Approximate optimal control of fractional stochastic hemivariational inequalities of order (1, 2] driven by Rosenblatt process

We study the approximate optimal control for a class of fractional stochastic hemivariational inequalities with non-instantaneous impulses driven by Rosenblatt process in a Hilbert space. Firstly, a suitable definition of piecewise continuous mild solution is introduced, and by using stochastic analysis, properties of \(\alpha \) $\alpha$ -order sine and cosine family and Picard type approximate sequences, we show the existence and uniqueness of approximate mild solutions for the inequality problems of fractional order (1, 2] under the non-Lipschitz conditions. Secondly, we provide the existence conditions of approximate solutions to optimal control problems driven by the presented control systems with the help of a new minimizing sequence method. Finally, an example is provided to illustrate the theory.