Abstract and Applied Analysis

Volume 2010 (2010), Article ID 405321, 18 pages

http://dx.doi.org/10.1155/2010/405321

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

^{1}College of Science, Guizhou University, Guiyang, Guizhou 550025, China^{2}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 where the state variable takes values in a reflexive Banach space , is the infinitesimal generator of a -semigroup on the state space , is -periodic step disturbance of the system and . Control variable , the input is a bounded linear operator. There is only one time sequences satisfing and , , , , , and represent, respectively the right and left limits of at . is a given bounded linear operator; is the dimensional output of the system (1.1).

We, then, study the following steady-state control of infinite dimensional semilinear system with impulses where is -periodic continuous function.

Suppose that control signal is the output of the dimensional pulse-width sampler controller, and is the input of the dimensional pulse-width sampler controller, which is the output of some dynamical controller

where is a matrix, is a matrix, is determined by the dynamic characteristics of the controller, and is called the feedback matrix which will be chosen in the latter (see Theorem 3.4 and Theorem 3.8). The output signal and the input signal of the pulse-width sampler satisfy the following dynamic relation:

where is called the sampling period of the pulse-width sampler which is the same as the period of and ,

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 dimensional vector is called the duration ratio of the pulse-width sampler in the th sampling period,

We defined a closed cube in as
then we have , for

*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 dimensional vector
and a corresponding periodicity rectangular-wave control signal defined by
such that the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), has a corresponding -periodic trajectory , , then the control signal (1.8) is called the *steady-state control* with respect to the disturbance . The -periodic trajectory is called steady-state corresponding to steady-state control and the constant vector 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 such that
then system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), corresponding to the disturbance 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 is called *stead-state stabilizability* if we can choose a suitable and 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 denote the space of linear operators from to , denote the space of bounded linear operators from to , denote the space of bounded linear operators from to , and denote the space of bounded linear operators from to . It is obvious that , , and is the Banach space with the usual supremum norm.

Define , where . We introduce is continuous at , is continuous from left and has right hand limits at , and Set

It can be seen that endowed with the norm , is a Banach space.

We introduce the following assumption [H1].

(i) [H1.1] is the infinitesimal generator of a -semigroup on with domain . (ii) [H1.2] There exists such that . (iii) [H1.3] For each , and .We first recall the homogeneous linear impulsive periodic system and the associated Cauchy problem

If and is an invariant subspace of , 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 represented by , where

given by

The operator is called impulsive evolution operator associated with .

The properties of the impulsive evolution operator, associated with and , are collected here.

Lemma 2.1 (see [26, Lemma 2.1] [27]). *Let assumption [H1] hold. The impulsive evolution operator has the following properties. *(1)*For , , there exists a such that *(2)*For , , = .*(3)*For , , .*(4)*For , , .*(5)*For , there exists an , such that*

The exponential stability of the impulsive evolution operator will be used throughout the paper; we recall them as the following definitions and lemmas.

*Definition 2.2. *, is called exponentially stable if there exist and such that
Assumption [H2]: is exponentially stable, that is, there exist and such that
Two important criteria for exponential stability of a -semigroup are collected here.

Lemma 2.3 (see [26, Lemma 2.4]). *Assumptions [H1] and [H2] hold. There exists such that
**
Then is exponentially stable.*

Lemma 2.4 (see [26, Lemma 2.5]). *Assume that assumption [H1] holds. Suppose
**
If there exists such that
**
where
**
then is exponentially stable.*

*Remark 2.5 (see [26, Theorem 3.2]). *If is exponentially stable, then is inverse and

#### 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]:, , is -periodic step perturbation. [H4]: Control signal is -periodic, which is defined by the rectangular wave signal , 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 is exponentially stable; for every , system (1.1) has a unique -periodic -mild solution
**
where
**
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 has the following properties:
**
where is the resolvent set of the matrix , satisfies , then the following open-loop control system
**
has a unique -periodic -mild solution given by
*

*Proof. *By (3.3), we know that , that is . Thus exists and is bounded. It is not difficult to see that
where .

