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This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function equals to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre. Birkhauser, Boston (1999) 177-247. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-Valued Analysis 8 (2000) 111-126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples illustrate our approach.
@article{COCV_2014__20_1_95_0,
author = {Guigue, Alexis},
title = {Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {95--115},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013056},
mrnumber = {3182692},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013056/}
}
TY - JOUR AU - Guigue, Alexis TI - Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 95 EP - 115 VL - 20 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013056/ DO - 10.1051/cocv/2013056 LA - en ID - COCV_2014__20_1_95_0 ER -
%0 Journal Article %A Guigue, Alexis %T Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 95-115 %V 20 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013056/ %R 10.1051/cocv/2013056 %G en %F COCV_2014__20_1_95_0
Guigue, Alexis. Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 95-115. doi: 10.1051/cocv/2013056
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