  The numerical realization of the dynamic programming principle for
continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman
equations which require the minimization of a nonlinear mapping over the set of
admissible controls. This minimization is often performed by comparison over a
finite number of elements of the control set. In this paper we demonstrate the
importance of an accurate realization of these minimization problems and
propose algorithms by which this can be achieved effectively. The considered
class of equations includes nonsmooth control problems with
$\ell_1$-penalization which lead to sparse controls.
