Relatively exact controllability for fractional stochastic delay differential equations of order κ ∈ (1,2]

In this paper, our main purpose is to study a class of fractional stochastic delay differential equations (FSDDEs) of order ������ & ISIN; (1, 2]. Firstly, we present a concept of delay Grammian matrix involving delayed matrix functions of sine. Subsequently, the relatively exact controllability of linear FSDDEs is obtained by using Grammian matrix. Furthermore, based on Krasnoselskii's fixed point theorem, we explore the relatively exact controllability of the nonlinear addressed equations. In addition, with the aid of delay Gronwall inequality, Jensen inequality and Ito isometry, existence of optimal control for the Lagrange problem is derived. Finally, the theoretical conclusions are supported through two examples.