Controllability of a Class of Impulsive -Caputo Fractional Evolution Equations of Sobolev Type

In this paper, we investigate the controllability of a class of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive ψ-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results.


Introduction
Fractional calculus sweeps the board in science and engineering, such as chemistry, economics, biology control of dynamical systems, financing viscoelastic materials, signal processing, and so on. For more details on the theory and applications in this filed, one may see the monographs [1][2][3][4][5], and the references cited therein. The fractional calculus from the physical community, for instance, fractional calculus and anomalous diffusion have been studied intensively [6][7][8][9].
Controllability of fractional semilinear evolution systems in Banach spaces has been paid much attention. Many researchers have focused on this topic. We refer the readers to El-Borai [10,11], Balachandran and Park [12], Wang et al. [13][14][15], Zhou and Jiao [16,17], Sakthivel et al. [18], Debbouchra and Baleanu [19], Li et al. [20], Kumar and Sukavanam [21], and Lord et al. [22] and the references therein. In 2013, Fečkan et al. [23] investigated the controllability of q ∈ (0, 1)-order Caputo fractional functional evolution equations of Sobolev type in Banach space X : where φ ∈ C([−r, 0], X), A and E are linear operators, A is closed, E is bijective, and E −1 is compact. By utilizing the Schauder fixed point theorem and the properties of two new characteristic solution operators, the authors presented the exact controllability of system (1). Due to important and potential applications of impulse and delay, the study of dynamical systems with impulses and time delay has gained more and more attention. Impulsive fractional differential equations with delays have been widely applied to many fields, such as weather predicting, drug delivery processing, agricultural insect pests control, and some other optimization problems. We refer the readers to [24][25][26][27][28] and the references therein.
In 2021, Zhao [29] studied the exact controllability of a class of impulsive fractional nonlinear evolution equations with delay in Banach spaces: a.e. t ∈ I := [0, a], where γ ∈ (0, 1), A : D ⊂ X → X is a closed the linear unbounded operator on X with dense domain D. In Ref. [29], Zhao defined the mild solution of system (2) as follows: Unfortunately, (3) is not correct. In fact, from [30] we know that the mild solution of system (2) should be defined as below.
In recent decades, the generalizations of the fractional calculus operators have been done [31][32][33][34], since they are more general operators that allow for the discussion and analysis of a wide class of particular cases. Considering the Caputo fractional derivative of a function with respect to another function ψ, Almeida [35] generalized the definition of Caputo fractional derivative, in which the advantage of this new definition of the fractional derivative is that by choosing a suitable function ψ, a higher accuracy of the model could be achieved. For recent relevant work on generalized fractional derivatives, one may see refs. [36][37][38]. In ref. [39], Suechori and Ngiamsunthorn studied the following semilinear ψ-Caputo fractional evolution equations: where 0 < α < 1, T < ∞, A is the infinitesimal generator of a C 0 -semigroup of uniformly bounded linear operators {T(t)} t≥0 . Existence results of mild solutions to (5) have been obtained. These results generalize the previous work in which the classical Caputo fractional derivative is studied. Motivated by the above works, we consider the following impulsive ψ-Caputo fractional evolution equations of Sobolev type: a.e. t ∈ J , ∆x(t k ) = I k (x(t k )), k = 1, . . . , m, is the Caputo fractional derivative of a function x with respect to another function ψ. The operators A : D(A) ⊂ X → Y and E : D(E) ⊂ X → Y, where X and Y are two real Banach spaces, x(·) ∈ X and the control function u(·) ∈ U . The Banach space of admissible control functions is denoted by U involving a Banach space U, in which we define either U := L 2 (J, U) for 1 . . , m are appropriate functions which will be specified later. PC(J * , X) = {x : J * → X, x(t) is continuous at t = t k , and left continuous at t = t k , and x(t + k ) exists, k = 1, 2, . . . , m}. Obviously, PC(J * , X) is a Banach space with the norm In this paper, by means of two new characteristic solution operators and Schauder fixed point theorem, we present the controllability of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. This paper will be organized as follows. In Section 2, we will briefly recall some definitions and preliminaries. In Section 3, sufficient conditions ensuring exact controllability of the systems are provided. In Section 4, an example is given to illustrate our theoretical result. Finally, we give the conclusions in Section 5.
To the best of our knowledge, no such results in the literature studied theoretically the impulsive fractional evolution equations of Sobolev type containing the fractional derivative of a function with respect to another function. Our goal is to cover this gap in this paper. Our results extend the main results of Ref. [23].

