Existence and optimal control results for Caputo fractional delay Clark's subdifferential inclusions of order r∈(1,2) with sectorial operators

In this study, we investigate the effect of Clarke's subdifferential type on the optimal control results for fractional differential systems of order (Formula presented.) with delay. The main findings of this study are tested by using multivalued functions, sectorial operators, fractional derivatives, and the fixed point theorem. To begin, the existence of mild solutions is established and verified primarily by using a very well multivalued fixed point theorem and the characteristics of generalized Clarke subdifferential problems. Furthermore, we get a finding on the existence of optimal control for the presented control system under particular reasonable conditions. After that, we will move on to the time optimal control results for the given system. Finally, an example for drawing the theory behind the main conclusions is shown.