A study on controllability of impulsive fractional evolution equations via resolvent operators

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the (α,β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha ,\beta )$\end{document}-resolvent operator, we concern with the term u′(⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'(\cdot )$\end{document} and finding a control v such that the mild solution satisfies u(b)=ub\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(b)=u_{b}$\end{document} and u′(b)=ub′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u'(b)=u'_{b}$\end{document}. Finally, we present an application to support the validity study.

Controllability is also strongly related to the theory of realization and so-called minimal realization and canonical forms for linear time-invariant control systems such as the Kalman canonical form, the Jordan canonical form and the Luenberger canonical form. Moreover, it is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic.
In recent years, the controllability problems for various linear and nonlinear deterministic and stochastic dynamic systems have been studied in many publications using different method, we refer the reader to [2,6,24]. In addition, Kailasavalli et al. [14] acknowledged the existence and controllability of fractional neutral integro-differential systems with SDD with Banach contraction and resolvent operator technique as the main reference. Dabas et al. [10] studied the existence, uniqueness and continuous dependence of a mild solution for an impulsive neutral fractional order differential equation with infinite delay. Recently, Heping Ma and Biu Liu [20] interpreted the exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay in Banach spaces. Also Yan [27] discussed the approximate controllability of neutral integro-differential delay systems with inclusion type in Hilbert space by using the fixed point theorem of discontinuous multi-valued operators supported by the Dhage fixed point technique with the resolvent operator. Additionally, Yan and Jia [28] explained the approximate controllability of partial fractional neutral stochastic functional integrodifferential inclusions with state-delay.
Especially, the controllability of fractional evolution equations is also studied. In 2015, Liang and Yang [19] investigated the exact controllability for the fractional integrodifferential evolution equations in Banach spaces E involving noncompact semigroups and nonlocal functions without Lipschitz continuity, ⎧ ⎨ ⎩ D α u(t) + Au(t) = f (t, u(t), Gu(t)) + Bv(t), t ∈ J, where D α denotes the Caputo fractional derivative of order α ∈ (0, 1), -A : D(A) ⊂ E → E is the infinitesimal generator of a C 0 -semigroup T(t) (t ≥ 0) of uniformly bounded linear operator, the control function v is given in L 2 (J, U); U is a Banach space, B is a linear bounded operator from U to E; f is a given function and
In 2017, Lian, Fan and Li [18] investigated the approximate controllability for a class of semilinear fractional differential systems of order 1 < α < 2 of the form via the resolvent operator.
In 2019, Singh and Pandey [26] studied some controllability results for the abstract second order Sobolev type impulsive delay differential system of the form On the other hand, in recent years, much attention has been paid to establishing sufficient conditions for the controllability of linear fractional dynamical systems of order 0 < α < 1 by several authors; see a recent monograph [1,2,4,5,8,22,23] and various papers [2,6]. However, there is no work that reported on the problem of controllability of nonlinear fractional dynamical system of order 1 < α < 2, to the best of our knowledge, up until now the controllability for a class of impulsive fractional integro-differential evolution equation with fractional derivative of order α ∈ (1, 2] has not been investigated in the literature. Motivated by the above mentioned aspects, in this paper, we discuss the controllability for a class of impulsive fractional integro-differential evolution equation of the where c D α 0 + is the Caputo fractional derivative of order α ∈ (1, 2] with the lower limit zero, A : D(A) ⊂ E → E a closed linear operator and A generates a strongly continuous (α, β)resolvent family S α,β (t) (t ≥ 0) of uniformly bounded linear operator on a Banach space E.
functions satisfying certain assumptions that will be specified later.
and u(tk ) represent the right and left limits of u(t) at t = t k , respectively, the control function v is given in L 2 (J, U), U is a Banach space, B is a linear bounded operator from U to E, and the operators G and F are given by Throughout this work, we always assume that In this paper, we introduce a suitable concept of a mild solution of the system (1.1). Moreover, we investigate the controllability for the system (1.1), by using the Mönch fixed point theorem combined with (α, β)-resolvent operators.
The paper is organized as follows: The second part of the paper some notations and recall some basic known results. The third part we present a controllability result for the problem (1.1) of our concern. And the last section is provided an example to illustrate applications of the obtained results. Concluding part close this article.

