Trajectory Controllability and Ulam–Hyer's Stability for Higher‐Order Coupled Hilfer Fractional Neutral Stochastic Integro‐Differential Equations via Measure of Non‐Compactness

Abstract

ABSTRACT The present manuscript investigates the controllability and stability of coupled higher‐order Hilfer fractional neutral stochastic integro‐differential systems (HFNSIDSs) driven by fractional Brownian motion (FBM). To the best of our knowledge, the study of coupled higher‐order HFNSIDSs with abstract fractional initial conditions has not been addressed previously in the literature. By employing tools from fractional calculus, stochastic analysis, semigroup theory, and the Mönch fixed point theorem (MFPT), sufficient conditions are established for the existence and uniqueness of mild solutions in infinite‐dimensional spaces. The adoption of MFPT is particularly motivated by the utilization of the Hausdorff measure of noncompactness (HMNC), which ensures the relative compactness conditions required for the analysis. Furthermore, trajectory controllability (TC) is examined by invoking Gronwall's inequality, and a corresponding stability criterion is derived for the coupled HFNSIDSs. The novelty of this work lies in its integration of rigorous analytical techniques with comprehensive numerical simulations in both two‐ and three‐dimensional frameworks. Finally, a numerical example is presented to demonstrate the validity and applicability of the proposed theoretical results.