Approximate controllability of non-instantaneous impulsive fractional evolution equations of order $1

Abstract

The current article examines the approximate controllability problem for noninstantaneous impulsive fractional evolution equations of order 1 < α < 2 with statedependent delay in separable reflexive Banach spaces. In order to establish sufficient conditions for the approximate controllability of our problem, we first formulate the linearregulator problem and obtain the optimal control in feedback form. By using this optimal control, we deduce the approximate controllability of the linear fractional control system of order 1 < α < 2. Further, we derive sufficient conditions for the approximate controllability of the nonlinear problem. Finally, we provide a concrete example to validate the efficiency of the derived results .