Preliminaries
Let E and E 1 be two Banach space. For any Banach space E, the norm of E is defined by · E . The space of all bounded linear operator from E to E 1 is denoted by L(E, E 1 ) and L(E, E) is written as L(E). We denote by C(J, E) the Banach space of all continuous E-value function on interval J the norm u C = max t∈J u(t) . We use f L p to denote the L p (J, E) norm of f whenever f ∈ L p (J, E) for some p with 1 ≤ p < ∞. We consider the following spaces: Let Let PC 1 (J, E) be the spaces of all functions u ∈ PC(J, E), which are continuously differentiable on J , and the lateral derivatives are continuous on [0, b) and (0, b], respectively. Furthermore, for u ∈ PC 1 (J, E), we denote by u (t) the left derivative at t ∈ (0, b], and by u (0), the right derivative at zero. It is easy to see that the space PC 1 (J, E) is a Banach space with the norm In the following, let us recall some well-known definitions. For more details, see [16].

Definition 2.1
The fractional integral of order γ with the lower limit zero for a function f : [0, ∞) → R is defined as provided the right side is point-wise defined on [0, ∞), where (·) is the gamma function.

Definition 2.2
The Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [0, ∞) → R can be written as

Definition 2.3
The Caputo fractional derivative of order γ for a function f : [0, ∞) → R can be written as where n = [γ ] + 1 and [γ ] denotes the integer part of γ .

Lemma 2.1 ([3])
For q > 0, the general solution of the fractional differential equation c D q t u(t) = 0 is given by denotes the integer part of the real number q.
Now, we review some definitions and lemmas on fractional calculus. For β ≥ 0, let where (·) is the Gamma function . The finite convolution of f and g is denoted by closed linear operators defined on a Banach space E and α, β > 0. Let ρ(A) be the resolvent set of A, we say that the A is the generator of an (α, β)-resolvent family, if there exist ω ≥ 0 and a strongly continuous function S α,β : In this case, {S α,β (t)} t≥0 is called the (α, β)-resolvent family (also called the (α, β)resolvent operator function) generated by A.
Proof For any fixed t ≥ 0, it is easy to check that S α,2 (t), S α,α (t), S α,α-1 (t) are also linear operators since S α,1 (t) is a linear operator. For any u ∈ E, by Lemma 2.2, we have Now, we recall some properties of Hausdorff measure of noncompactness that will be used later.
Next, we are ready to construct a mild solution for the impulsive system (1.1).

Lemma 2.11 Assume
Proof The proof is similar to the proof in paper [12], here we omit it.
Based on Lemma 2.11, we will give the definition of mild solutions for the problem (1.1).

