Abstract and Applied Analysis

Volume 2013 (2013), Article ID 639492, 17 pages

http://dx.doi.org/10.1155/2013/639492

Research Article

Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

Zhenhai Liu and Maojun Bin

Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 21 July 2013; Revised 25 September 2013; Accepted 2 November 2013

Academic Editor: Naseer Shahzad

Copyright © 2013 Zhenhai Liu and Maojun Bin. 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.

We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce the -mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.

1. Introduction

The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman [1] first introduced its definition in 1963, controllability of the deterministic and stochastic dynamical control systems in finite-dimensional and infinite-dimensional spaces is well developed in different classes of approaches, and more details can be found in papers [2–4]. Some authors [5–7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems. In [5–7], to prove the controllability results for fractional-order semilinear systems, the authors made an assumption that the semigroup associated with the linear part is compact. But if -semigroup is compact or the operator is compact, then the controllability operator is also compact and hence the inverse of it does not exist if the state space is infinite dimensional [8]. Thus, it is shown that the concept of exact controllability is difficult to be satisfied in infinite-dimensional space. Therefore, it is important to study the weaker concept of controllability, namely, approximate controllability for differential equations. In these years, several researchers [9–17] have studied it for control systems.

In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems:

where is Caputo’s fractional derivative of and is the infinitesimal generator of a -semigroup of bounded operators on the Hilbert space ; the control function is given in ; is a Hilbert space; is a bounded linear operator from to ; is a given function satisfying some assumptions and is an element of the Hilbert space .

In [16], Sukavanam and Kumar researched approximate controllability of fractional-order semilinear delay systems:

where ; is a closed linear operator with dense domain generating a -semigroup ; the state takes values in the Banach space ; the control function takes values in ; is a bounded linear operator from to ; the operator is nonlinear. If is a continuous function, then is defined as for and .

In [18], Rykaczewski studied the approximate controllability of an inclusion of the form

where is a linear operator which generates a compact semigroup, is u.h.c. multivalued perturbation with weakly compact values, and the state takes values in the Hilbert space . is a Hilbert space of all admissible controls. is a continuous linear operator.

Fractional differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth; see [19–27] for example. As a consequence there was an intensive development of the theory of differential equations of fractional order. One can see the monographs of Kilbas et al. [28] and Podlubny [29] and the references therein. The definitions of Riemann-Liouville fractional derivatives or integrals with initial conditions are strong tools to resolve some fractional differential problems in the real world. Heymans and Podlubny [30] have verified that it was possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals, and such initial conditions are more appropriate than physically interpretable initial conditions. Furthermore, they have investigated that the impulse response with Riemann-Liouville fractional derivatives was seldom used in the fields of physics, such as viscoelasticity. In recent years, many authors [18, 27, 31] were devoted to mild solutions to fractional evolution equations with Caputo fractional derivative, and there have been a lot of interesting works. As for the study of the fractional differential systems with Caputo fractional derivative, we can refer to [27, 31, 32] for the existence results. Its approximate controllability was considered in [9, 13–16]. The approximate controllability of Caputo fractional inclusion systems has been investigated by [10]. We know that differential inclusions are strong tools to solve some problems in various fields of engineering, physics, and optimal control; see [10, 32–35]. However, the approximate controllability for the impulsive fractional differential evolution inclusion with Riemann-Liouville fractional derivatives is still an untreated topic in the literature.

Motivated by the above work, in this paper, we consider the following system:

where and denotes the Riemann-Liouville fractional derivative of order with the lower limit zero. is a nonempty, bounded, closed, and convex multivalued map. is the infinitesimal generator of a -semigroup () on a Banach space . , , , and denote the right and the left limits of at , . The control function takes value in , , and is a Banach space; is a linear operator from to .

