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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 374938, 7 pages
http://dx.doi.org/10.1155/2013/374938
Research Article

Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

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

Received 27 February 2013; Accepted 23 June 2013

Academic Editor: G. M. N'Guérékata

Copyright © 2013 Xian-Feng Zhou 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

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

1. Introduction

Recently, a variety of problems such as the existence, uniqueness of mild solution for the initial value problem, periodic boundary value problems, antiperiodic boundary value problems, and Ulam stability for impulsive fractional differential equations have been considered due to their important role in modeling natural phenomena such as medicine, biology, and optimal control; see the paper [116].

The concept of controllability plays an important role in the analysis and design of control systems. With the developments of theories of impulsive fractional differential equations, there have been a few papers devoted to the controllability of impulsive fractional differential systems; see [1720]. In [17], the author discussed the controllability of impulsive fractional linear time-invariant systems through constructing a suitable control input in time domain. By fixed point theorem, the controllability of integrodifferential systems was investigated in [1820]. It should be mentioned that the controllability for linear fractional dynamical systems has been investigated by several scholars [2126] while the theory of controllability for impulsive fractional linear time-invariant systems is still in the initial stage [17].

The impulsive fractional differential equations (systems) which had been investigated earlier often have the form or and so forth, where is the Caputo fractional derivative of order with lower limit zero, , is jointly continuous, , satisfies , , and represent, respectively, the right and the left limits of at , , , , are the known constant matrices, , , are vectors with appropriate dimensions.

However, the function defined on is continuous everywhere except for finite number of points , , at which the limits and exist with . If there exists some such that ,  , and , then does not exist since is meaningless at the impulsive moment . That is to say is meaningless. As a result, investigating (1)–(6) is meaningless.

Motivated by this fact, this paper is concerned with the complete controllability of the impulsive fractional linear time-invariant system with delay in -dimensional Euclidean space, where , , denotes the Caputo’s derivative of order with the lower limit , , , , , , , , are known constant matrices with appropriate dimensions, the state variable , the initial function , the delay , , the control input , the output , , .

In this paper, the methods used is to construct a suitable control input function in time domain. The results obtained is sufficient and necessary, which are convenient for computation.

2. Preliminaries

In this section, we begin with some notations, definitions, and lemmas. Throughout this paper, or  denotes the Caputo’s derivative of order with the lower limit for the function , or denotes integral of order with lower limit for the function , denotes the Laplace transform of the function , and “” denotes the norm of the matrix “,” “” denotes the transpose of the matrix “”. Let be the Banach space of all continuous functions from into with the norm . Let the Banach space be and the norm .

Definition 1 (see [27]). The fractional integral of order with the lower limit for a function is defined as Provided that the right-hand side is pointwise defined on , where is the Gamma function.

Definition 2 (see [27]). The Caputo’s derivative of order with the lower limit for a function can be written as Particularly, when , it holds

The Laplace transform of is where is the Laplace transform of .

In particular, for , it holds

Definition 3 (see [27]). The two-parameter Mittag-Leffler function is defined as
The Laplace transform of Mittag-Leffler function is where denotes the real parts of .

In addition, the Laplace transform of is

Lemma 4 (see [28]). Let . If , then where denotes the set of continuous functions on .

3. Main Results

Definition 5 (complete controllability). The system (8)–(12) is said to be completely controllable on the interval if, for any (), , and , there exists an admissible control input such that the state variable of the system (8)–(12) satisfies .
Using the Laplace transform method, we can easily obtain the following lemma.

Lemma 6. The movement orbit of the state variable of the system (8)−(12) can be written as

Theorem 7. The system (8)–(12) is completely controllable on if and only if the controllability matrices are nonsingular, .

Proof. Sufficiency. Suppose that is nonsingular; then is well defined, .
For , it follows from the formula (23) that For all , choosing and inserting (26) into (25) yields . Thus, the system (8)–(12) is completely controllable on .
Similarly, for , it follows from the formula (23) that
Since the system (8)–(12) is completely controllable on , there exists a control input such that . By (27), it follows that
For all , choosing together with (28) yields . Thus, the system (8)–(12) is completely controllable on .
By similar arguments, we can prove that the system (8)–(12) is completely controllable on ,  .
Consequently, the system (8)–(12) is completely controllable on .
Necessity. Suppose that the system (8)–(12) is completely controllable on .
If is singular, then there exists a nonzero vector such that That is Then we have on . By the assumption that the system (8)–(12) is completely controllable on , the system (8)–(12) is completely controllable on , . There exist control inputs and such that By (34), we have Inserting (35) into (33) yields Multiplying on both side of (36) yields By (32) and (37), we have . Thus, . This is a contradiction.
If is singular for some , then there exists a nonzero vector such that That is Then, it follows that on . By formula (23) and the assumption that the system (8)–(12) is completely controllable, there exist control inputs and such that and
Similarly, there exists a control input such that By (42), we have Inserting (43) into (41) yields Multiplying on both side of (44) yields Combining (45) with (40) yields . Thus, . This is a contradiction.
Thus, is nonsingular for . This completes the proof.

Theorem 8. The system (8)–(12) is completely controllable on if and only if

Proof. Necessity. Suppose that system (8)–(12) is completely controllable on . Then, the system (8)–(12) is completely controllable on . Then, for any , there exists a control input such that . By the formula (23), it follows that By Cayley-Hamilton theorem, we have where are functions in , . Combining the formula (48) and the equality (47), we have where , . For arbitrary state and initial function , the system (8)–(12) is completely controllable on if and only if there exists a control input such that (47) or (49) holds. Obviously, for arbitrary initial function and , the sufficient and necessary condition to have a control input satisfying (49) is that Sufficiency. Suppose that . In order to prove that the system (8)–(12) is completely controllable on , it is sufficient to prove that the system (8)–(12) is completely controllable on ,  , respectively.
The formula (23) together with (48) yields (49). By the assumption that , the system (8)–(12) is completely controllable on .
Now we prove that the system (8)–(12) is completely controllable on . The complete controllability of the system (8)–(12) on implies that there exists a control input such that . Inserting into the formula (23), we have, for , Thus, it follows By (48) it follows taht where , . Similar to the previous arguments, we can conclude that system (8)–(12) is completely controllable on .
Repeating the process on , respectively, we can prove that the system (8)–(12) is completely controllable on , . In conclusion, the system (8)–(12) is completely controllable on . This completes the proof.

Remark 9. From Theorem 8, we can conclude that the complete controllability of the system (8)–(12) is unrelated to the matrix and initial function . The matrices determine if the the system (8)–(12) possesses complete controllability.

4. Examples

Example 1. Consider the system (8)–(12). Choose , ,  , , , , , . Now, we employ Theorems 7 and 8 to prove if that the system (8)–(10) is completely controllable, respectively.
By computation, we have By the formula (24) we have It is obvious that and are nonsingular. By Theorem 7, the system is completely controllable.

On the other hand, By Theorem 8, the system is completely controllable.

Example 2. Consider the time-invariant system (8)–(12). Choose
By computation, we have By Theorem 8, the system is completely controllable.

Acknowledgments

The authors would like to thank the referee for his or her valuable comments, which help us to improve the quality of the paper. This paper is supported by National Natural Science Foundation of China (11071001), Research Fund for Doctoral Program of Educational Ministry of China (20103401120002 and 20123401120001), Program of Natural Science Research in Anhui Universities (KJ2011A020 and KJ2013A032), Scientific Research Starting Fund for Dr. of Anhui University (023033190001, 023033190181), and the 211 Project of Anhui University (KJQN1001, 023033050055).

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