Optimal Control of PDEs using Occupation Measures and SDP Relaxations

TL;DR

This paper develops a method using occupation measures and SDP relaxations to solve nonlinear optimal control problems with PDE constraints, providing converging bounds and numerical examples involving diffusion and wave equations.

Contribution

It introduces a hierarchy of semidefinite relaxations for PDE-constrained optimal control problems with polynomial data, ensuring convergence to the optimal value.

Findings

Converging hierarchy of SDP relaxations for PDE control problems.

Numerical examples demonstrate effectiveness on diffusion and wave equations.

Lower bounds obtained from relaxations approximate true optimal values.

Abstract

This paper addresses the problem of solving a class of nonlinear optimal control problems (OCP) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then, we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of semidefinite programming relaxations whose optimal values yield lower bounds for the initial OCP. We illustrate our method by two numerical examples, involving a diffusion partial differential equation and a wave equation. We also report on the related experiments.