In this article, we study the stochastic fractional optimal control problem for a system governed by a class of non-instantaneous impulsive stochastic fractional differential equation in the infinite-dimensional spaces. We utilize the fractional calculus, stochastic analysis theory and a suitable fixed point approach to present the solvability of the stochastic fractional system. Then, the existence of optimal state-control pairs of the Lagrange problem is derived without uniqueness of solutions of the stochastic system. Finally, the main results are validated with the aid of an example.

This paper is devoted to the study of a new class of non-instantaneous impulsive stochastic differential equations driven by mixed fractional Brownian motion in separable Hilbert spaces. Based on the stochastic analysis theory, analytic semigroup theory of linear operators, fractional powers of operators, and a fixed point technique, a new set of sufficient conditions are derived to ensure the existence and uniqueness of mild solutions for the proposed stochastic system. Moreover, we also investigate the asymptotic behaviour of mild solutions and controllability results for the proposed stochastic system. Finally, an example is given to demonstrate the applicability of our main results.

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter $\widehat{\mathcal{H}}\in (1/2,1)$. Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.

The goal of this paper is to consider a new class of $\phi$-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the $\phi$-Hilfer fractional control system. Our main results are well supported by an illustrative example.