  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.
