Optimal Controls for Fractional Stochastic Functional Differential Equations of Order $$\alpha \in (1, 2]$$ α ∈ ( 1 , 2

Abstract

In this paper, we study optimal control problems for a class of fractional stochastic functional differential equations of order \(\alpha \in (1, 2]\) in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence of mild solutions are investigated by using the fractional calculus, stochastic analysis theory, and fixed point theorems with the strongly continuous \(\alpha \)-order cosine family. Then, the existence conditions of optimal pairs of systems governed by a fractional stochastic partial differential equations of order \(\alpha \in (1, 2]\) with infinite delay are presented. The results are obtained under the mixed Lipschitz and Caratheodory conditions. Finally, an example is presented to illustrate the main results.