Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses.The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order systemwith respect to the approximate controllability of the corresponding linear system in aHilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.


Introduction
Neutral differential equations appear as mathematical models in electrical networks involving lossless transmission, mechanics, electrical engineering, medicine, biology, ecology, and so forth.Neutral differential equations are functional differential equations in which the highest order derivative of the unknown function appears both with and without derivatives.Second order neutral differential equations model variational problems in calculus of variation and appear in the study of vibrating masses are attached to an electric bar.
Impulsive differential equations are known for their utility in simulating processes and phenomena subject to short term perturbations during their evolution.Discrete perturbations are negligible to the total duration of the process which have been studied in [1][2][3][4][5][6].
However noninstantaneous impulses are recently studied by Ahmad [7].Stimulated by their numerous applications in mechanics, electrical engineering, medicine, ecology, and so forth, noninstantaneous impulsive differential equations are recently investigated.
Recently, much attention is paid to partial functional differential equation with state dependent delay.For details see [7][8][9][10][11][12].As a matter of fact, in these papers their authors assume severe conditions on the operator family generated by , which imply that the underlying space  has finite dimension.Thus the equations treated in these works are really ordinary and not partial equations.The literature related to state dependent delay mostly deals with functional differential equations in which the state belongs to a finite dimensional space.As a consequence, the study of partial functional differential equations with state dependent delay is neglected.This is one of the motivations of our paper.
The papers [13,14] study existence of differential equation via measure of noncompactness.Measure of noncompactness significantly removes the need to assume Lipschitz continuity of nonlinear functions and operators.
In recent years, controllability of infinite dimensional systems has been extensively studied for various applications.In 2 International Journal of Partial Differential Equations the papers [15,16] the authors discuss the exact controllability results by assuming that the semigroup associated with the linear part is compact.However, if the operator  is compact or  0 -semigroup () is compact then the controllability operator is also compact.Hence the inverse of it does not exist if the state space  is infinite dimensional [17].
Another available method in the literature involves the invertibility of operator (+Γ  0 ), where Γ  0 is the controllability Gramian and a limit condition which is difficult to check and apply in practical real world problems.See for details [18].Also it is practically difficult to verify their condition directly.This is one of the motivations of our paper.
However our work is a continuation of coauthor Sukavanam's novel approach in article [19].We extend our work [20][21][22] in this paper.
The organization of the paper is as follows.In Section 3 we study the existence and uniqueness of mild solution of the second order equation modelled in the form where  is the infinitesimal generator of a strongly continuous cosine family {() :  ∈ R} of bounded linear operators on a Banach space .The history valued function   : (−∞, 0] → ,   () = ( + ) belongs to some abstract phase space B defined axiomatically; , ,  1  ,  2  ,  = 1, . . .,  are appropriate functions.0 =  0 =  0 <  1 ≤  1 ≤  2 , < ⋅ ⋅ ⋅ , <   ≤   ≤  +1 =  are prefixed numbers.In Section 5 we study the approximate controllability of where  is the infinitesimal generator of a strongly continuous cosine family {() :  ∈ R} of bounded linear operators on a Hilbert space .The history valued function   : (−∞, 0] → ,   () = ( + ) belongs to some abstract phase space B defined axiomatically; ,  are appropriate functions. is a bounded linear operator on a Hilbert space .

Preliminaries
In this section some definitions, notations, and lemmas that are used throughout this paper are stated.The family {() :  ∈ R} of operators in () is a strongly continuous cosine family if the following are satisfied: For more details see book by Fattorini [28] and articles [35][36][37].In this work we use the axiomatic definition of phase space B, introduced by Hale and Kato [30].
Definition 1 (see [30]).Let B be a linear space of functions mapping (−∞, 0] into  endowed with seminorm ‖ ⋅ ‖ B and satisfy the following conditions: (B) The space B is complete.

Existence and Uniqueness of Mild Solution
We define mild solution of problem (1) as follows.
To prove our result we always assume  :  × B → (−∞, ] is a continuous function.The following hypotheses are used.is Lipschitz continuous such that there exists positive constant   such that International Journal of Partial Differential Equations
Step 2. To prove that Γ is a -contraction.

Theorem 16. If the assumptions (Hg) and (HR) hold then the corresponding neutral system
with  ≡ 0 is approximately controllable.
By using Lemma 15 we denote   the map associated to this decomposition and construct  2 = N  and  1 = ().Also set   = ‖  ‖.

Examples
Example 1.In this section we discuss a partial differential equation applying the abstract results of this paper.In this application, B is the phase space  0 ×  2 (ℎ, ) (see [10]).
Hence by assumptions (a)-(c) and Theorem 10 it is ensured that mild solution to the problem (51) exists.
14.The system (24) is said to be approximately controllable on [0, ] if R  () is dense in .The corresponding linear system is approximately controllable if R(0) is dense in .Lemma 15.