Sample average approximations in optimal control of uncertain systems

TLDR: An optimality function is provided for both the uncertain optimal control problem and its approximation, and it is shown that the approximation based on the sample average approach is consistent in the sense of Polak.

Abstract: This paper focuses on an optimal control problem in which the objective is to minimize the expectation of a cost functional with stochastic parameters. The inclusion of the stochastic parameters in the objective raises new theoretical and computational challenges not present in a standard nonlinear optimal control problem. In this paper, we provide a numerical framework for the solution of this uncertain optimal control problem by taking a sample average approximation approach. An independent random sample is taken from the parameter space, and the expectation is approximated by the sample average. The result is a family of standard nonlinear optimal control problems which can be solved using existing techniques. We provide an optimality function for both the uncertain optimal control problem and its approximation, and show that the approximation based on the sample average approach is consistent in the sense of Polak. We illustrate the approach with a numerical example arising in optimal search for a moving target.