Main results
In this section, we will establish the sufficient conditions for the controllability of the system (1.1). For arbitrary u ∈ PC 1 (J, E), we denote the final stages of u by u b = u(b) and Let B r 0 := {u ∈ PC 1 (J, E) : u PC 1 ≤ r 0 , where r 0 = max{r 1 , r 2 } such that u PC ≤ r 1 , u PC ≤ r 2 }. However, to achieve such a result, we assume certain conditions: (H0) A generates a strongly continuous (α, β)-resolvent family S α,β (t) (t ≥ 0) of the uniformly bounded linear operator on a Banach space E. That is, there exists a constants M ≥ 1 such that S α,β (t) ≤ M for all t ≥ 0 and there exists a positive (i) for a.e. t ∈ J, the function f (t, ·, ·, ·) : E × E × E → E is continuous, and for each (x, y) ∈ E × E, the function f (·, x, y, z) : J → E is strongly measurable; (ii) for any r 0 > 0, there exist a constant q 2 ∈ (0, α) and functions for any countable subsets D 1 , D 2 , D 3 ∈ PC(J, E). (H2) The function I k , J k : PC(J, E) → E, for k = 1, 2, . . . , m, satisfies: (i) There exists a nondecreasing function L j k : R + → R + (j = 1, 2) such that for all u ∈ E and k = 1, 2, . . . , m.
(ii) There exist constants M j > 0 such that, for any countable subsets D j ⊂ E, and for all j = 1, 2 and i = 1, 2, . . . , m. (H3) (i) The function g, h : PC → E is Lipschitz continuous and bounded in E, that is, there exists aconstants c 1 , c 2 ≥ 0 and c 3 , c 4 ≥ 0 such that for all u, v ∈ PC(J, E) .
(H4) Linear operator W : where v u 1 and v u 2 are defined in (3.6). (i) W has an inverse operator W -1 which takes values in L 2 (J, U)\ ker W , and there exist two constants M 1 > 0, (ii) There exist a constant q 1 ∈ (0, α) and a function K w ∈ L 1 q 1 (J, R + ) such that for any bounded subset D 4 ⊂ E. For the sake of brevity, we introduce the notations Theorem 3.1 Assume that the assumptions (H0)-(H4) are satisfied, then the system (1.1) is controllable on J provided that max(λ 1 , λ 2 ) < 1, where Proof Consider the operator Q :

α (ts) Bv(s) + f s, u(s), Gu(s), Fu(s) ds, t ∈ J. (3.4)
Furthermore, by Lemma 2.3, we get (Qu) : where the control v is defined by (3.6) and v u 1 (t) and v u 2 (t) are given by v u Now, the objective is to prove that the operator Q has a fixed point. The proof will be carried out in three steps.
Step 1: ∃r 0 > 0; Q(B r 0 ) ⊂ B r 0 . For this step, it will be carried out by contradiction. Suppose this is not true. Then, for each r 0 > 0, there exists u r 0 (·) ∈ B r 0 and, for some t ∈ J such that (Qu r 0 )(t) > r 0 , we have
Step 2: We show that Q is continuous on B r 0 . To show this, let {u n } ∞ n=1 , {u n } ∞ n=1 ⊂ B r 0 be a sequence such that u n → u, u n → u in B r 0 . Then there exists a number r 0 > 0 such that u n PC ≤ r 0 , u n PC ≤ r 0 and u PC ≤ r 0 and u PC ≤ r 0 for all n ≥ 1, and we define Then we obtain Mb J k u n (t k ) -J k u (t k ) where (3.14) By continuity of f , I k , J k , and the Lebesgue dominated convergence theorem combined with (3.13), (3.14), we get Qu n -Qu PC → 0, as n → ∞. Then, in a similar manner to above, we get . (3.16) By continuity of f , I k , J k , and the Lebesgue dominated convergence theorem combined with (3.15), (3.16), we get (Qu n ) -(Qu) PC → 0, as n → ∞.
Step 3: We will prove that Q satisfies Mönch's condition. To this end, let us assume that D and D are countable subsets of B r 0 and D ⊂ conv({0} ∪ Q(D)) and D ⊂ conv({0} ∪ (Q(D)) ). Then we show that α PC 1 (D) = 0.
By equicontinuity of S α,β (t) and absolute continuity of the Lebesgue integral, we conclude that the right side of the above inequality tends to zero as l 2 → l 1 independently of u. Thus, Q(D) shows equicontinuity on J k for all k = 0, 1, 2, . . . , m. Now, by Lemma 2.9 and (H1)(iii), (H2)(iii) and (H3)(ii), we have Similarly, we obtain where u n (t) = 2 π sin nt, n = 1, 2, . . . , is the orthogonal set of eigenfunctions corresponding to the eigenvalues λ n = -n 2 of A. Then A will be a generator of the (α, β)-resolvent family such that S α,β (t)u = ∞ n=1 cos n 2 t 1 + n 2 (u, u n )u n .