Approximate controllability and optimal control of impulsive fractional semilinear delay differential equations with non-local conditions

In this paper we study the approximate controllability and existence of optimal control of impulsive fractional semilinear delay differential equations with non-local conditions. We use Sadovskii's fixed point theorem, semigroup theory of linear operators and direct method for minimizing a functional to establish our results. At the end we give an example to illustrate our analytical findings.