Optimal controllability of stochastic multi-term Hilfer fractional systems with delayed controls

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.