An investigation on boundary controllability for Sobolev-type neutral evolution equations of fractional order in Banach space

: The main focus of this paper is on the boundary controllability of fractional order Sobolev-type neutral evolution equations in Banach space. We show our key results using facts from fractional calculus, semigroup theory, and the ﬁxed point method. Finally, we give an example to illustrate the theory we have established


Introduction
In recent decades, fractional calculus has played a significant role in mathematics.Some physical problems cannot be addressed using integer-order differential equations, while fractional-order differential equations can.Fractional differential equations have received a lot of attention and are utilized widely in engineering, physics, chemistry, biology, and a variety of other subjects.Fractional calculus notions have lately been effectively applied to a wide range of domains, and scientists are increasingly realizing that the fractional system may well correspond to many occurrences in regular sciences and engineering.Rheology, liquid stream, scattering, microscopic structures, viscoelasticity, and optics are just a few of the significant fractional calculus issues that are now being studied.Although diagnostic structures are often difficult to come by, the efficacy of mathematical evaluation methodologies for fractional systems in these disciplines has impressed some academics.Readers can check [1-3, 6, 7, 14-18, 22, 25-31, 33, 35, 37-41, 44, 52, 54-59].
The use of controllability notation in the research and design of control systems is beneficial.Fractional derivatives of various significations can be used to address these types of difficulties.It may be used in a range of sectors, including economics, chemical outgrowth control, biology, power systems, space technology, engineering, electronics, physics, robotics, transportation, chemistry, and so on.The topic of controllability is particularly important in control theory.If the control system is controllable, it can manage a variety of issues such as stability, pole assignment, and optimum control.Boundary controllability plays an important role in the analysis and design of control systems.The researchers in the recent years derived results on controllability for a variety of systems like neutral systems, integrodifferential equations, impulsive systems, fixed delay systems, and time-varying delay systems, etc. Solving these types of seeds has become a significant work for young scholars, one can refer to [1-18, 20, 21, 23, 27-29, 31-33, 38-53].
Fattorini proved the controllability condition on the first and second-order boundary control systems by replacing boundary controls with distributed controls [13].By assuming exact controllability of the linear system and approximate controllability of linearization, the authors [21] studied global controllability for the abstract semilinear system.[2] obtained results for approximate boundary controllability of stochastic control systems of fractional order with Poisson jump and fractional Brownian motion are cited by the authors.Zhou et al. [59] derived various conditions for the existence of mild solutions with the help of fixed point theorems and fractional power of operators for neutral fractional-order evolution equations having nonlocal conditions.Also, authors in [22] established results for the neutral integrodifferential fractional-order system having nonlocal conditions and finite delays in abstract space with the help of the measure of noncompactness.In [4] authors established some sufficient conditions for boundary controllability of integrodifferential system of Sobolev-type with the help of Banach contraction principle and theory of strongly continuous operators.With the help of Schauder's fixed point theorem, Ahmed [3] established sufficient conditions for boundary controllability of integrodifferential fractional-order non-linear system in abstract space.Inspired by the above and recent work, to the best of our knowledge there is no article dealing with boundary controllability for Sobolev-type neutral evolution equations of fractional order using this technique.We obtained sufficient conditions for boundary controllability.The results are advanced and weighed as an improvement to the control theory for fractional-order control systems.
The paper is structured in the following manner: In segment 2, we propose a few elementary definitions.In segment 3, we obtained results for boundary controllability.In segment 4, we discussed an example to understand theoretical results.

