We address the problem of optimal control of a nonsmooth dynamical system described by an evolution variational inequality. We consider both the discrete-time and continuous-time versions of the problem and we relax the problem in the space of measures. We show that there is no gap between the original finite-dimensional problem and the relaxed problem. We show the convergence of the relaxed discrete-time optimal control in measures to continuous-time optimal control in measures. This paves the way to a sound implementation of the moment sum-of-squares hierarchy to solve numerically the optimal control of nonsmooth dynamical systems.