2 1 ) By (3.18), (3.20) and (3.21), there exists a constant 𝑀 > 0 such that ‖ ‖ 𝐺 ( 𝛼 ) − 𝐺 𝛼 ‖ ‖ ≤ 𝑀 ‖ ‖ 𝐾 2 ‖ ‖ ‖ ‖ 𝛼 − 𝛼 ‖ ‖ , 𝛼 , 𝛼 ∈ Ω . ( 3 . 2 2 )

By Lemma 3.3, we have the following result immediately.

Theorem 3.4. Under the assumptions Theorem 3.2, one can choose a suitable ‖ 𝐾 2 ‖ such that the systems (1.1), (1.3)–(1.5) have a unique steady-state and the fixed point of 𝐺 is just the conducting vector.

Proof. Let 𝑥 ( 𝑡 , 𝛼 ) be the 𝑇 0 -periodic 𝑃 𝐶 -mild solution of system (1.1) corresponding to 𝛼 ∈ Ω , then 𝑇 𝑥 ( 0 ) = 𝑥 0 𝑇 = 𝑆 0 , 0 𝑥 ( 0 ) + 𝑇 0 0 𝑆 𝑇 0 , 𝜃 ( 𝑓 ( 𝜃 ) + 𝐶 𝑢 ( 𝜃 , 𝛼 ) ) 𝑑 𝜃 , ( 3 . 2 3 ) that is, 𝑇 𝑥 ( 0 ) = 𝐼 − 𝑆 0 , 0 − 1 𝑇 0 0 𝑆 𝑇 0 , 𝜃 ( 𝑓 ( 𝜃 ) + 𝐶 𝑢 ( 𝜃 , 𝛼 ) ) 𝑑 𝜃 . ( 3 . 2 4 ) By virtue of [H3], we can suppose that ‖ 𝑓 ( 𝑡 ) ‖ ≤ 𝑓 0 , 𝑡 ≥ 0 , then ‖ ‖ 𝑇 ‖ 𝑥 ( 0 ) ‖ ≤ 𝐼 − 𝑆 0 , 0 − 1 ‖ ‖ ‖ ‖ 𝑆 𝑇 0 ‖ ‖ , 𝜃 𝑇 0 0 ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) 𝑞 + 𝑓 0 𝑑 𝜃 ≤ 𝐾 ‖ 𝑄 ‖ ‖ 𝐶 ‖ L 𝑏 ( ℝ 𝑞 , 𝑋 ) 𝑞 + 𝑓 0 𝑇 0 ≡ 𝑀 3 . ( 3 . 2 5 ) It comes from 𝐺 ( 𝛼 ) = 𝐼 − 𝑒 𝐽 𝑇 0 − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 𝐾 1 + 𝑆 ( 𝑡 , 0 ) 𝑥 ( 0 ) 𝑑 𝑠 𝐼 − 𝑒 𝐽 𝑇 0 − 1 𝑇 0 0 𝑒 𝐽 ( 𝑇 0 − 𝑠 ) 𝐾 2 𝐾 1 𝑡 0 𝑆 ( 𝑡 , 𝑠 ) ( 𝑓 ( 𝑠 ) + 𝐶 𝑢 ( 𝑠 , 𝛼 ) ) 𝑑 𝑠 𝑑 𝑠 ( 3 . 2 6 ) that ( ‖ 𝐺 𝛼 ) ‖ ≤ 𝑀 4 ‖ ‖ 𝐾 2 ‖ ‖ , ( 3 . 2 7 ) where 𝑀 4 = ‖ ‖ 𝐼 − 𝑒 𝐽 𝑇 0 − 1 ‖ ‖ ‖ ‖ 𝑒 𝐽 𝑇 0 ‖ ‖ ‖ ‖ 𝐾 1 ‖ ‖ L 𝑏 ( 𝑋 , ℝ 𝑝 ) 𝑇 0 𝑀 3