Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay

: The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii’s ﬁxed point theorem, we get sufﬁcient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then deﬁned. Next, we studied whether the deﬁned problem is wellposed. Finally, we apply our results to examine the controllability of the second order state dependent delay impulsive equation. Abstract differential equations and partial differential equations of the ﬁrst order were investigated for the state-dependent delay condition in the papers provided by Hernandez, and et al. [17] and the research articles of [18–21]. The existence of solutions to the impulsive differential equation of second order with state-dependent delay has recently been investigated, refered to [22,23]. Also, the state dependent delay term for the parabolic and other ODE systems contributed by Krisztin et al. [24] and the cited papers [25–27] have been studied for the current research.


Introduction
The investigation of this paper guarantees that the defined problem is controllable on any interval. Such problems come from some physical applications as a natural generalization of the classical initial value problems. Consider the Cauchy problem with state-dependent delay and control: ∆y(t k ) = I k (x t k ), k = 1, 2, . . . , n, (3) ∆y (t k ) = J k (x t k ), k = 1, 2, . . . , n.
conditions. Next, we investigated the existence and uniqueness of the controllability of neutral impulsive differential equations with nonlocal conditions in prior work. The controllability of a nonlocal neutral impulsive differential equation under compactness conditions, Lipschitz conditions, and mixed-type conditions is obtained using the property of measure of noncompactness and Darbo-Sadovskii's fixed point theorem, read the cited papers [38,39]. We now extend our research to investigate the controllability and wellposedness of an impulsive functional abstract second order differential equation with state dependent delay. The study of controllability and wellposedness of impulsive functional abstract second-order differential equations with state dependent delay is a nearly unexplored area in the literature, which is the motivating factor behind this paper.
By using the ideas of cited papers [25,40], we investigate that problem (1)-(4) is controllable and also show the wellposedness of the problem (1)- (4). We use this fact in Theorem 1 that shows the controllability condition in C 1 Lip ([−γ, b]; X) exists. The wellposedness of the problem (1)-(4) is proved in Proposition 1. Consider the condition σ j (0, ϕ) = 0, j = 1, 2 from the above results. The term F (t, y σ 1 (t,y t ) , y σ 2 (t,y t ) ) act as class of equations and it is represented as G (t, y(ρ 1 (t, y t )), y (ρ 2 (t, y t ))). The instance ρ(0, ϕ) < 0 is studied for the above cases. Finally, we illustrate few applications on impulsive differential equations of second order with delay condition.

Basic Preliminaries
The cosine family (C(t)) t∈R is defined in X and the norm y A = x + A x is described from the domain D of A . The sine function (S(t)) t∈R is associated to (C(t)) t∈R defined by S(t)x := t 0 C(s)xds, for x ∈ X and t ∈ R. Furthermore, C 0 is a non-negative constant and it's norm described as Here, we describe the phase space B axiomatically, using notions and annotations prepared in [41] and suitably modified to the impulsive differential equations. More particularly, B : (−∞, 0] → X denotes the vector space of functions provided with a norm denoted as · B and thus, the following axioms hold: (I) If y : (−∞, µ + a] → X, a > 0, is a function such that y µ ∈ B and x| [µ,µ+a] ∈ C([µ, µ + a], X), then, for every t ∈ [µ, µ + a), the following properties hold: , K 1 is continuous, K 2 is locally bounded and H, K 1 , K 2 are independent of y(·).
(II) The space B is complete.
The space E = {x ∈ X : C(·)x is continuosly differentiable} awarded with the norm From the literature of Kisiński [42], since E is a Banach space, A S(t) ∈ L (E , X), for every t in the real line R and if s converges to 0 then A S(s)x converges to 0, for every x ∈ E . Now, We define the abstract differential equation of second order Here, η is an integrable function defined from [0, b] to X and x 1 , x 2 ∈ X. The mild solution of (5) and (6) is described as and for x 1 ∈ E , the C 1 function u(·) defined on [0, b], We study the further feature on abstract Cauchy problem of second order and Cosine functions from the reference of the papers [43,44] . We use the state-dependent delay from [25]. The function y (·) : [0, a] → B noted as y (·) is defined by y (·) (s) = y s , for y ∈ C([−γ, a]; X) and 0 < a ≤ b. This consequence and the Lemma 1 are used for our main results of the article.

Main Results
First, the existence of controllability of the problem (1)-(4) is studied. We define the mild solution and controllable of the problem (1)-(4). Then we prove the wellposedness of problem (1)-(4). In view of simplification, suppose that σ 1 (·) = σ 2 (·) and we termed that · B for · C([−γ,0];X) . In our argument, we take σ 1 = σ 2 as in the remaining of this work. We present following definitions and assumptions for further study.
Following conditions are useful to prove the main results.

Remark 1.
Also, we define the following constants for our convenience Now, we can obtain the first main result.
This shows the contraction map G, which implies the map G is a condensing operator on the interval [0, a] such that y ∈ C Lip ([−γ, a]; X). From Lemma 2, G has a fixed point in X. Thus, any fixed point of G is a mild solution of (1)-(4). Hence the system (1)-(4) is controllable on [−γ, a].

Wellposedness
The metric space (M, · M ) is continuous and is defined on C([−γ, 0]; X). Now, suppose that ω (0) exists for every ω ∈ M. Definition 3. If for every pair of open and bounded convex subset S of M, there is any a S > 0, in order to allow all ω ∈ S and there is exactly one mild solution y ω ∈ C 1 ([−γ, a S ]; X) of (1) with the conditions y 0 = ω, y (0 + ) = ω (0) and y ω − y φ C 1 ([0,a S ];X) → 0 as ω − φ M → 0, then the problem (1) is called wellposed on the neighborhood of M.
The remaining results, M B denotes that Further, we show the consequence of wellposedness and it is proved from Theorem 1. Now, we introduce the following hypothesis to demonstrate our next result of condition (H σ,ϕ ).

Examples
Let X = L 2 ([0, π]) and define D(A ) = {x ∈ H 2 (0, π) : x(0) = x(π) = 0}. The infinitesimal generator A : D(A ) → X of a strongly continuous cosine function (C(t)) t∈R on X given by A x = x and A x = − ∑ ∞ n=1 n 2 x, y n y n , for x ∈ D(A ). Furthermore, A has a distinct eigenvalues of the form −n 2 , n ∈ N, and its associated eigen functions defined as y n (ζ) := 2 π sin(nζ). Additionally, it holds the characteristics defined below: (i) The sequence {y n : n ∈ N} creates an orthonormal basis of X.
(ii) The cosine family C(t)x = ∑ ∞ n=1 cos(nt) x, y n y n and sine family S(t)x = ∑ ∞ n=1 sin(nt) n x, y n y n , for x ∈ D(A ). Now, analyze the following problem for all t ∈ [0, b] and ς ∈ [0, π].
We have to show that the existence of a solution of the problem (33)-(37) by using any spcified starting phase to end phase in a Banach space X.
Clearly, we show that F ,σ are Lipschitz. We get the next consequence from Theorem 1.

Conclusions
The conditions for controllability and wellposedness of an abstract second-order differential system with state-dependent delay are investigated in this paper. The use of the work is shown by demonstrating its application to dynamical systems. We came to the conclusion that the existence of controllability and wellposedness achieves the goal of a stated problem and its remedies. It will be more fascinating to study the equivalent system's trajectory controllability and numerical estimation. We plan to extend it to a fractional order system with state dependent indefinite delay and integral impulsive conditions in the future.