The paper investigates control problems for a class of nonlinear elliptic variational-hemivariational inequalities with constraint sets. Based on the well posedness of a variational-hemivariational inequality, we prove some results on continuous dependence and existence of optimal pairs to optimal control problems. We obtain some continuous dependence results in which the strong dependence and weak dependence are considered, respectively. A frictional contact problem is given to illustrate our main results.