New discussion on optimal feedback control for Caputo fractional neutral evolution systems governed by hemivariational inequalities

The main focus of this article is to investigate the existence of feedback optimal control for the neutral fractional evolution systems in Hilbert spaces in the sense of the Caputo fractional derivatives. In order to establish the necessary conditions for the proposed problem, we apply the semigroup property, the fixed point theorem of multivalued maps, and the properties of generalized Clarke's subdifferentials. Then, by using the Filippov theorem and the Cesari property, a set of sufficient conditions is formulated to ensure the existence of a feasible pair for the feedback control systems. Finally, we apply our main results to obtain the optimal feedback control pair. In the end, an example is given to illustrate our theory.