Consider
which is the unique solution of the following equation:
Let
it comes from Lemma 3.1 that
It is easy to verify that
is just the -periodic -mild solution 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 given by
where is the -periodic -mild solution of system (1.1) corresponding to . Then we have the following result.

Lemma 3.3. *Under the assumptions of Theorem 3.2, there exists a constant such that
*

*Proof. *Suppose and are the -periodic -mild solution of system (1.1) corresponding to and , respectively, then
Thus,
For , we obtain
where
By elementaly computation,
where
(i)For . Without loss of generality, we suppose that , then we have
(ii)For . For example, , , we have
By (3.18), (3.20) and (3.21), there exists a constant such that

By Lemma 3.3, we have the following result immediately.

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

*Proof. *Let be the -periodic -mild solution of system (1.1) corresponding to , then
that is,
By virtue of [H3], we can suppose that , , then
It comes from
that
where
Using Lemma 3.3 and (3.27), it is not difficult to verify that is a contraction map when

By the application of contraction mapping principle, has a unique fixed point . Obviously, the -periodic -mild solution of system (1.1) corresponding to is just the unique steady-state.

Next, we investigate the steady-state control of system (1.2), (1.3)–(1.5). We need to introduce the following assumption [H5].

(i) [H5.1] is measurable for , and for any , , there exists a positive constant such that(ii) [H5.2] is -periodic in . That is, , .

Lemma 3.5. *Under the assumptions [H1], [H4] and [H5], the impulsive evolution operator is exponentially stable, that is, there exists a constant and such that
**
where , then system (1.2) has a unique -periodic -mild solution corresponding to control given by
**
and is also exponentially stable.*

*Proof. *Suppose that () is the -mild solution of system (1.2) corresponding to initial value (), respectively, then
By Gronwall inequality, we can deduce
Define a map given by
Then we can verify that
It comes from
that is a contraction map on . Thus, by the application of contraction mapping principle again, has a unique fixed point satisfying
Using (2), (3), (4) of Lemma 2.1, one can verify that
By virtue of (3.34), (3.38), and (3.39), we know that is just the -periodic -mild solution of (1.2) which is exponentially stable.

Similar to the proof of Lemma 3.1, using Lemma 3.5, we can obtain the following result immediately.

Lemma 3.6. *Under the assumptions of Lemma 3.5, if also satisfies (3.3), then the open-loop control system**
has a unique -periodic -mild solution . *

In order to discuss the existence of steady-state control of system (1.2), we define a map given by

where is the periodic solution of system (1.2) corresponding to . Then we have the following results.

Lemma 3.7. *Under the assumptions of Lemma 3.6, there exists a constant such that
*

*Proof. *Suppose that and are the -periodic -mild solutions corresponding to and with the initial value and , respectively, then
Thus,
Furthermore,
For , we have
By Gronwall inequality again, we obtain
Integrating from to , we obtain
where
Thus,
Choosing a constant
then,

Using Lemma 3.7, we have the following result.

Theorem 3.8. *Under the assumptions of Lemma 3.7, there exists a constant such that , if is sufficiently small, then system (1.2), (1.3)–(1.5) has a unique steady-state and the fixed point of is just the conducting vector.*

*Proof. *Let be the -periodic -mild solution of system (1.2) corresponding to , then
Further,
Let
It comes from
that
It is not difficult to see that is a contraction map when
By application of contraction mapping principle again, has a unique fixed point . Obviously, the -periodic -mild solution of system (1.2) corresponding to is just the unique steady-state.

Finally, an example is given for demonstration. Consider the following system and the output satisfies

where , , and are constants.

Let ; define

Then can generate an exponentially stable -semigroup in and , . We only choose a suitable positive number , then all the assumptions are met in Theorem 3.4, our results can be used to system (3.59).

#### Acknowledgment

The authors acknowledge the support from the National Natural Science Foundation of China (no.10961009), Introducing Talents Foundation for the Doctor of Guizhou University (2009, no.031). Youth Teachers Natural Science Foundation of Guizhou University (2009, no.083).

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