The purpose of this paper is to provide some suitable sufficient conditions for the existence of mild solutions and approximate controllability results for the impulsive fractional abstract Cauchy problems with Riemann-Liouville fractional derivatives. The main tools used in our study are fixed point theorem, semigroup theory for multivalued maps, and the theory from fractional differential equations. The rest of this paper is organized as follows. In Section 2, we present some preliminaries to prove our main results. In Section 3, by applying some standard fixed point principles, we prove the existence of the mild solutions for semilinear fractional differential equations, and the approximate controllability of the system (4) is proved. In Section 4, we give an example to illustrate our main results.

2. Preliminaries

In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach space will be denoted by . denotes the space of bounded linear operators from to . For the uniformly bounded -semigroup (), we set . Let denote the Banach space of all value continuous functions from to with the norm . Let with the norm

Obviously, the space is a Banach space.

To define the mild solutions of (4), we also consider the Banach space , exist, and with , with the norm

It is easily known that the space is a Banach space.

Let be a metric space. We use the notations

Firstly, let us recall the following basic definitions from fractional calculus. For more details, one can see [28, 29].

Definition 1. The integral

is called Riemann-Liouville fractional integral of order , where is the gamma function.

Definition 2. For a function given in the interval , the expression

where and denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .

Lemma 3 (see [28]). Let , , and let be the fractional integral of order . If and , then one has the following equality:

In order to study the -mild solutions of (4) in Banach space , we give the following results which will be used throughout this paper.

Lemma 4. Let , and let be the fractional integral of order . If and , then one has the following equality:

where , .

Proof. If , then, by Lemma 3, we easily get

If , since

then, by (13), we have

Thus, by (13) and (14), we get

Similarly, if , , we can get

The proof is completed.

The Laplace transform formula for the Riemann-Liouville fractional integral is defined by

where is the Laplace of defined by

Lemma 5. Let and , ; if , , and is a solution of the following problem:

then, satisfies the following equation:

where ,

where is a probability density function defined on ; that is,

Proof. We observe that can be decomposed to , where is the continuous mild solution for

and is the -mild solution for

Indeed, by adding together (23) and (24), it follows by (19). Since is continuous, then , . On the other hand, any solution of (19) can be decomposed to (23) and (24). So we show the results by the following.

At first, we calculate the mild solution of (23).

Apply Riemann-Liouville fractional integral operator on both sides of (23); then, by Lemma 3, we get

That is,

Let ; taking the Laplace transformations

to (26), we obtain

Consider the one-sided stable probability density

whose Laplace transformation is given by

Hence, it follows from (28) and (30) that

According to the above work, we get

Now, we can invert the Laplace transform to (20) and obtain

Let

Then, we get

Now we calculate the -mild solution of (24).

Applying Riemann-Liouville fractional integral operator on both sides of (24), then by Lemma 4, we get

The above equation (36) can be rewritten as

where

Let ; taking the Laplace transformation to (37), we obtain

That is,

Notice that the Laplace transform of is . Thus one can calculate the mild solution of (24) as

By the above work, the -mild solution of (19) is given by

That is,

where ,

where is a probability density function defined on ; that is,

This completes the proof of the lemma.

According to Lemma 5, we give the following definition.

Definition 6. A function is called a mild solution of (4) if and there exits such that a.e. on and

where ,

where is a probability density function defined on ; that is,

Due to the work of the paper [31], we have the following result.

Lemma 7. The operator has the following properties.

(i)For any fixed , is linear and bounded operator; that is, for any ,

(ii) () is strongly continuous.

Now, we also introduce some basic definitions on multivalued maps. For more details, see [36–38].

A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if . is bounded on for any bounded set of ; that is, .

is called upper semicontinuous (u.s.c.) on if, for each , the set is a nonempty closed subset of , and if, for each open set of containing , there exists an open neighborhood of such that .

is said to be completely continuous if is relatively compact for every .

If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e., , , imply ).

We say that has a fixed point if there is a such that .

A multivalued map is said to be measurable if for each the function defined by is measurable.

Definition 8. The system (4) is said to be exactly controllable on , if, for all , there exists a control () such that the mild solution of (4) satisfies and .

