ABSTRACT In this paper, we address the feedback control problem for fractional differential equations of order , driven by fractional Brownian motion (fBm) with Hurst parameter in Hilbert spaces. We initially establish the existence of mild solutions through the application of advanced analytical techniques involving fractional calculus, cosine operator theory, stochastic analysis, and Schauder's fixed‐point theorem. Under suitable assumptions, we further demonstrate the existence of feasible state‐control pairs, which serve as a foundation for constructing optimal feedback control pairs. To validate the theoretical developments, a comprehensive illustrative example is provided, confirming the applicability of the proposed framework.
Dhanush et al. (2026) studied this question.