Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

Wediscuss the functional control systems governed by differential equationswithRiemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.


Introduction
As we all know, the study of fractional calculus theory can be traced back to the end of the seventeenth century.In recent years, a considerable interest has been paid to functional evolution equations with fractional derivatives since they are of importance in describing the natural phenomenon including the models in stochastic processes, finance, and physics (see [1][2][3][4][5][6][7][8][9]).On the other hand, the notion of controllability plays a central role in the study of the theory of control and optimization.Therefore, there are a lot of works on the controllability, approximate controllability, and optimal control of linear and nonlinear differential and integral systems in various frameworks (see [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and references therein).
Recently, the theory of resolvent families was formulated rapidly for the application of differential and integral equations, including the concepts of integrated solution operators [29], fractional resolvent operators [30], and (, )regularized resolvent operators [19].Furthermore, there are extensive studies for the control systems governed by Caputo fractional evolution equations via resolvent theory (see, for instance, [10-12, 15, 27, 31]).However, for the controllability of functional differential systems governed by Riemann-Liouville fractional derivatives there are few results to be shown so far.
In [32,33], the authors established the general theory of fractional resolvent and studied its application to the well posed problem for the evolution equation below: () =  () +  () ,  ∈ (0, ] , where   is the standard Riemann-Liouville derivative with 0 <  ≤ 1 and operator  : () ⊂  →  generates a semigroup (),  ≥ 0, on an abstract space . :  ≐   ([0, ], ) →   ([0, ], ) belongs to the space B(,   ([0, ], )),  is a Banach space, and the control function  belongs to .Motivated by the above-mentioned papers, we try to solve the approximate controllability for functional differential equation ( 2) again but under the assumption that operator  : () ⊂  →  is the infinitesimal generator of a resolvent   (),  > 0, on the general Banach space .Under this general condition, the difficulty on the well-posedness is how to deal with the singularity of resolvent operators and solutions at zero.We deal with this problem by utilizing the new space  1− ([0, ], ) and the theory of fractional resolvent developed in [32,33].In the present article, we first obtain the uniqueness and existence of mild solutions for functional differential equation ( 2) via fractional resolvent and topological approach.Then we will give the sufficient conditions for approximate controllability to equation ( 2) by means of the iterative and approximate method.Our main result seems to be more general and extends some recent related theorems.
In this paper, we first review some basic definitions and give some necessary lemmas.We establish the uniqueness and existence results of mild solutions for functional differential equation (2) in the third part.We will solve the approximately controllable problem of functional control system (2) in the last section.

Preliminaries
This section is devoted to introducing some necessary concepts and auxiliary results which are used in the remainder of this article.
As usual, let Γ() ≐ ∫ ∞ 0  −  −1  denote the Gamma function and [] the integer part of the real number  > 0. We give the following definitions.
Then  is closed with dense domain and the following hold.
(b) For  > 0, one has Lemma 7. Let  > 0 be fixed and operator A be the generator of a -order fractional resolvent {  ()} >0 in Banach space  and Then there is a positive number  > 0 satisfying Proof.According to the definition of -order fractional resolvent, for every  ∈  there is a positive number Moreover, since  1−   () ∈ B(),  ∈ [0, ], it follows from the uniform boundedness principle that there is a positive number  > 0 independent of  satisfying sup ∈[0,] ‖ 1−   ()‖ ≤ .
Now, we turn to the concept of mild solutions and approximate controllability of functional equation (2).In [33], Fan proved that the convolution   *  exists and defines a continuous function on [0, ] when the resolvent is uniformly integrable and the function  is continuous on [0, ].In fact, by the Proposition 1.3.4 in [35] and the Young inequality, we can also prove that the above convolution exists and defines a continuous function on [0, ] under the assumption that   () is just a fractional resolvent and  ∈  1 ([0, ], ).Thus, we have the following definitions.

The Result of Uniqueness and Existence for Mild Solutions
In this part, we study the existence and uniqueness of solutions for functional equation (2).To prove our result, we suppose the following conditions.(H1)  is the infinitesimal generator of an analytic order fractional resolvent of continuous linear operators {  ()} >0 on Banach space .

Theorem 10. Under the conditions (H1)-(H3), the functional equation (2) has one and only one mild solution in the space
Obviously, it is well-defined.To prove our result, it is enough to verify the mapping  has a unique fixed point in space  1− ([0, ], ).We next verify that   is a contraction map on  1− ([0, ], ) for sufficiently large integer number .

The Result of Approximate Controllability
In this part, new sufficient conditions for the approximate controllability of the functional equation ( 2) are derived and proved by means of the iterative and approximate approach.For this purpose, we define operator  : where   () = (; 0,  0 ,   ), 0 <  ≤ .
Because condition (H3  ) implies (H3), the existence result is still true if condition (H3) is replaced by (H3  ) in Theorem 10.Next, to prove the result of approximate controllability of functional equation ( 2), we need two lemmas below.
Proof.The mild solution () = (; 0,  0 , ) Thus, for 0 ≤  ≤ , one has Set By a generalized Gronwall inequality for convolution type integral equations (see Corollary 2 in [36]), we obtain It follows that Now, let us take  1 ,  2 ∈  1− ([0, ], ) to be the mild solutions for the control system (2) with the control functions  1 ,  2 ∈ , respectively.Then, by the same way, we can get This ends the proof.
Remark 15.In this article, we consider the control system with Riemann-Liouville derivative under the general assumption that linear operator  : () ⊂  →  generates a fractional resolvent   (),  > 0 on the Banach space .By the resolvent theory, fixed point theorem, and iterative and approximate approach, we obtain the results of existence and uniqueness for mild solutions and approximate controllability of functional control system.From this point of view, our results extend the theorems in [17,18], where the authors assumed that the operator  generates  0 -semigroups.As for examples, we refer the reader to [17,18,33] since the examples therein still hold in our framework.