The present paper represents a continuation of Migorski et al. (J Elast 127:151-178, 2017). There, the analysis of a new class of elliptic variational-hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force.