Exact Controllability Results for Sobolev-Type Hilfer Fractional Neutral Delay Volterra-Fredholm Integro-Differential Systems

: This manuscript mainly focuses on the exact controllability of Sobolev-type Hilfer fractional neutral delay Volterra-Fredholm integro-differential systems. The principal ﬁndings of this discussion are established by using the theories on fractional calculus, the measure of noncompactness and Mönch ﬁxed point technique. Initially, the exact controllability of the system is presented and then we improve the discussion to the system with nonlocal conditions. Finally, abstract and ﬁlter systems are provided for the illustration.


Introduction
In many physical processes, fractional differential equations incorporating not only one fractional derivative but also several fractional derivatives are heavily concentrated. The meaning of fractional systems has recently attracted a lot of attention due to its astonishing applications in showcasing the wonders of science and engineering. The use of fractional order differential equations allows for the management of a wide range of issues in a variety of fields, including fluid flow, electrical systems, visco-elasticity, electro-chemistry, and so on. The monographs [1][2][3][4][5][6][7][8][9] and the research articles [8][9][10][11][12][13] show the interlinking in the same way that the separation between classical and fractional differential representations seems to. Applications of the differential systems can be found in [14][15][16]. Neutral structures with delays or without delays, in particular, serve as a summary association of a large number of partial neutral structures that appear in problems involving heat flow in substances, visco-elasticity, and a variety of natural processes. Neutral systems appear in many areas of applied mathematics; as a result, the most successful neutral structures have gotten a lot of attention in the current generation; readers can look at [12,13,[17][18][19][20][21].
Recently, in [22,23], the author initiated another kind of derivative of fractional order, that including Riemann-Liouville and Caputo fractional derivative. In [24], the authors proved the existence of mild solution for evolution equation with Hilfer fractional derivative which generalized the famous Riemann-Liouville fractional derivative by using the semigroup theory, measure of noncompactness and fixed point approach. In [25], the authors proved the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems by using fractional calculus and Bohenblust-Karlin's theorem.
In [18,26,27], the authors proved the existence and controallbility of various extensions related with Hilfer fractional derivative by using semigroup theory, measure of noncompactness and various fixed point theorems.
In [28], the authors discussed the approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral delay differential system by using Bohenblust-Karlin's fixed point theorem. In [29], the authors proved the existence of nonlocal functional integro-differential equations via Hilfer fractional derivative by using Mönch fixed point theorem. In [13], the authors discusssed the existence of Sobolev-type Hilfer fractional neutral integro-differential equations with infinite delay by uisng Mönch fixed point theorem. In [30], the authors proved the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions by using Bohenblust-Karlin's fixed point theorem. The existence and exact controllability described in our paper have still to be investigated, and it is the motivation of this article.
Assume that the Hilfer fractional neutral delay Volterra-Fredholm integro-differential system of Sobolev-type has the following form where D ν,ζ 0 + stands for the Hilfer fractional derivative, 0 ≤ ν ≤ 1; 1 2 < ζ < 1; and x(·) takes value in Banach Space X with · . The histories u : (−∞, 0] → R l , u (s) = u( + s), s ≤ 0 with phase space R l . F 1 : I × R l → X, F 2 : I × R l × X × X → X, e : I × I × R l → X and f : I × I × R l → X are appropriate functions. u(·) is a control function and B is a bounded linear operator from V → X.
The rest of the paper is organized as follows: 1. Section 2: Theoretical notions linked to fractional calculus and the measure of noncompactness are presented.
Section 4: Our findings are expanded to include the concept of nonlocal situations. 4.
Section 5: Finally, abstract and filter systems are provided for the illustration of the obtained theory.

Preliminaries
We now provide some fundamental theories, lemmas, and facts to discuss our main results. C(I, X)the spaces of all continuous functions. Let us take [46].
(iv) A α T(t) is bounded on X and there exists M α > 0 such that Definition 2 ([47]). The operators define A : D(A) ⊂ X → X and J : D(A) ⊂ X → X satisfy the following: (J 1 ) A and J are closed linear operators.
Additionally, because of (J 1 ) and (J 2 ) J −1 is closed, by (J 3 ) and by referring closed graph theorem, we obtain the boundedness of AJ −1 : X → X. Define J −1 = J m and J = J m .

