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.

Accepted:

DOI: 10.1051/cocv/2015041

Keywords: Optimal control, relaxed control, impulsive control, semidefinite programming

^{1}; Henrion, Didier

^{2, 3, 4}; Kruıžík, Martin

^{5, 6}

@article{COCV_2017__23_1_95_0, author = {Claeys, Mathieu and Henrion, Didier and Kru{\i}\v{z}{\'\i}k, Martin}, title = {Semi-definite relaxations for optimal control problems with oscillation and concentration effects}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {95--117}, publisher = {EDP-Sciences}, volume = {23}, number = {1}, year = {2017}, doi = {10.1051/cocv/2015041}, mrnumber = {3601017}, zbl = {1358.49026}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015041/} }

TY - JOUR AU - Claeys, Mathieu AU - Henrion, Didier AU - Kruıžík, Martin TI - Semi-definite relaxations for optimal control problems with oscillation and concentration effects JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 95 EP - 117 VL - 23 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015041/ DO - 10.1051/cocv/2015041 LA - en ID - COCV_2017__23_1_95_0 ER -

%0 Journal Article %A Claeys, Mathieu %A Henrion, Didier %A Kruıžík, Martin %T Semi-definite relaxations for optimal control problems with oscillation and concentration effects %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 95-117 %V 23 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015041/ %R 10.1051/cocv/2015041 %G en %F COCV_2017__23_1_95_0

Claeys, Mathieu; Henrion, Didier; Kruıžík, Martin. Semi-definite relaxations for optimal control problems with oscillation and concentration effects. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 1, pp. 95-117. doi : 10.1051/cocv/2015041. http://archive.numdam.org/articles/10.1051/cocv/2015041/

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