Time-optimal control of Hilfer fractional neutral stochastic hemivariations inequalities with mixed fBm

In this manuscript, we explore the time optimal control problem for Hilfer fractional neutral stochastic integro-differential evolution hemivariational inequalities driven by mixed fractional Brownian motion (fBm) in Hilbert spaces. We start by showing that existence results by using methods from fractional calculus, stochastic analysis, multivalued analysis, the Bohnenblust-Karlin's fixed point theorem, and important properties of fractional Brownian motion. After setting suitable assumptions, we derive the time optimal control results for the system. Finally, we provide a specific example to illustrate and confirm the theoretical findings.