Investigation on the Approximate Controllability of Neutral Hilfer Fractional Differential Inclusion With Infinite Delay and Noninstantaneous Impulses

ABSTRACT The main aim of this paper is to establish a set of sufficient conditions for the existence of integral form mild solution and approximate controllability of a nonlinear multivalued neutral Hilfer fractional differential system in a Hilbert space. In the proposed control problem, we deal with a system that is under the noninstantaneous impulsive effects and infinite delay. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Dhage's fixed‐point theorem. Further, the approximate controllability results are established using piecewise defined control function and under the consideration that the corresponding linear system is approximate controllable. The main findings have been established by using the facts concerning the semigroup theory of operators, fixed‐point technique, fractional calculus, and the theory of multivalued maps. Finally, an example is given to illustrate the abstract results.