Sobolev type time fractional differential equations and optimal controls with the order in $(1,2)$

Abstract

This paper is mainly concerned with controlled time fractional differential equations of Sobolev type in Caputo and Riemann-Liouville fractional derivatives with the order in $(1,2)$ respectively. By properties on some corresponding fractional resolvent operators family, we first establish sufficient conditions for the existence of mild solutions to these controlled time fractional differential equations of Sobolev type. Then, we present the existence of optimal controls of systems governed by corresponding time fractional differential equations of Sobolev type via setting up approximating minimizing sequences of suitable functions twice.