Solving Unbounded Optimal Control Problems With the Moment-SOS Hierarchy

The behaviour of the moment-sums-of-squares (moment-SOS) hierarchy for polynomial optimal control problems on compact sets has been explored to a large extent. Our contribution focuses on the case of non-compact control sets. We describe a new approach to optimal control problems with unbounded controls, using compactification by partial homogenization, leading to an equivalent infinite dimensional linear program with compactly supported measures. Our results are closely related to the results of a previous approach using DiPerna-Majda measures. However, our work provides a sound proof of the absence of relaxation gap, which was conjectured in the previous work, and thereby enables the design of a moment-sum-of-squares relaxation with guaranteed convergence.