We deal with the optimal control of systems driven by differential inclusions with anti-periodic conditions in Banach spaces. The operators in the differential inclusions, describing the control system, all depend on control variables. By a solution of the control system we mean a ‘trajectory-control’ pair. We first prove that, for fixed control variables, the set of solutions is non-empty for the corresponding differential inclusion with anti-periodic conditions. We then show the functional-topological properties of the set of solutions. Finally, we investigate the existence of solutions to the optimal control problem. These abstract results can be used to study optimal control systems of evolutionary hemi-variational inequalities.