Optimal control of linear PDEs using occupation measures and SDP relaxations

Abstract

This paper addresses the problem of solving a class of optimal control problems (OCPs) 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.