Controllability Analysis of Impulsive Fractional Stochastic Integro‐Differential Equations Under Hemivariational Inequalities With Numerical Results

ABSTRACT This article explores the solvability and approximate controllability of a new class of neutral impulsive stochastic integro‐differential systems. These systems are uniquely characterized by their use of fractional calculus to model real‐world behavior, the inclusion of hemivariational inequalities and impulsive terms to capture nonlinearities and sudden changes, and a history‐dependent operator to account for memory effects. The research demonstrates solvability through a fixed‐point framework that combines stochastic analysis, the generalized Clarke subdifferential, and fractional calculus. A numerical example illustrates the practical application of these findings, incorporating a comprehensive framework that uses fractional finite differences for the Caputo derivative, Monte Carlo sampling for stochastic forcing, and finite difference methods for spatial derivatives. The study highlights the effectiveness of selecting a Gaussian spectral decay function in the Monte Carlo approximation for enhancing numerical stability and accuracy, thereby advancing numerical techniques for these complex stochastic models.