Preliminaries
In this section, we recall some basic definitions and lemmas that will be used later.
Definition 2 ([35,40]). Let n − 1 < α < n, f ∈ C n ([a, b]) and ψ ∈ C n ([a, b]) be an increasing function with ψ (t) = 0 for all t ∈ [a, b]. The left ψ-Caputo fractional derivative of order α of a function f is defined by We will give some properties of the fractional integral and the fractional derivatives of a function with respect to another function. Lemma 1 ([35]). Let f ∈ C n ([a, b]) and n − 1 < α < n. Then we have In special case, given α ∈ (0, 1), we have for all s. From Ref. [40], we have the following property of the generalized Laplace transform of the Caputo fractional operators with respect to function ψ.
For problem (6), throughout this paper, the following assumptions on the operators A and E are satisfied.
(H1) A and E are linear operators, and A is closed.
By (H3) we know that E is closed. In fact, E −1 is closed and injective, then the inverse is also closed. Note (H1)-(H3) and the closed graph theorem that the boundedness of the linear operator −AE −1 : By Definitions 1 and 2, and Lemma 1, the impulsive problem (6) could be written as the following fractional integral equation if the integral in (10) exists.
Proof. (i) For t ≥ 0, by (11) and Definition 3, we get On the other hand, by Lemma 2, one has Combing (13) with (14), we obtain (ii) For t ≥ t i , similar to the proof of (i), we can prove that (ii) holds, so we omit it here.
If t ∈ J n = (t n , t n+1 ], then we obtain by Lemma 3 that For t = 0, one has Moreover, we have Thus, expression (17) is a solution of problem (6).

Definition 4.
For each u ∈ U and φ ∈ C(E), a function x ∈ PC(J * , X) is called a mild solution of (6) if (17) holds. From [35], we can obtain easily that the following properties of S

Main Results
According to the exact controllability considered in Ref. [41], we give the following definition. (6) is the exact controllability on J = [0, b] if for any initial function φ ∈ C(E) and x 1 ∈ D(E), there has a control u ∈ U such that the mild solution x of (6) on [−r, b] satisfies x(b) = x 1 .
(H7) B : U → Y is a bounded linear operator and a linear operator W : U → D(E) defined by The right inverse operator W −1 : D(E) → U is bounded, i.e., WW −1 = I D(E) , and thus there exist two constants M 1 , M 2 > 0 such that B ≤ M 1 and W −1 ≤ M 2 , then by determining M 2 we could define the norm · D(E) on D(E).
If α ∈ (0, 1), then we have If α ∈ ( 1 2 , 1), then one has by (H6) that for any t ∈ J, where Obviously, Wu ∈ D(E) and W is well defined. In fact, by Lemma 5 and (21), one has For an arbitrary function x(·), by means of the above assumptions, it is suitable to define the following control formula: In the following, we will prove that, in view of the control u in (23), the operator P defined by from PC(J * , D(E)) into PC(J * , D(E)), has a fixed point. It is obvious that this fixed point is just a solution of system (6). Moreover, we can check that For each number K > 0, set Clearly, B K is a bounded, closed, convex subset in PC(J, D(E)).

Lemma 6.
Assume that (H1)-(H7) are satisfied. Then there exists a K ≥ max φ , N 3 1−ρ where and , then Since K ≥ φ , by (25), we note that the control u defined in (23) satisfies where Thus, for t ∈ J n , we derive by (21) and (26) that From (18), we have We complete the proof.
The proof is completed. Proof. Let x ∈ B K and t , t ∈ J n , t < t . One has Obviously, J 7 → 0 as t → t . By Lemma 3(iv), S α,ψ E (t, s) and T α,ψ E (t, s) are continuous in the uniform operator topology for t ≥ s ≥ 0, and u(·) is bounded by (28). Then one can check the terms J 1 , J 2 , J 4 , J 6 , J 8 → 0 as t → t . By virtue of Lebesgue's dominated convergence theorem, we obtain J 3 , J 5 → 0 as t → t . Hence, P B K is equicontinuous and bounded.
Proof. From Lammas 6-8 and the Arzela-Ascoli theorem, we obtain that P B K is precompact in PC(J * , X). Thus P is a completely continuous operator on PC(J * , X). By the Schauder fixed point theorem, P has a fixed point in B K . Each fixed point of P is a mild solution of the system (6) on J such that (P x)(t) = x(t) ∈ X. Hence, the system (6) is controllable on J.

Conclusions
In this study, we constructed a mild solution for a class of impulsive Caputo fractional evolution equations of Sobolev type based on generalized Laplace transform with respect to the ψ-function. By using the boundedness and compactness of two new introduced characteristic solution operators and the fixed point technique, we derive some new controllability results for ψ-fractional impulsive functional evolution equations of Sobolev type. The obtained results generalized the non-impulse and classical Caputo fractional derivative cases. Finally, an example is given to illustrate the effectiveness and feasibility of our criterion.
In the future, we will consider the nonlinear impulsive ψ-Hilfer fractional evolution equations of Sobolev type, and study the controllability of the mild solution for such equations. Data Availability Statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest:
The authors declare that there are no conflicts of interest.