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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 639492, 17 pages
Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions
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.
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.
The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman  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 . 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 , 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 , 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 , 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.  and Podlubny  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  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 . 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.
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
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 ). 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 , .
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
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 is a probability density function defined on ; that is,
Due to the work of the paper , 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.
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 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 .
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 ). 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 ). 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
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 ,