  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.
