  This work focuses on numerical solutions of optimal control problems. A time
discretization error representation is derived for the approximation of the
associated value function. It concerns Symplectic Euler solutions of the
Hamiltonian system connected with the optimal control problem. The error
representation has a leading order term consisting of an error density that is
computable from Symplectic Euler solutions. Under an assumption of the pathwise
convergence of the approximate dual function as the maximum time step goes to
zero, we prove that the remainder is of higher order than the leading error
density part in the error representation. With the error representation, it is
possible to perform adaptive time stepping. We apply an adaptive algorithm
originally developed for ordinary differential equations. The performance is
illustrated by numerical tests.
