A sharp error analysis for the DG method of optimal control problems

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 first show the existence of a local solution to the discretized problem. We then provide sharp estimates for the $L^2$-error of the approximate solutions. The convergence rate of the error depends on the regularity of the optimal solution $\overline{u}$ and its adjoint state with the degree of piecewise polynomials. Numerical experiments are presented supporting the theoretical results.