Definition 9. The system (4) is said to be approximately controllable on the interval , if, for all , one has , where is the reachable set of system (4) with the initial values at the terminal time .

It is convenient at this point to introduce two relevant operators:

where denotes the adjoint of and is the adjoint of . It is straightforward that the operator is a linear bounded operator.

We consider the following linear fractional differential system:

Lemma 10. The linear fractional differential system (51) is approximately controllable on if and only if as in the strong operator topology.

The proof of this lemma is a straightforward adaptation of the proof of [3].

Lemma 11. Let be a Banach space and let be integrably bounded. If, for all , there is a relatively weakly compact set such that for every , then is relatively weakly compact in .

Lemma 12 (Lasota and Opial [39]). Let be a compact real interval and let be a Banach space. The multivalued map is measurable to for each fixed , u.s.c. to for each , and for each the set is nonempty. Let be a linear continuous mapping from to ; then, the operator

is a closed graph operator in .

Lemma 13 (see [37]). Let be a bounded, convex, and closed subset in the Banach space and let be a u.s.c. condensing multivalued map. If, for every , is a closed and convex set in , then has a fixed point.

3. Main Results

In this section, we present our main result on approximate controllability of system (4). To do this, we first prove the existence of solutions for fractional control system. Secondly, we show that, under certain assumptions, the approximate controllability of (4) is implied by the approximate controllability of the corresponding linear system.

For convenience, let us introduce some notations:

Before stating and proving our main results, we introduce the following assumptions.

 : is compact, . : is a multivalued map satisfying which is measurable to for each fixed , u.s.c. to for each , and for each the set

 is nonempty. : There exist a function , , and a nondecreasing continuous function , such that

 for a.e. , for all , and for each , there exists , such that

Theorem 14. If the conditions are held, then the system (4) has a mild solution.

Proof. We consider a set

on the space . We easily know that is a bounded, closed, and convex set in . For , for all , , we take the control function as

where

By this control, we define the operator as follows:

We will show that, for all , the operator has a fixed point. For the sake of convenience, we subdivide the proof into several steps.

Step  1. For each , the operator is convex for each .

In fact, if , then, for each , , there exist such that

Let ; then, for each , , we have

Since is convex (because has convex values), ; thus, .

Step  2. For each , there is a positive constant , such that .

If this is not true, then there exists such that, for every , there exists a such that ; that is,

for some .

By using Holder’s inequality and , we have

Then, we obtain

Thus,

Dividing both sides by and taking the low limit as , we get

which is a contradiction to . Thus, for each , there exists such that maps into itself.

Step  3. is closed for each .

Indeed, for each given , let such that in . Then, there exists such that, for each ,

where

Because of [40, Proposition 3.1], is weakly compact in which implies that converges weakly to some in . Thus, , and

Then, for each ,

Thus, we show that .

Step  4. is u.s.c and condensing.

We decompose as , where the operators and are defined by

According to [41, Corollary ], we will prove that is a contraction operator, while is a completely continuous operator.

Let us begin proving that is a contraction operator. For any , we obtain

Then, is a contraction operator, since .

Next, we prove that is u.s.c and completely continuous. We subdivide the proof into several claims.

Claim  1. There exists a positive constant such that .

By employing the technique used in Step 2, one can easily show that there exists such that .

Claim  2. is a family of equicontinuous functions.

Let . For each , , there exists such that

Then, we have

By using Hölder’s inequality and assumption , we get

It is easy to see that tends to zero independently of as . Note that, from Lemma 10, is continuous in the uniform operator topology for ; we can directly obtain and tending to zero independently of as . Applying the absolute continuity of the Lebesgue integral, we have , , and tending to zero independently of as .

Therefore, is equicontinuous.

Claim  3. The set is relatively compact for each .

Let be fixed. For and , there exists such that, for each ,

For all and for all , define

By the compactness of (), we obtain the set

which is relatively compact in for all and . Moreover, we have

The right-hand side of the above inequality tends to zero as . Therefore, there are relatively compact sets arbitrarily close to the set , . Hence the set , is also relatively compact in . As a consequence of Claims 1–3 together with the Arzola-Ascoli theorem, we can conclude that is completely continuous.

