On the Approximate Controllability of Fractional Evolution Equations with Generalized Riemann-Liouville Fractional Derivative

We discuss the approximate controllability of fractional evolution equations involving generalized Riemann-Liouville fractional derivative. The results are obtained with the help of the theory of fractional calculus, semigroup theory, and the Schauder fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the abstract theory.


Introduction
Many social, physical, biological, and engineering problems can be described by fractional partial differential equations.In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations.In the last two decades, fractional differential equations (see, e.g., [1][2][3][4] and the references therein) have attracted many scientists, and notable contributions have been made to both the theory and applications of fractional differential equations.Several researchers have studied the existence results of initial and boundary value problems involving fractional differential equations.The motivation for those works arises from both the development of the theory of fractional calculus itself and the applications of such constructions in various fields, including physics, chemistry, aerodynamics, and electrodynamics of complex medium.Recently, Zhou and Jiao [5] discussed the existence of mild solutions of fractional evolution and neutral evolution equations in an arbitrary Banach space in which the mild solution is defined using the probability density function and semigroup theory.Using the same method, Zhou et al. [6] gave a suitable definition of a mild solution for an evolution equation involving a Riemann-Liouville fractional derivative.Using sectorial operators, Shu and Wang [7] gave a definition of a mild solution for fractional differential equations with order 1 <  < 2 and established existence results.Agarwal et al. [8] studied the existence and dimension of the set of mild solutions of semilinear fractional differential equations inclusions.Hilfer [9] proposed a generalized Riemann-Liouville fractional derivative, for short, which includes Riemann-Liouville fractional derivative and Caputo fractional derivative.Very recently, Gu and Trujillo [10] investigated a class of evolution equations involving Hilfer fractional derivatives.
Recently, the approximate controllability of fractional semilinear evolution systems in abstract spaces has been studied by many researchers.In [11], Sakthivel et al. studied the approximate controllability of semilinear fractional differential systems.Kumar and Sukavanam [12,13] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and the Schauder fixed point theorem.Balasubramaniam et al. [14] derived sufficient conditions for the approximate controllability of impulsive fractional integrodifferential systems with nonlocal conditions in Hilbert space.Using the analytic resolvent method and the continuity of a resolvent in the uniform operator topology, Fan [15] derived existence and 2 Journal of Function Spaces approximate controllability results of a fractional control system.Liu and Bin [16] studied existence of mild solutions and approximate controllability results for impulsive fractional abstract Cauchy problems involving Riemann-Liouville fractional derivatives.More recently, Mahmudov [17] formulated and proved a new set of sufficient conditions for the approximate controllability of fractional neutral type evolution equations in Banach spaces by using Schauder's fixed point theorem.However, the approximate controllability of fractional evolution equations with Hilfer fractional derivative has not yet been studied.
Motivated by the aforementioned papers, we study the approximate controllability of a class of fractional evolution equations: where  ], 0 + is the Hilfer fractional derivative, 0 ≤ ] ≤ 1, 0 <  < 1, the state (⋅) takes value in a Hilbert space , and  is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {(),  > 0} in .The control function  takes values in a Hilbert space ,  ∈  2 ([0, ], ), and  :  →  is a linear bounded operator.The function  : [0, ] ×  →  will be specified in later sections.
The focus of this paper is the study of the approximate controllability of fractional semilinear differential equations in Hilbert spaces.We will explore approximate controllability using techniques from [18].The method is inspired by viewing the problem of approximate controllability as the limit of optimal control problems and replacing it via the convergence of resolvent operators (the resolvent condition, (R)).
Definition 1 (see [1]).The fractional integral of order  > 0 with the lower limit  for a function  : [, ∞) → R is defined as provided that the right-hand side is pointwise defined on [0, ∞), where Γ is the gamma function.
Definition 2 (see [9]).The Hilfer derivative of order 0 ≤ ] ≤ 1 and 0 <  < 1 with lower limit  is defined as for functions such that the expression on the right-hand side exists.
(i) For any fixed  > 0, S ], () and P  () are linear and bounded operators, and

Journal of Function Spaces 3
In this paper we adopt the following definition of mild solution of the initial-value problem (1); see [10].Definition 5. A solution (⋅; ) ∈ ([0, ], ) is said to be a mild solution of (1) if for any  ∈  2 ([0, ], ) the integral equation is satisfied, for all 0 ≤  ≤ .
Let (; ) be the state value of ( 12) at the terminal time  corresponding to the control .We introduce the set R() = {(; ) :  ∈  2 ([0, ], )}, which is called the reachable set of system ( 12) at terminal time , and denote its closure in  by R().Definition 6.The system ( 1) is said to be approximately controllable on [0, ] if R() = ; that is, given an arbitrary  > 0 it is possible to steer from the point  0 to within a distance  from all points in the state space  at time .
Remark 7. (i) When ] = 0, the fractional equation ( 12) simplifies to the classical Riemann-Liouville fractional equation which has been studied by Zhou et al. in [6].In this case (ii) When ] = 1, the fractional equation ( 12) simplifies to the classical Caputo fractional equation which has been studied by Zhou and Jiao in [5].In this case where S  () is defined in [5].

Main Results
To investigate the approximate controllability of system (12), we impose the following conditions: (H1) (),  > 0, is compact; (H2) the function  : [0, ]× →  satisfies the following: (a) (, ⋅) :  →  is continuous for each  ∈ (0, ], (b) for each  ∈ , (⋅, ) : (0, ] →  is strongly measurable; (H3) there is a constant  1 ∈ (0, ) and  ∈ Consider the following linear fractional differential system: The approximate controllability for the linear fractional system ( 16) is a natural generalization of approximate controllability of linear first order control system.It is convenient at this point to introduce the following controllability and resolvent operators associated with (16): respectively, where  * denotes the adjoint of  and P *  () is the adjoint of P  ().It is straightforward to show that the operator   0 is a linear bounded operator for 1/2 <  ≤ 1.We also impose the following resolvent condition: (R) for every ℎ ∈ , ( + Γ  0 ) −1 (ℎ) converges to zero as  → 0 + in strong topology.
In order to formulate the controllability problem in the form in which the fixed point theorem is readily applicable, it is assumed that the corresponding linear system is approximately controllable.It will be shown that system (1) is approximately controllable provided that we can show for all  > 0 there exists a continuous function  ∈ ([0, ], ) such that where Based on this observation, our goal is to find conditions for the solvability of (18).Note also that it will be shown that the control in (18) drives the system (1) from  0 to provided that the system (18) has a solution.For all  > 0, consider the operator Φ  :  ], ([0, ], ) →  ], ([0, ], ) defined as follows: Define  (1−])(1−) (Φ  ) () as follows: It will be shown that for all  > 0 the operator Φ  :  ], ([0, ], ) →  ], ([0, ], ) has a fixed point.To prove this we will employ the Schauder fixed point theorem.
For  > 0 sufficiently small, we have Since the compactness of T  () ( > 0) implies the continuity of T  () ( > 0) in the uniform operator topology, it can be easily seen that lim ℎ → 0 +  7 = 0.
The case  = 0 and 0 < ℎ ≤  follows from (23).Thus, the set {Φ   :  ∈   } is an equicontinuous family of functions in  ], ([0, ], ).Proof.Let 0 <  ≤  be fixed and let  be a real number satisfying 0 <  < .For  > 0 define the operator Φ ,  on   by Since () is a compact operator, the set {(Φ ,  )() :  ∈   } is relatively compact in .Moreover, for each  ∈   , we have A similar argument as before yields where we have used the equality (51) The rest of the proof is similar to the proof of Lemma 9.