Controllability of fractional order non-autonomous differential equation with a delay in control

Abstract

This paper deals with a non-autonomous differential equation of fractional order with delay in control. Our aim is to discuss sufficient conditions for controllability and optimal controllability of the differential equation with delay. We used the semigroup theory of bounded linear operators, the two parameter evolution operator, M\"{o}nch's fixed point theorem, and measure of non-compactness to examine these results. An example is included to demonstrate the effectuality of the obtained result.