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

¹ Avenue Edmond Cordier 19, 1160 Auderghem, Belgium
² CNRS-LAAS, 7 Avenue du colonel Roche, 31400 Toulouse, France
³ Université de Toulouse, LAAS, 31400 Toulouse, France
henrion@laas.fr
⁴ Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 26 Prague, Czech Republic
⁵ Institute of Information Theory and Automation of the Academy of Sciences of the Czech Republic, Pod vodárenskou veíž 4, 182 08, Prague, Czech Republic
⁶ Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, CZ-166 29 Prague, Czech Republic

Received: 28 October 2014
Accepted: 5 August 2015

Abstract

Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the control 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 introduced originally by DiPerna and Majda in the partial differential equations literature 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 nonconvex optimal control problems with polynomial vector field and semialgebraic state constraints.