Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

Abstract

We obtain sufficient conditions for the approximate controllability of a fractional nonlinear hybrid differential system. The results are obtained by using resolvent and sectorial operators technique via Dhage fixed point theorem.

1. Introduction

Fractional differential systems have described many practical dynamical phenomena more efficiently than the corresponding integer-order systems; hence they have attracted the attention of many researchers in such fields (see [1–10] and references cited therein).

One of these systems is the fractional control system with all its branches such as stability, controllability, and observability. In the recent years, many investigations on the controllability problems of fractional behaviour have extensively appeared with various applications on linear and nonlinear systems. Particularly, the researches have focused on exact (complete) and approximate controllability (see the articles [11–17] and the references therein).

The fractional control systems involving a linear closed (unbounded) operator which generates resolvent operators were considered recently by many authors [15, 18–20]. The lack of the semigroup property of the generated resolvent operator was the most popular difficulty that has been faced by the interested researchers. However, some authors used the idea of analytic sectorial operators to overcome this problem. For more details, we refer the reader to the papers [20–23] and references therein. To the best of our knowledge, there is not any investigation in the controllability problem via resolvent operators applied on hybrid systems such as the system that has been discussed in the article [24].

In this article, we study the approximate controllability for a fractional hybrid differential system of the formwhere is the Caputo fractional derivative of order such that , and are two real Hilbert spaces, is the infinitesimal generator of a resolvent operator , is a bounded linear operator, , denotes the -order fractional integral, and are given functions such that does not vanish on

2. Preliminaries

Let be the space of all -valued continuous functions defined on with the norm , and let be the space of -valued Bochner integrable functions defined on with the norm , where

Now, let us recall some basic preliminaries on fractional calculus [25] and operator theory [26].

Definition 1. The fractional-order integral of a function (or ) of order is defined by

Definition 2. The Caputo fractional derivative of order of a function (or ) is defined by

Definition 3. The Laplace transform of a function is given by

The Laplace transform of the Caputo fractional derivative is given by The Laplace transform of the fractional integral is given by

The inverse Laplace transform of a function is given byfor some suitable path to ensure the existence of the integral.

The resolvent operator of an operator is defined as

The resolvent set is the set of all regular values of such that is injective, bounded linear operator.

The following fixed point theorem, which is due to Dhage [27], is essential tool for the proof of the main result.

Theorem 4. Let be a nonempty bounded closed convex subset of a Banach algebra Let and be continuous operators satisfying the following:(a) is completely continuous,(b) is Lipschitzian with a Lipschitz constant ,(c) implies for all , and(d), where .Then the operator equation has a solution in .

3. Fractional Control Systems via Resolvent Operators

Let be a linear operator defined on the subspace , the domain of to the space An operator is said to be closed if and only if its domain is a complete space with respect to the norm An operator is said to be densely defined if its domain is dense in . The denseness of the domain is necessary and sufficient for the existence of the adjoint. The adjoint operator of unbounded operators can be defined as bounded operators. For more details on these topics, the reader may refer to [26, 28].

Next, we introduce some information about solution operators [29].

Consider system

which has an integral solution given by

Definition 5. Let be a closed and densely defined operator on . A family of bounded linear operators in is called a solution operator (or -resolvent) generated by if the following conditions are satisfied:(S1) is strong continuous on and , where is the identity operator.(S2) and for all and (S3) is a solution of the integral equation (10).

Moreover, a solution operator is called compactif for every , is a compact operator. If is a solution operator of system (9), then by (S3), we deduce that

where consists of all for which the limit exists. We call as the infinitesimal generator of or simply we say that generates the solution operator .

Definition 6. Let be a closed linear operator. is said to be a sectorial operator of type if there exist , and such that the solution operator of exists throughout the sector and

Hereafter, we assume that is a sectorial operator of type that generates the solution operator In this case, we can write the solution operator of system (9) as

with being a suitable path in a sector

Lemma 7. The linear fractional systemhas an integral solution given by

Proof. Letting , for any , then system (14) is equivalent to systemApplying the Laplace transform to system (16), we have and that impliesTherefore, Now, taking the inverse Laplace transform, we get the solution (15). This finishes the proof.

We define a mild solution for system (1).

Definition 8. A function is called a mild solution of system (1) if it satisfies for any .

