Semi-definite relaxations for optimal control problems with oscillation and concentration effects

TL;DR

This paper develops a unified semi-definite relaxation framework using measures to handle oscillation and concentration effects in complex optimal control problems, enabling numerical solutions for non-convex cases.

Contribution

It introduces a novel approach combining measures from PDE literature with semi-definite relaxations to address oscillation and concentration phenomena simultaneously in optimal control.

Findings

Hierarchies of semi-definite relaxations can be constructed for complex control problems.

The method effectively models oscillation and concentration effects in a unified framework.

Numerical solutions are feasible for non-convex polynomial control systems.

Abstract

Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phe-nomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the con-trol input and typically featuring concentration phenomena, interpreting the control as a measure of time with a discrete singular component modeling discontinuities or jumps of the state trajectories. In this contribution, we use measures intro-duced originally by DiPerna and Majda in the partial differential equations litera-ture to model simultaneously, and in a unified framework, possible oscillation and concentration effects of the optimal control policy. We show that hierarchies of semi-definite relaxations can also be constructed to deal numerically with noncon-vex optimal control problems with polynomial vector field and semialgebraic state constraints.