The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. Under some reasonable assumptions, we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity. Those results enable us to design a new continuation algorithm, presented at the end of the paper, that handles automatically changes in the structure of the trajectory.
Mots-clés : optimal control, first-order state constraint, strong regularity, sensitivity analysis, touch point, homotopy method
@article{COCV_2008__14_4_825_0,
author = {Bonnans, Joseph Fr\'ed\'eric and Hermant, Audrey},
title = {Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {825--863},
publisher = {EDP-Sciences},
volume = {14},
number = {4},
year = {2008},
doi = {10.1051/cocv:2008016},
mrnumber = {2451799},
zbl = {1148.49026},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008016/}
}
TY - JOUR AU - Bonnans, Joseph Frédéric AU - Hermant, Audrey TI - Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 825 EP - 863 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008016/ DO - 10.1051/cocv:2008016 LA - en ID - COCV_2008__14_4_825_0 ER -
%0 Journal Article %A Bonnans, Joseph Frédéric %A Hermant, Audrey %T Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 825-863 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008016/ %R 10.1051/cocv:2008016 %G en %F COCV_2008__14_4_825_0
Bonnans, Joseph Frédéric; Hermant, Audrey. Stability and sensitivity analysis for optimal control problems with a first-order state constraint and application to continuation methods. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 825-863. doi : 10.1051/cocv:2008016. http://www.numdam.org/articles/10.1051/cocv:2008016/
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