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Set-based integration methods allow to prove properties of differentialsystems, which take into account bounded disturbances. The systems (eithertime-discrete, time-continuous or hybrid) satisfying such properties are saidto be "robust". In the context of optimal control synthesis, the set-basedmethods are generally extensions of numerical optimal methods of two classes:first, methods based on convex optimization; second, methods based on thedynamic programming principle. Heymann et al. have recently shown that, forcertain systems of low dimension, the second numerical method can give bettersolutions than the first one. They have built a solver (Bocop) that implementsboth numerical methods. We show in this paper that a set-based extension of amethod of the second class which uses a guaranteed Euler integration method,allows us to find such good solutions. Besides, these solutions enjoy theproperty of robustness against uncertainties on initial conditions and boundeddisturbances. We demonstrate the practical interest of our method on an exampletaken 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 moreefficiently an optimal control at the price of losing robustness.

Robust optimal control using dynamic programming and guaranteed Euler's method