Preliminaries
Assume that Y and Z be two real Banach spaces with • and | • |.Assume that σ be a closed linear and densely defined operator with domain D(σ) ⊆ Y and R(σ) ⊆ Z.Consider Q be a linear operator with D(Q) ⊆ Y and R(Q) ⊆ X, a Banach space together • X .
Assume that the boundary control of neutral evolution equations of Sobolev-type with fractional order of the form (2.1) In the above, S : D(S) ⊂ Y → R(S) ⊂ Z is a linear operator, the control function u ∈ L 2 (J, U), a Banach space of admissible control function with U as a Banach space, B 1 : U → X is a linear continuous operator and ) is a set of all continuous function defined from J to J. G and F are the appropriate functions to be specified later and c D α , 0 < α < 1 is in the Caputo sense.Let y( ) = Sx( ) for x ∈ Y, then (2.1) and (2.2) can be written as where Q = QS −1 : Z → X is a linear operator.The operator A : Y → Z given by AS −1 w = σS −1 w for w ∈ D(AS −1 ), (see [4,23]).The hypothesis (H1)-(H3) and the closed graph theorem imply the boundedness of the linear operator AS −1 : Z → Z and AS −1 generates an analytic compact semigroup of uniformly bounded linear operators {T ( ) : ≥ 0}.This means that there exists a M ≥ 1 such that T ( ) ≤ M. Without loss of generality, we assume that 0 ∈ λ(AS −1 ).This allow us to define the fractional power (−A) q , for 0 < q < 1 as a closed linear operator on its domain D((−A) q ) with inverse (−A) −q .Theorem 2.2.[34] (1) Y q = D((−A) q ) is a Banach space with the norm x q = (−A) q x , x ∈ Y.
(3) For all > 0, (−A) q T ( ) is bounded on Y and there exists a positive constant C q such that (−A) q T ( ) ≤ C q q .
(4) If 0 < β < q ≤ 1, then D(−A) q → D(−A) β and the embedding is compact whenever the resolvent operator of A is compact.
Let us recall the following known definitions.
Definition 2.3.[35] The fractional integral of order α > 0 with the lower limit zero for a function f can be defined as provided the right-hand side is pointwise defined on [0, ∞), where Γ is the Gamma function.
Definition 2.4.[35] The Caputo derivative of order α with the lower limit zero for a function f can be written as: If f is an abstract function with the values in Y, then the integrals in the above definition are taken in Bochner's sense.
Lemma 2.5.A measurable function We now present the following results on the controllability.(A2) There exists a linear continuous operator S : . Moreover, there exists a positive function M 0 > 0 such that AS −1 T ( ) ≤ M 0 (see [3,18]).
Let y( ) be the solution of the systems (2.3) and (2.4).Then we define a function z( ) = y( ) − Bu( ).From the assumptions it follows that z( ) ∈ D(AS −1 ).Hence, the systems (2.3) and (2.4) can be written in terms of A and B as For more details, see [1,19].From the systems (2.5) and (2.6), we present the integral form of the systems (2.3) and (2.4) in the following way: (see [4,14,22,59]) and hence, the mild solution of the systems (2.1) and (2.2) is presented in the following way: x where ξ α (θ) is a probability density function defined on (0, ∞) and [4,14] The systems (2.1) and (2.2) are said to be controllable on the interval J if for every x 0 , x 1 ∈ Y, there exists a control u ∈ L 2 (J, U) such that the solution x(•) of the systems (2.1) and (2.2) satisfies x(b) = x 1 .
(iv) For any x ∈ Y, β ∈ (0, 1) and δ ∈ (0, 1), we have Remark 2.9.For any x ∈ Y, β ∈ (0, 1) and δ = 1, we have Further, assume the following assumptions: induces an invertible operator W defined on L 2 (J, U)/KerW, and there exists K 1 , K 2 and (A5) F : J × Y m+1 → Y is continuous and there exists β ∈ (0, 1) and M 1 , M 2 > 0 such that (−A) β F fulfills the subsequent condition: holds for ( , and there exists > 0 such that lim (A7) We assume Proof.For our convenience, we use the following Using the assumption (A5), for x(•), we define We now define P as follows: has a fixed point and this fixed point is then a solution of (2.1) and (2.2).So, we have to prove that P has a fixed point.For every k > 0, we set Then for every k, B k is clearly a bounded closed convex set in Y. From Lemma 2.8 and Eq (2.9) yields ) is integrable on J, by Lemma (2.5), P is well defined on B k .From (A6)(ii), one can get 0 We conclude that for k > 0 such that PB k ⊆ B k .If it fails, then there exist a function , and Px k ( ) > k, for some (k) ∈ J, where (k) denotes that is independent of k.Then, one can get Dividing by k on both sides of the above inequality and letting k → +∞, one can get . Therefore, The above equation contradicts Eq (2.11).Thus, for k > 0, PB k ⊆ B k .Now, we need to verify P has a fixed point on B k , which implies (2.1) and (2.2) have a mild solution.We decompose P as P = P 1 + P 2 , where P 1 and P 2 are determined on B k by for 0 ≤ ≤ b.We have to verify P 1 is a contraction mapping if P 2 is compact.For checking P 1 fulfills the contraction condition, we assume x 1 , x 2 ∈ B k .Then, for every ∈ J and by hypothesis (A5) and Eq (2.10), one can get Hence Thus, and by assumption 0 < L * < 1, we see that P 1 is a contraction.To prove that P 2 is compact, firstly we prove that P 2 is continuous on B k .Assume {x n } ⊆ B k with x n → x in B k , then for every ν ∈ J, w n (ν) → w(ν) and by (A6)(i), one can get G(ν, w n (ν)) → G(ν, w(ν)), when n → ∞.By the dominated convergence theorem, one can get when n → ∞, i.e., P 2 is continuous.Now, we need to verify {P 2 x : x ∈ B k } is an equicontinuous family of functions.For this, we assume > 0 be small, 0 < 1 < 2 , then Observe that We see that We check (P 2 x)( 2 ) − (P 2 x)( 1 ) tends to zero independently of x ∈ B k when 2 → 1 , with sufficiently small because of the compactness of S α ( ), for > 0 (see [34]) implies the continuity of S α ( ) for > 0 in in the uniform operator topology.We can verify that P 2 x, x ∈ B k is continuous at = 0. Therefore, P 2 maps B k into a family of equicontinuous functions.We need to verify that V( ) = {(P 2 x)( ) : x ∈ B k } is relatively compact in Y. Assume that 0 < ≤ b be fixed, 0 < < , for arbitrary δ > 0, for x ∈ B k , we determine every , 0 < < and for all δ > 0. Additionally, for each x ∈ B k , one can get Hence, there are relative compact sets arbitrary close to V( ), > 0. Therefore, V( ), > 0 is also relatively compact in Y. Consequently, with the help of Arzela-Ascoli theorem it can be say that P 2 is compact.The above evidence demonstrates that P = P 1 + P 2 is a condensing mapping on B k , and by the Theorem 2.10, x(•) exists for P on B k and the systems (2.1) and (2.2) have a mild solution.
Remark 3.2.Many authors have recently investigated the boundary controllability of fractional evolution differential systems utilizing fractional theories, mild solutions, Caputo fractional derivatives, and fixed-point techniques.Very particularly, in [1][2][3][4][5], the authors discussed the existence and boundary controllability outcomes for integer and fractional order systems with and without delay by referring to multivalued functions, various fixed point theorems, fractional calculus, and nonlocal conditions.One can extend our current study to the integro-differential systems, Volterra-Fredholm integro-differential systems with integer and fractional order settings by using well-known fixed point theorems.

