This paper investigates the optimal controllability of stochastic multi-term Hilfer fractional differential systems with finite delays in the control function. The considered model combines multi-order fractional dynamics, stochastic perturbations, and delayed control actions, which provide a unified framework to describe complex dynamical processes with memory and uncertainty. By employing the theory of fractional calculus and the concept of a $(\delta ,{\beta }_{\kappa })$-resolvent family, we establish the existence and uniqueness of mild solutions for the proposed system. Balder's theorem and variational techniques demonstrate the existence of an optimal state-control pair minimising a quadratic cost functional. A diffusion-type fractional stochastic model with delayed control is presented to illustrate the applicability of the theoretical findings. The obtained results not only extend several existing works on fractional stochastic control systems, but also provide new insights into the analysis of Hilfer-type operators with multiple fractional orders and control delays, which are relevant in viscoelastic materials, heat transfer processes, and neural network models with memory effects.

This paper studies a semilinear fractional stochastic differential equation with multiple constant point delays in control. We transform the controllability problem into a fixed point problem. We obtain sufficient condition for the controllability by using Schauder's fixed point theorem. In addition, we discuss the optimal controllability of the problem. Some examples are given to illustrate the main result.

In this paper, we study an abstract system of fractional delay differential equations of order $1<q<2$ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.