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The current study addresses the control problems posed by a semilinear neutral integro-differential equation with memory. The primary objectives of this study are to investigate the existence of a mild solution and approximate controllability of both linear and semilinear control systems in Banach spaces. To accomplish this, we begin by introducing the concept of a resolvent family associated with the homogeneous neutral integro-differential equation without memory. In the process, we establish some important properties of the resolvent family. Subsequently, we develop approximate controllability results for a linear control problem by constructing a linear-quadratic regulator problem. This involves establishing the existence of an optimal pair and determining the expression of the optimal control that produces the approximate controllability of the linear system. Furthermore, we deduce sufficient conditions for the existence of a mild solution and approximate controllability of a semilinear system in a reflexive Banach space with a uniformly convex dual. Additionally, we delve into the discussion of approximate controllability for a semilinear problem in general Banach space, assuming a Lipschitz type condition on the nonlinear term. Finally, we implement our findings to examine the approximate controllability of certain partial differential equations, demonstrating their practical relevance.

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder's fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we verify our results to examine the approximate controllability of the non-autonomous wave equation with non-instantaneous impulses and finite delay in the application section.

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects and lack of compactness of the operator $h(\cdot)\mapsto \int_{0}^{\cdot}\mathrm{U}(\cdot,s)h(s)\mathrm{d} s : \mathrm{L}^{1}([0,T];\mathbb{Y}) \rightarrow \mathrm{C}([0,T];\mathbb{Y}),$ where $\mathbb{Y}$ is a Banach space and $\mathrm{U}(\cdot,\cdot)$ is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.