  We study an optimal control problem in which both the objective function and
the dynamic constraint contain an uncertain parameter. Since the distribution
of this uncertain parameter is not exactly known, the objective function is
taken as the worst-case expectation over a set of possible distributions of the
uncertain parameter. This ambiguity set of distributions is, in turn, defined
by the first two moments of the random variables involved. The optimal control
is found by minimizing the worst-case expectation over all possible
distributions in this set. If the distributions are discrete, the stochastic
min-max optimal control problem can be converted into a convensional optimal
control problem via duality, which is then approximated as a finite-dimensional
optimization problem via the control parametrization. We derive necessary
conditions of optimality and propose an algorithm to solve the approximation
optimization problem. The results of discrete probability distribution are then
extended to the case with one dimensional continuous stochastic variable by
applying the control parametrization methodology on the continuous stochastic
variable, and the convergence results are derived. A numerical example is
present to illustrate the potential application of the proposed model and the
effectiveness of the algorithm.
