Improved order in Hilfer fractional differential systems: Solvability and optimal control problem for hemivariational inequalities

The main objective of this study is to analyze the optimal control results and solvability concerning Hilfer fractional differential hemivariational inequalities within the range of (1<ϱ<2). Initially, we explore the solvability of a mild solutions for the Hilfer fractional hemivariational inequality. Subsequently, we investigate the optimal control strategies for the given problems by employing cost functionals, cosine operators, mild solutions, the fixed point method for multivalued functions, and a generalized Clarke subdifferential approach. To illustrate our findings, we provide a practical example and a filter diagram.