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

Abstract

We discuss the functional control systems governed by differential equations with Riemann-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.

1. 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 [1–9]). 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 [10–28] 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:where is the standard Riemann-Liouville derivative with and operator generates a -order fractional resolvent . On the other hand, in [17], the authors considered the following control system governed by functional differential equations with Riemann-Liouville derivative in an abstract space :where is the standard Riemann-Liouville derivative with and operator generates a semigroup , , on an abstract space . belongs to the space , 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 , , 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 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.

2. Preliminaries

This section is devoted to introducing some necessary concepts and auxiliary results which are used in the remainder of this article.

Let be two Banach spaces with norm and be the set of nonnegative real numbers. We denote by all the continuous linear operators from Banach space into itself, by the set of all the continuous functions from the interval to Banach space with , and by the set of all Bochner integrable functions from the interval to Banach space with , where .

Let be the set of all absolutely continuous functions from the interval to Banach space , and . Let ; we consider the Banach space with . This space is of importance in dealing with the well-posedness of the functional equation (2).

As usual, let denote the Gamma function and the integer part of the real number . We give the following definitions.

Definition 1 (see [3, 34]). The Riemann-Liouville fractional integral of order for a function is defined by if the integral above exists point-wisely.

Definition 2 (see [3, 34]). Let . The Riemann-Liouville fractional derivative of order for a function is defined by if the derivative of -order above exists point-wisely.

Lemma 3 (see [3, 34]). Let and . Suppose and . Then, one has

Definition 4 ([32, Definition  3.1]). Let . A family of operators belonging to the space is called a -order fractional resolvent if(a) is continuous in the strong operator topology on , and for .(b) for any .(c) for all .

Definition 5. The infinitesimal generator of the fractional resolvent is defined by with

Lemma 6 ([32, Theorem  3.2]). Let be the infinitesimal generator of the fractional resolvent on Banach space . Then is closed with dense domain and the following hold. (a) and for , .(b)For , one has

Lemma 7. Let be fixed and operator A be the generator of a -order fractional resolvent in Banach space and Then there is a positive number satisfying .

Proof. According to the definition of -order fractional resolvent, for every there is a positive number dependent on satisfying Moreover, since , it follows from the uniform boundedness principle that there is a positive number independent of satisfying .

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 when the resolvent is uniformly integrable and the function is continuous on . 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 under the assumption that is just a fractional resolvent and . Thus, we have the following definitions.

Definition 8. A mapping is called a mild solution of functional differential equation (2) if for , the integral equation is satisfied.

Let be a mild solution of functional equation (2) with the control function belonging to the space and initial value in Banach space . Define the reachable set of functional equation (2) at time by .

Definition 9. The functional equation (2) is called approximate controllability on if .

3. 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 on Banach space .

(H2) There are nonnegative function and a real number satisfying for and .

(H3) There is a nonnegative real number satisfying for and .

Now, we can give the existence result of mild solutions for the functional equation (2).

Theorem 10. Under the conditions (H1)–(H3), the functional equation (2) has one and only one mild solution in the space for every belonging to the control space .

Proof. Define the mapping byObviously, it is well-defined. To prove our result, it is enough to verify the mapping has a unique fixed point in space . We next verify that is a contraction map on for sufficiently large integer number .
According to Lemma 3, there is a constant satisfying . Thus, for any and , one hasFurther,By repeating the above process, one has Thus, Note that the Mittag-Leffler function is uniformly convergent; one has sufficiently large integer , which implies is a contraction map on the space . Hence, it follows from the generalized Banach contraction principle that has one and only one fixed point in . This completes the proof.

4. 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 and a continuous linear mapping , respectively, by Let the pair be the mild solution of (2) with the control function . We also denote the pair by and write the terminal state by So the reachable set of the functional equation (2) at time is Thus the approximate controllability of functional equation (2) means the set is dense on space . In other words, we have the following expression of approximate controllability.

Definition 11. Let . We called the functional equation (2) approximately controllable on if, for any and , there is a control function satisfying where .

Now, we give the following assumptions.

(H3′) There is a nonnegative number satisfying for all and .

(H4) For given and , there is a function satisfying where is a positive real number independent of .

(H5) The following inequality holds: where is the Mittage-Leffler function and comes from Lemma 3.

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.

Lemma 12. Let , be two pairs associated with the control system (2). Then under the hypotheses (H1), (H2), and (H3′) the following results hold: where , .

Proof. The mild solution of control system (2) in satisfies Thus, for , one hasSet . Thus, By a generalized Gronwall inequality for convolution type integral equations (see Corollary  2 in [36]), we obtain It follows that Now, let us take to be the mild solutions for the control system (2) with the control functions , respectively. Then, by the same way, we can get This ends the proof.

Lemma 13. Under the condition (H1), for any there exists such that .

Proof. For and , let Then, we have which means that .
Moreover, it follows from the analyticity and the strong continuity of that . This completes the proof.

Now, we can give the result about the approximate controllability of this section.

Theorem 14. Suppose conditions (H1), (H2), (H3′), (H4), and (H5) hold. Then the functional equation (2) is approximately controllable on the interval .

Proof. Since is the infinitesimal generator of analytic -order fractional resolvent of continuous operator on space , the domain of operator is dense in Banach space . Thus, by the definition of approximate controllability, it suffices to prove that . Next, we should prove that for any and , there is a control function withwhere satisfiesIt follows from the analyticity of fractional resolvent that for , which implies that for . Then, by Lemma 13 there is a function satisfying Thus, for every and , by means of hypothesis (H4), we can find a function satisfying where . Further, for , we determine again by condition (H4) and Lemma 12 with the following two properties: where .
Thus, we may define in , which has the following property By the same way, we obtain the sequence satisfyingwhere Since the condition (H5) is satisfied, it follows that the sequence is a Cauchy sequence in and thus we can obtain a function with Note that the mapping is a continuous linear operator. Therefore for every , we can find a real integer number satisfyingConsequently, by inequalities (43) and (46) we derive where . 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 , 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 -semigroups. As for examples, we refer the reader to [17, 18, 33] since the examples therein still hold in our framework.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by Special Fund for Public Welfare Research of Ministry of Land and Resources of China (Grant no. 201411007).