  In this paper, we are concerned with a nonlinear optimal control problem of
ordinary differential equations. We consider a discretization of the problem
with the discontinuous Galerkin method with arbitrary order $r \in
\mathbb{N}\cup \{0\}$. Under suitable regularity assumptions on the cost
functional and solutions of the state equations, we provide sharp estimates for
the error of the approximate solutions. Numerical experiments are presented
supporting the theoretical results.
