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We study an optimal control problem in which both the objective function andthe dynamic constraint contain an uncertain parameter. Since the distributionof this uncertain parameter is not exactly known, the objective function istaken as the worst-case expectation over a set of possible distributions of theuncertain parameter. This ambiguity set of distributions is, in turn, definedby the first two moments of the random variables involved. The optimal controlis found by minimizing the worst-case expectation over all possibledistributions in this set. If the distributions are discrete, the stochasticmin-max optimal control problem can be converted into a convensional optimalcontrol problem via duality, which is then approximated as a finite-dimensionaloptimization problem via the control parametrization. We derive necessaryconditions of optimality and propose an algorithm to solve the approximationoptimization problem. The results of discrete probability distribution are thenextended to the case with one dimensional continuous stochastic variable byapplying the control parametrization methodology on the continuous stochasticvariable, and the convergence results are derived. A numerical example ispresent to illustrate the potential application of the proposed model and theeffectiveness of the algorithm.

A robust optimal control problem with moment constraints on distribution: theoretical analysis and an algorithm