Definition 3 ([4]
). The left sided Riemann-Liouville fractional integral of order ζ having lower limit c for F : [c, +∞) → R is presented as if the right side is pointwise determined on [c, +∞), where Γ(·) denotes gamma function.

Lemma 4 ([20]
). The operators P ν,ζ and S ζ satisfies the following: (i) For t ≥ 0, P ν,ζ and S ζ are linear and bounded, that is, for every u ∈ X, Definition 7 ([49,50]). The Measure of noncompactness of Hausdorff T (·) determined on every bounded subset of X by T ( ) = inf{ > 0 : can be covered by a finite number of balls of radii lesser than }.

Definition 8 ([35]
). Let F + be the positive cone of an order Banach space (F, ≤). The value E of F + is said to be measure of noncompactness on X of D determined on the set of all bounded subsets of X if and only if E(co ) = E( ) for all bounded subsets ⊆ X, where co is a closed convex hull of .
(i) Monotone if and only if for all bounded subsets , 1 , 2 of X we get: ). Assume F be closed convex subset of X and 0 ∈ F, K : F → X is continuous and that satisfies Mönch's condition, that is, (P ⊆ F is countable, P ⊆ co({0} ∪ K(P)) ⇒ P is compact ). Then K has a fixed point in F.

Existence
The reason for this part is to examine the existence of the fractional system (1)- (2).
(ii) There exists q 1 ∈ (0, q) and 1 ∈ L 1 q 1 (I, R + ) and the integrable function κ : (iii) There exists q 2 ∈ (0, q) and 2 ∈ L 1 q 2 (I, R + ) such that for any bounded subset D 1 ⊂ X and G 1 ⊂ R l , (H 3 ) The function e : I × I × R l → X satisfies the following: which satisfies the following: (ii) For q 5 ∈ (0, q) and for every bounded subset D ∈ X, there exists 4 We present the following for our convenience: Theorem 2. Suppose that the hypotheses (H 0 )-(H 5 ) are satisfied, then the fractional system (1) and (2) is controllable if Proof. We now define the operator Φ : R l → R l by For φ ∈ R l , we presentβ as follows: (3), if and only if p satisfies p 0 = 0 and Hence (R l , · c ) is a Banach space. For r > 0, fix F r = {p ∈ R l : w c ≤ r}, thus F r ⊆ R l is uniformly bounded, and for p ∈ F r , by referring Lemma 1, we have IntroduceΥ : R l → R l by Clearly, Υ having a fixed point and which is similar to Υ. To prove Υ having a fixed point, we subdivide the whole proof as follows: Step 1: We state that there exists r > 0 such that Υ(F r ) ⊆ F r .
If it is not correct, then there exists p r (·) ∈ F r and t ∈ I such that Υ(p r )(t) > r, i.e., Υ(p r ) / ∈ F r . Fix r > 0, and assume {F r = u ∈ C : u c ≤ r}. Clearly, F r is a closed, bounded and convex set of C. Now, we need to check there exists r > 0 such that φ(F r ) ⊆ F r . If it fails, then p r ∈ F r . However, φ(u r ) / ∈ F r . Hence, Using assumptions (H 2 ) and (H 3 ) and Lemma 4, we have By referring the Hölder's inequality and Lemma 4, we have By referring the Hypotheses (H 1 )-(H 5 ), we get Combining all the above results V 1 -V 5 , we get If we divide (6) by r, and assuming r → ∞, we have 1 ≤ 0, and this contradicts with(4), then, φ(F r ) ⊆ F r .
Step 2: Υ is continuous on F r .
The right-hand side of T 1 to T 12 tends to '0' as δ → 0. On implementing the absolute continuity of the Lebesgue integral dominance convergence theorem for inequality, we conclude that T 1 to T 12 gives '0' when t 2 − t 1 → 0. Therefore, Υ(F r ) is equicontinuous on I.
Step 4: Now, we need to prove that the Mönch's condition holds.
Hence by using Mönch's condition, we have this implies T (H) = 0. Therefore, Υ has a fixed point y ∈ F r , in view of Lemma 6. Next, u = p +β is a mild solution of the fractional system (1)-(2) satisfying u(c) = u 1 , then the fractional system (1)-(2) is controllable on X.

