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

TL;DR

This paper addresses a robust optimal control problem with moment-based distribution ambiguity, deriving theoretical conditions and proposing an algorithm to solve the resulting min-max optimization, with extensions to continuous variables and numerical validation.

Contribution

It introduces a novel approach to robust control under distributional ambiguity defined by moments, including theoretical analysis and an algorithm for both discrete and continuous cases.

Findings

The algorithm effectively solves the min-max control problem.

Theoretical convergence results are established for the continuous case.

Numerical example demonstrates practical applicability and robustness of the method.

Abstract

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.