Claim  4. has a closed graph.

Let (), , (). We will prove that . Since , there exists , such that, for each ,

We must prove that there exists , such that, for each ,

Since (), we can obtain

Consider the linear continuous operator

Clearly it follows from Lemma 13 that is a closed graph operator. Moreover, we have

Since , it follows from Lemma 13 that

Therefore, has a closed graph. Since is a completely continuous multivalued map with compact value, we have that is u.s.c.

Thus is u.s.c and condensing. Therefore, applying Lemma 13, we conclude that has a fixed point on . Thus, the fractional control system (4) has a mild solution on .

The proof is complete.

The following result concerns the approximate controllability of that problem (4). We assume that the following assumption be held.

 : There exists a positive constant such that for all .

Theorem 15. Assume that assumptions , , , and are satisfied and the linear system (24) is approximately controllable on . Then system (4) is approximately controllable on .

Proof. By employing the technique used in Theorem 14, we can easily show that, for all , the operator has a fixed point in , where . Let be a fixed point of in . Any fixed point of is a mild solution of (4); this means that there exists such that, for each ,

Define

Noting that , we get

By assumption ,

Consequently the sequence is uniformly bounded in . Thus, there is a subsequence, still denoted by , that converges weakly to, say, in . Denoting

we see that

By the Ascoli-Arzela theorem, we can show that the linear operator is compact; consequently the right-hand side of (92) tends to zero as .

This implies that

This proves the approximate controllability of system (4).

Remark 16. In [9], Ganesh et al. have studied the approximate controllability of fractional integrodifferential evolution equations. If our problem (4) can be changed as

where denotes the Riemann-Liouville fractional derivative, the operator is defined by . By a similar way, we can get the approximate controllability of the system (94).

Remark 17. In [15], Rathinasamy and Yong have researched the approximate controllability of fractional differential equations with state-dependent delay. Our problem (4) can be adopted to the impulsive fractional differential inclusion system with state-dependent delay:

where is the Riemann-Liouville fractional derivative, is the infinitesimal generator of a -semigroup on a Banach space , the state takes values in , the control function is given in (), is a Banach space of admissible control, and is a bounded linear operator. with is a multivalued map. For a continuous function , is the element of defined by , , and . Further, are appropriate nonlinear functions.

4. An Example

In this final section, we give an example to illustrate our abstract results.

Example 1. Consider the following fractional partial differential inclusion with control

where is the Riemann-Liouville fractional partial derivative of order , .

Let , define; the operator by with the domain

Then,

where , , , is the orthogonal set of eigenvectors of . It is easily shown that operator generates a strongly continuous semigroup on , which are compact, analytic, and self-adjoint in .

It can be easily seen that the linear system corresponding to (96) is approximately controllable on [42].

Define ; let be defined by . Then system (96) can be written in the abstract form given by (4). We assume that satisfy the following conditions.

 : is measurable to for each fixed , u.s.c. to for a.e. , and for each the set

 is nonempty. : There exists a positive constant such that for all , since, for any , we have

Obviously, the hypotheses of Theorem 15 are fulfilled. Thus, the system (96) is approximately controllable on .

5. Conclusions

In this paper, we study the approximate controllability for impulsive fractional differential inclusions with Riemann-Liouville fractional derivatives in Banach spaces. We mainly use fixed point theorem and semigroup theory; we obtain some new mild solutions for this kind of problems (4). An illustrative example is also discussed to show the effectiveness of the results in this paper. In the near future, we will consider Riemann-Liouville fractional partial differential problems, which will be more complicated.

Acknowledgments

The project was supported by NNSF of China Grant nos. 11271087 and 61263006, the Scientific Research Foundation of Guangxi University for Nationalities no. 2012QD014, and the Innovation Project of Guangxi University for Nationalities no. gxun-chx2013083.

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