We introduce some preliminaries about controllability (see [3, 11–13, 18–20, 27]). We assume that is a mild solution (we call it now as state function) of the fractional differential system (1) corresponding to a control .

Definition 9. System (1) is said to be approximately controllable on if for every desired final state and , there exists a control such that satisfies

The set is called the reachability set of system (1). Therefore, the fractional system (1) is said to be approximately controllable on if , where denotes the closure of . If the used control function is fixed, the symbol is used instead of

We define the controllability operator as

Then is a bounded linear operator defined on . The adjoint operator of is given by

The controllability Gramnian is defined by

Following the idea, as in [20], the suggested control function for system (1) can be written in the form. where

4. Approximate Controllability

We prove the approximate controllability of the fractional control system (1) by using the mild solution (20) and the control defined by (26). More precisely, we prove the existence of at least one state satisfying (20) and (26) following the same arguments presented in [20], but using Dhage fixed point theorem. For this lets and where is given by (26).

If is compact -semigroup, then the Cauchy operator defined as is also compact. Unfortunately, the resolvent operator does not have the property of semigroups which leads to the impossibility of obtaining the compactness of the Cauchy operator . However, we can prove the continuity of the solution operator in the case of analytic operators by which we can prove the compactness of the Cauchy operator .

Let be a fixed positive real number such that Clearly, is a bounded closed and convex set. We need the following assumptions:(H1) is compact analytic operator such that (H2) is continuous and there exists a positive constant such that , for all (H3) is continuous and there exists a function such that , for all (H4) is a linear bounded operator and there exists such that .(H5)Let , such that (H6),

Theorem 10. Assume that conditions (H1)-(H6) are satisfied. Then system (1) has a mild solution on .

Proof. We show the operators and satisfying the hypotheses of Dhage fixed point theorem. For the sake of clarity, we split the proof into two main steps.
Step 1. Firstly, we prove the continuity of and . Let be a sequence in with in . By the hypotheses (H2) and (H3), we obtain the convergence of and to and , respectively, for any . HenceThus, by Lebsegue Dominated Convergence Theorem and the fact that , for some and for any , we have ThenAs and using again dominated convergence theorem we have Thus and are continuous on Next, we show that is bounded on . In fact, for all , we haveThen, the inequality holds for all .
The last thing in this step, we show that is a compact operator. It is sufficient to prove that is compact. But this has been proved in many articles see, for example, ([20]: Theorem 3.3) by using the Ascoli-Arzela theorem. Hence we conclude that is compact. Therefore, is completely continuous. By following the same arguments presented in [13], we can prove.....
Step 2. The hypothesis (H3) shows that the operator is Lipschitz with Lipschitzian constant Next, we show that whenever for all For this, letting and , we haveIn consequence, this implies that

If is chosen large enough such that , then we ensure that Therefore, all hypotheses of Dhage Theorem are satisfied; then there exists a fixed point satisfying the operator equation , which is a solution of system (1).

Next result, we investigate the approximate controllability of the fractional control system (1). We introduce the following extra conditions:(H7) as in the strong operator topology.(H8)The sequence is bounded in

Theorem 11. Assume that conditions (H1)-(H8) are satisfied. Then, the fractional system (1) is approximately controllable on .

Proof. In virtue of Theorem 10, there exists a mild solution such thatwhereTherefore,Hence,Now, by condition (H2), we havewhich implies that the sequence is bounded in the Hilbert space Together with (H8), there exist subsequences of and in converging weakly to some points , respectively. LetThus, Using the compactness of , we can deduce that the mappingfrom to is compact. So, we obtain thatas This implies that as In view of (40) and condition (H5), we obtain thatHence as , which implies that the fractional system (1) is approximately controllable on . This finishes the proof.

Example 12. Consider the fractional control system Let and , where and are absolutely continuous, , Then is an infinitesimal generator of an analytic semigroup , , which can be written in the formwhere is the orthonormal basis for It can be shown that is also a generator of a compact analytic operator , , given bywhere is the Wright function (see [19, 21]). Setting , , , and , then system (48) is equivalent to system (1) for any The operator satisfies the hypothesis (H1) such that . Simple calculations lead to , , and Therefore, if we choose such that and that (H6) and (H7) are both satisfied, then, using Theorems 10, and 11, we ensure that system (48) is approximately controllable on

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that they have no conflicts of interest.