Nonlocal Conditions
Physical problems prompted the development of evolution equations with nonlocal conditions. In [54,55], the authors explored nonlocal issues for the first time in 1990, obtaining the existence and uniqueness of mild solutions for nonlocal differential equations of integer order. For more details on the systems with integer of fractional orders, one can refer [29,30,32,35,45,54,55]. Assume that nonlocal Hilfer fractional delay Volterra-Fredholm integro-differential system has the following form where 0 < t 1 < t 2 < t 3 < · · · < t n ≤ d, g : R n l → R l and satisfies the following: (H 6 ) g : R n → R is continuous, there exists K i (g) > 0 such that for all u i , w i ∈ R l and consider K g = sup{ g(u 1 , u 2 , · · · , u n ) : u i ∈ R l }. Definition 10. A function u : (−∞, c] → X is said to be a mild solution of (9) and (10) if u 0 = φ + g(u t 1 , u t 2 , u t 3 , · · · , u t n )(0)) ∈ R l on (−∞, 0] and Theorem 3. Assume that the hypotheses (H 0 )-(H 6 ) are satisfied, then the fractional system (9) and (10) is controllable if , for some 1 2 < ζ < 1.

Filter System
An advanced filter is a framework that performs mathematical operations on an inspected, digitized sign to decrease or upgrade certain highlights of the prepared signal. Propelled by the plans examined in [13,19,29,56,57], we presented a filter design for our framework which is shown in Figure 1. Figure 1 portrays the rough pattern of block diagram which helps to improve the viability of arrangement with least measure of sources of input and which is presented as follows. Product modulator (PM)-1 receives inputs u t and F 1 generates the output as F 1 (t, u t ). PM-2 receives A and F 1 (t, u t ) generates AF 1 (t, u t ). PM-4 receives u τ and f generates f (t, τ, u τ ). PM-4 receives u t and F 2 produces F 2 (t, u t ). PM-5 receives u τ and e generates e(t, τ, u τ ). PM-6 receives F 2 (t, u t ), e(t, τ, u τ ) and f (t, τ, u τ ) generates PM-7 receives x(t) and B generates Bx(t). PM-8 receives [φ(0) − F 1 (0, φ(0))] and P ν,ζ (t) at time t = 0, generates P ν,ζ (t). The integrators execute the integral of S ζ (t) AS ζ (t)F 1 (t, u t ) + F 2 t, u t , t 0 e(t, τ, u τ )dτ, c 0 f (t, τ, u τ )dτ + Bx(t) , over t.
Additionally, Inputs S ζ (t), AF 1 (t, u t ) are joined and multiplying with the output on the interval (0, t). S ζ (t), F 2 t, u t , t 0 e(t, τ, u τ )dτ, c 0 f (t, τ, u τ )dτ are joined and multiplying with the output on the interval (0, t). S ζ (t), Bx(t) are joined and multiplying with the output on the interval (0, t).
Finally, if we shift all the outputs from the integrators to summer network, then, the output of u(t) is achieved, which is bounded and controllable.

Conclusions
The exact controllability of Sobolev-type Hilfer fractional neutral integro-differential systems via measure of noncompactness is the topic of our article. The main conclusions of our paper are based on theoretical ideas such as fractional calculus, the measure of noncompactness, and the fixed-point approach. First, we looked at the exact controllability of mild solutions for fractional evolution systems. Then we expanded on our findings to consider the system in nonlocal conditions. Finally, we presented theoretical and practical applications to aid in the efficacy of the discussion.