Journal of Applied Mathematics

Volume 2013 (2013), Article ID 146010, 9 pages

http://dx.doi.org/10.1155/2013/146010

Research Article

Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses

Hai Zhang,1,2 Jinde Cao,1,3 and Wei Jiang⁴

¹Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China

²Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China

³Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

⁴School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039, China

Received 8 April 2013; Accepted 14 May 2013

Academic Editor: Francisco J. Marcellán

Copyright © 2013 Hai Zhang 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 is concerned with the controllability of linear fractional differential systems with delay in state and impulses. The factors of such systems including fractional derivative, impulses, and delay are taken into account synchronously. The expression of state response for such systems is derived, and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.

1. Introduction

In this paper, we consider the controllability of linear fractional differential systems with state delay and impulses as follows:

where denotes an order Caputo’s fractional derivative of , , , , and are the known constant matrices and satisfy , , is a positive constant, is the state variable, is the control input, is the initial state function, where denotes the space of all continuous functions mapping the interval into , is continuous for , and

represent the right and left limits of at and the discontinuous points

where , , and which implies that the solution of system (1) is left continuous at .

The subject of fractional differential equations is gaining much importance and attention (see [1–11] and references therein). Fractional differential equations have been proved to be an excellent tool in the modelling of many phenomena in various fields of engineering, physics, and economics. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. At the same time, time delay is one of the inevitable problems in practical engineering applications, which has an important effect on the stability and performance of system. In the last few years, the results with regard to the fractional delay differential systems have been presented in [12–15].

Although most dynamical systems are analyzed in either the continuous or discrete-time domain, many real systems in physics, chemistry, biology, engineering, and information science may experience abrupt changes as certain instants during the continuous dynamical processes. This kind of impulsive behaviors can be modeled by impulsive systems. The basic theory of impulsive differential equations can be found in the monographs of Baĭnov and Simeonov [16], Benchohra et al. [17], and the paper of Fečkan et al. [18].

On the other hand, controllability is the most fundamental concept in modern control theory, which has close connections to pole assignment, structural decomposition, quadratic optimal control, and so forth. Some important results concerning the control theory for various kinds of systems have been obtained in [19–36] and references therein. Kalman et al. [19] have investigated the controllability of linear dynamical systems based on the algebraic approach. Wonham and Morse [20] have discussed the pole assignment problems of linear systems based on the geometric approach. In [21–24], the authors have discussed the controllability of integer derivative delay systems. In [25, 26], the controllability of the descriptor (singular) systems has been considered. Impulsive control systems with integer derivative have been investigated in [27–29]. For integer derivative control systems with state delay and impulses, Zhang et al. [27] have derived the sufficient conditions for the controllability based on the fixed point theorem. It is worth pointing out that notable contributions have been made to fractional control systems in [30–36]. The different techniques have been developed to investigate the control problems of fractional differential systems, such as fractional sliding manifold approach [30], fixed point theorems [31–34], functional analysis method [33, 34], and algebraic method [35, 36]. To the best of our knowledge, there are no relevant reports on the controllability of fractional differential systems with state delay and impulses as treated in the current literature. In this paper, the factors of control systems including the Caputo's fractional derivative, impulses, and delay are taken into account synchronously. The purpose of this paper is to establish the sufficient and necessary conditions of controllability for system (1) based on the algebraic approach. The recent research surge in developing the theory of fractional control systems has motivated and inspired our present work.

This paper is organized as follows. In Section 2, we recall some definitions and preliminary facts, and the expression of state response for system (1) is derived. In Section 3, the sufficient and necessary conditions of controllability criteria are established. In Section 4, some examples are given to illustrate the effectiveness and applicability of controllability criteria. Finally, some concluding remarks are drawn in Section 5.

2. Preliminaries

Throughout this paper, denote by the space of all piecewise left continuous functions mapping the interval into .

Let us recall some definitions and preliminary facts. For more details, one can see [1–4].

Definition 1. The Riemann-Liouville's fractional integral of order with the lower limit zero for a function is defined as

provided the right side is pointwise defined on , where is the Gamma function.

Definition 2. The Caputo's fractional derivative of order for a function is defined as

Definition 3. The Mittag-Leffler function in two parameters is defined as

where , , and , denotes the complex plane.

Definition 4. The Laplace transform of a function is defined as

where is -dimensional vector-valued function.

