Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

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. Formore 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 form

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][12][13][14][15][16][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][19][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][21][22][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].
Now, let us recall some basic preliminaries on fractional calculus [25] and operator theory [26].

Journal of Mathematics
Definition .The fractional-order integral of a function  ∈ (, )(or  1 (, )) of order  > 0 is defined by Definition .The Caputo fractional derivative of order  ∈ (0, 1) of a function  ∈  1 (, ) (or   (, )) is defined by ( Definition .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  1− 0 is given by The inverse Laplace transform of a function  = L{} is given by for 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  ∈ C 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.

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 .Let  be a closed and densely defined operator on .A family {  ()} ≥0 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 R + and   (0) = , where  is the identity operator.
Moreover, a solution operator   () is called compact if for every  > 0,   () 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   ().
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. e linear fractional system has an integral solution given by Proof.Letting () ̸ = 0, for any  ∈ , then system ( 14) is equivalent to system Applying the Laplace transform to system (16), we have and that implies Therefore, Now, taking the inverse Laplace transform, we get the solution (15).This finishes the proof.
We define a mild solution for system (1).
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 .
The set  = { (; ) ∈  :  ∈  2 (, ) ,  is the mild solution of (1) with control } , (21) 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   :  2 (, ) →  as Then   is a bounded linear operator defined on  2 (, ).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

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 Φ () =   ()  0 ( (0,  0 )) where  is given by (26).
If   is compact  0 -semigroup, then the Cauchy operator Ψ : (, ) → (, ) 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 Ψ.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 Ψ.

Theorem 10. Assume that conditions (H )-(H ) are satisfied. en system ( ) has a mild solution on 𝐽.
Proof.We show the operators Θ and Φ satisfying the hypotheses of Dhage fixed point theorem.For the sake of clarity, we split the proof into two main steps.
Step .Firstly, we prove the continuity of Θ and Φ. (33) Then, the inequality ‖Φ‖ ≤  holds for all  ∈   .The last thing in this step, we show that Φ : (, ) → (, ) 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 Φ is compact.Therefore, Φ is completely continuous.By following the same arguments presented in [13], we can prove.....
Next result, we investigate the approximate controllability of the fractional control system (1).We introduce the following extra conditions: (H7) ( + ) −1 → 0 as  → 0 + in the strong operator topology.
Theorem 11.Assume that conditions (H )-(H ) are satisfied.en, the fractional system ( ) is approximately controllable on .
Example .Consider the fractional control system Therefore, if we choose  such that  < 3 and that (H6) and (H7) are both satisfied, then, using Theorems 10, and 11, we ensure that system (48) is approximately controllable on [0, 1].