  Set-based integration methods allow to prove properties of differential
systems, which take into account bounded disturbances. The systems (either
time-discrete, time-continuous or hybrid) satisfying such properties are said
to be "robust". In the context of optimal control synthesis, the set-based
methods are generally extensions of numerical optimal methods of two classes:
first, methods based on convex optimization; second, methods based on the
dynamic programming principle. Heymann et al. have recently shown that, for
certain systems of low dimension, the second numerical method can give better
solutions than the first one. They have built a solver (Bocop) that implements
both numerical methods. We show in this paper that a set-based extension of a
method of the second class which uses a guaranteed Euler integration method,
allows us to find such good solutions. Besides, these solutions enjoy the
property of robustness against uncertainties on initial conditions and bounded
disturbances. We demonstrate the practical interest of our method on an example
taken from the numerical Bocop solver. We also give a variant of our method,
inspired by the method of Model Predictive Control, that allows us to find more
efficiently an optimal control at the price of losing robustness.
