Optimal control of a class of mixed hemivariational inequalities involving set-valued maps

The purpose of this paper is to study optimal control of a generalized class of hemivariational inequalities involving a set-valued mapping. We use a Galerkin-type method and the pseudomonotonicity notion in the sense of Brézis for bifunctions. A recent paper by Steck (J. Optim. Theory Appi. 181 (2019), 318-323) has shown the interest of this notion and showed how it is strictly weaker than the notion called Ky Fan hemicontinuity which has been used in many recent works related to the problem studied in this paper. The results obtained in this paper are new and improve considerably many existing results in literature.