Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

and Applied Analysis 3 Let (X, d) be a metric space. We use the notations P cl (X) = {Y ∈ P (X) : Y is closed} , P b (X) = {Y ∈ P (X) : Y is bounded} , P cV (X) = {Y ∈ P (X) : Y is convex} , P cp (X) = {Y ∈ P (X) : Y is compact} . (7) Firstly, let us recall the following basic definitions from fractional calculus. For more details, one can see [28, 29]. Definition 1. The integral I α t f (t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f (s) ds, α > 0, (8) is called Riemann-Liouville fractional integral of order α, where Γ is the gamma function. Definition 2. For a function f(t) given in the interval [0,∞), the expression D α t f (t) = 1 Γ (n − α) ( d dt ) n ∫ t 0 (t − s) n−α−1 f (s) dt, (9) where n = [α] + 1 and [α] denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α. Lemma 3 (see [28]). Let α > 0,m = [α]+1, and let x m−α (t) = I m−α t x(t) be the fractional integral of order m − α. If x(t) ∈ L 1 (J, X) and x m−α (t) ∈ AC m (J, X), then one has the following equality: I α t D α t x (t) = x (t) − m ∑ k=1 x (m−k) m−α (0) Γ (α − k + 1) t α−k . (10) In order to study the PC-mild solutions of (4) in Banach space PC 1−α (J, X), we give the following results which will be used throughout this paper. Lemma 4. Let 0 < α ≤ 1, and let x 1−α (t) = I 1−α t x(t) be the fractional integral of order 1 − α. If x(t) ∈ PC 1−α (J, X) and x 1−α (t) ∈ PC(J, X), then one has the following equality:


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][3][4].Some authors [5][6][7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems.In [5][6][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  0 -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][10][11][12][13][14][15][16][17] have studied it for control systems.
In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems: () =  () +  () +  (,  ()) ,  ∈  = [0, ] , where     is Caputo's fractional derivative of 0 <  < 1 and  is the infinitesimal generator of a  0 -semigroup () of bounded operators on the Hilbert space ; the control function (⋅) is given in  2 (, );  is a Hilbert space;  is a bounded linear operator from  to ;  :  ×  →  is a given function satisfying some assumptions and  0 is an element of the Hilbert space .
In [18], Rykaczewski studied the approximate controllability of an inclusion of the form ẋ () ∈  () +  (,  ()) +  () ,  ∈  = [0, ] , 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][20][21][22][23][24][25][26][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][14][15][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][33][34][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.
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.

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  0 -semigroup () ( ≥ 0), we set  := sup ∈[0,∞) ‖()‖   () < ∞.Let (, ) denote the Banach space of all  value continuous functions from  = [0, ] to  with the norm Obviously, the space  1− (, ) 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].
is called Riemann-Liouville fractional integral of order , where Γ is the gamma function.
Definition 2. For a function () given in the interval [0, ∞), the expression where  = [] + 1 and [] denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .
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  > 0; 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  1− -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  > 0; taking the Laplace transformation to (37), we obtain That is, Notice that the Laplace transform of Thus one can calculate the mild solution of (24) as By the above work, the  1− -mild solution of ( 19) is given by That is, where  = 1, 2, . . ., , where   is a probability density function defined on (0, ∞); that is, This completes the proof of the lemma.
Now, we also introduce some basic definitions on multivalued maps.For more details, see [36][37][38] We say that  has a fixed point if there is a  ∈  such that  ∈ ().
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 Γ  0 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 (, Γ  0 ) → 0 as  → 0 + 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 W ⊂  1 (, ) be integrably bounded.If, for all  ∈ , there is a relatively weakly compact set () ⊂  such that () ∈ () for every  ∈ W, then W is relatively weakly compact in  1 (, ).
Lemma 13 (see [37]).Let  be a bounded, convex, and closed subset in the Banach space  and let  :  → 2  \ {0} be a u.s.c.condensing multivalued map.If, for every  ∈ , () is a closed and convex set in , then  has a fixed point.

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.2):  is a multivalued map satisfying  :  ×  → P ,V () which is measurable to  for each fixed  ∈ , u.s.c. to  for each  ∈ , and for each  ∈  1− (, ) the set  , = { ∈  1 (, ) :  () ∈  (,  ())} (54) is nonempty.
Step 4. Φ  is u.s.c and condensing.We decompose Φ  as , where the operators Φ 1  and Φ 2  are defined by According to [41, Corollary 2.2.1], we will prove that Φ 1  is a contraction operator, while Φ 2  is a completely continuous operator.
Next, we prove that Φ 2  is u.s.c and completely continuous.We subdivide the proof into several claims.Claim 1.There exists a positive constant  such that Φ 2  (  ) ⊆   .
By employing the technique used in Step 2, one can easily show that there exists  > 0 such that Φ 2  (  ) ⊆   .
By using Hölder's inequality and assumption (3), we get  (79) By the compactness of (  ) (   > 0), we obtain the set The right-hand side of the above inequality tends to zero as  0. Therefore, there are relatively compact sets arbitrarily close to the set Π(),  > 0. Hence the set Π(),  > 0 is also relatively compact in .As a consequence of Claims 1-3 together with the Arzola-Ascoli theorem, we can conclude that Φ 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 Γ ∘   is a closed graph operator.Moreover, we have Since   →  * , it follows from Lemma 13 that Therefore, Φ 2  has a closed graph.Since Φ 2  is a completely continuous multivalued map with compact value, we have that Φ 2  is u.s.c.Thus Φ  = Φ 1  + Φ 2  is u.s.c and condensing.Therefore, applying Lemma 13, we conclude that Φ  has a fixed point (⋅) on   0 .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.
Proof.By employing the technique used in Theorem 14, we can easily show that, for all 0 <  < 1, the operator Φ  has a fixed point in   0 , where  0 = ().Let   (⋅) be a fixed point of Φ  in   0 .Any fixed point of Φ  is a mild solution of (4); this means that there exists   ∈  ,  such that, for each  ∈   , Consequently the sequence {  } is uniformly bounded in  1/ (, ).Thus, there is a subsequence, still denoted by {  }, that converges weakly to, say,  in  1/ (, ).Denoting This proves the approximate controllability of system (4).

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

Lemma 7 .
The operator   () has the following properties.
It is easy to see that  2 tends to zero independently of  ∈   as  2 →  1 .Note that, from Lemma 10,   () is continuous in the uniform operator topology for  > 0; we can directly obtain  1 and  3 tending to zero independently of  ∈   as  2 →  1 .Applying the absolute continuity of the Lebesgue integral, we have  4 ,  5 , and  6 tending to zero independently of  ∈   as  2 →  1 .Therefore, Φ 2  (  ) ⊂  1− (, ) is equicontinuous.