Example
Let us assume that Ω be a bounded, open subset of R n .Consider Γ be a sufficiently smooth boundary of Ω. Assume that the following fractional differential system: In the above, u ∈ L 2 (Σ), χ 0 ∈ L 2 (∆), F, G ∈ L 2 (Q) and c ∂ α is a Caputo fractional partial derivative of order 0 < α < 1.We can formulate the above problem as the boundary control problem (2.1) and (2.We conclude now F, G fulfill the hypotheses (A5) and (A6).Additionally, W −1 also exists.Assume b and the remaining constants fulfill the hypotheses (A5)-(A7).Therefore, all the requirements of the Theorem (3.1) are fulfilled and (2.1) and (2.2) are controllable.

Conclusions
In this article, we mainly focused on the boundary controllability of fractional order Sobolev-type neutral evolution equations in Banach space.We show our key results using facts from fractional calculus, semigroup theory, and the fixed point method.Finally, we give an example to illustrate the theory we have established.In the future, we will focus on the boundary controllability of Hilfer fractional-order neutral evolution equations and integrodifferential equations in Banach space by using the fixed point theorem approach.

(
A1) D(σ) ⊂ D(Q) and the restriction of Q to D(σ) is continuous relative to graph norm of D(σ).
.11) Theorem 2.10.[36, Sadovskii fixed point theorem] Let φ be a condensing operator on a Banach space Y, that is φ is continuous and takes bounded sets into bounded sets, and µ(φ(B)) ≤ µ(B) for every bounded set B of Y with µ(B) > 0. If φ(γ) ⊂ γ for a convex, closed and bounded set γ of Y, then φ has a fixed point in Y. (Here µ(•) denotes Kuratowski's measure of non compactness).