Volume 23, Number 1, January-March 2017
|Page(s)||95 - 117|
|Published online||13 October 2016|
Semi-definite relaxations for optimal control problems with oscillation and concentration effects∗
1 Avenue Edmond
Cordier 19, 1160
2 CNRS-LAAS, 7 Avenue du colonel Roche, 31400 Toulouse, France
3 Université de Toulouse, LAAS, 31400 Toulouse, France
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
Accepted: 5 August 2015
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.
Mathematics Subject Classification: 49M20 / 49J15 / 49N25 / 90C22
Key words: Optimal control / relaxed control / impulsive control / semidefinite programming
© EDP Sciences, SMAI 2016
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