Remark 5. If , then

Lemma 6 (see [2]). Let be complex plane, for any , , and ; then

holds, where represents the real part of the complex number and denotes the identity matrix.

In order to obtain the state response of system (1), we firstly consider the representation of solution for linear fractional delay differential systems without impulses as follows:

Lemma 7. Let ; if is continuous and exponentially bounded, then the solution of system (10) can be represented as

and , .

Proof. Applying the method of steps which has been presented in [12], then there exists a unique solution to system (10).

For , taking the Laplace transform with respect to in both sides of system (10), we obtain

Then (12) can be written as

From Definition 4 and Lemma 6, then (13) is equivalent to

The convolution theorem of the Laplace transform applied to (14) yields the form

Applying the inverse Laplace transform, we obtain

Therefore, we have the stated result.

Lemma 8. Let and ; then state response of system (1) can be represented as follows.

For ,

For ,

For ,

For , ,

Proof. If , then the conclusion obviously holds. If , then, from Lemma 7,

If , applying the idea used in [18], we have

If , then

If , then the same argument implies the following expression:

Thus, the proof is completed.

3. Controllability Criteria for System (1)

In this section, we establish the sufficient and necessary conditions of controllability criteria for system (1) based on the algebraic approach.

Definition 9. System (1) is called controllable on ; for any initial function and any state , there exists a control input , such that the corresponding solution of (1) satisfies .

Theorem 10. System (1) is controllable on if and only if the Gramian matrix

is nonsingular for some , where is the Mittag-Leffler function and denotes the matrix transpose.

Proof. We firstly prove sufficiency of Theorem 10. If is nonsingular, then is well defined. For any initial state , when , we take the control function as

Substituting in (18) and inserting (26) yield

Thus system (1) is controllable on , .

For , we take the control function as

Substituting in (19) and inserting (28) yield

Thus system (1) is controllable on , .

For , , we take the control function as

Substituting in (20) and inserting (30), then the same argument implies . Therefore system (1) is controllable on .

Next, we prove necessity of Theorem 10. Suppose is singular, without loss of generality; for , , there exists a nonzero vector such that

That is,

Then it follows

on . Since system (1) is controllable, there exist control inputs and such that

Combining (34) and (35) yields

Multiplying on both sides of (36), we get

According to , we have . Thus . This contradiction therefore completes the proof.

Theorem 10 presents a geometric type criterion. By the algebraic transform and computation, we can obtain an algebraic criterion which is similar to the famous Kalman's rank condition [19].

Theorem 11. System (1) is controllable on if and only if

Proof. According to Cayley-Hamilton theorem, can be represented as

For ,

Let

Then combining (40) with (41) yields

where , . Note that, for arbitrary and , the sufficient and necessary condition to have a control input satisfying (42) is that

For ,

Combining (41) with (44) yields

Note that, for arbitrary and , the sufficient and necessary condition to have a control input satisfying (45) is that

Thus, the proof is completed.

Remark 12. System (1) is controllable if and only if the resolvent condition as holds (here is the respective Gramian matrix in the nonfractional, nondelay, and nonimpulsive case) since this is equivalent to the rank condition in the finite dimensional case [19, 35, 36].

4. Illustrative Examples

In this section, we give two examples to illustrate the presented criteria.

Example 13. Consider the controllability of linear fractional differential systems with state delay and impulses as follows:

Now, we apply Theorem 10 to prove that system (47) is controllable on . Let us take

By computation, we have

Substituting in (25) and combining (25) with (49)–(51) yield

Obviously, is nonsingular. Thus by Theorem 10, system (47) is controllable on .

Example 14. Consider the controllability of linear fractional differential systems with state delay and impulses as follows:

Now, we apply Theorem 11 to prove that system (53) is controllable on . Let us take

Then one can obtain

Thus by Theorem 11, system (53) is controllable on .

5. Conclusions

In this paper, the controllability criteria for linear fractional differential systems with delay in the state and impulses have been investigated. The sufficient and necessary conditions for the controllability of such systems have been established. Furthermore, both the proposed criteria and illustrative examples have shown that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.

Acknowledgments

The authors are very grateful to the Associate Editor, Professor Francisco J. Marcellán, and the two anonymous reviewers for their helpful and valuable comments and suggestions, which significantly contributed to improving the quality of this paper. This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Key Program of Educational Commission of Anhui Province of China under Grant no. KJ2011A197.

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