Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

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, show existence mild solution non-autonomous evolution inclusions separable reflexive spaces help family and generalization Leray-Schauder fixed point theorem multi-valued maps. order to establish sufficient conditions our problem, first consider linear-quadratic regulator problem obtain optimal feedback form, which contains resolvent operator consisting duality mapping. With control, prove linear system hence derive problem. Moreover, rectify several shortcomings related works available literature, namely, proper identification spaces, characterization phase space presence effects, lack compactness h(?)??0?U(?,s)h(s)ds:L1([0,T];Y)?C([0,T];Y), where Y is U(?,?) family, etc. Finally, provide concrete example illustrate efficiency results.