# Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels

• Volume: 20, Issue: 1, page 95-115
• ISSN: 1292-8119

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## Abstract

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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.

## How to cite

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Guigue, Alexis. "Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 95-115. <http://eudml.org/doc/272960>.

@article{Guigue2014,
abstract = {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.},
author = {Guigue, Alexis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {multiobjective optimal control; Pareto optimality; viability theory; convergent numerical approximation; dynamic programming; set-valued map; set-valued return function; viability kernel; external stability; recession cone},
language = {eng},
number = {1},
pages = {95-115},
publisher = {EDP-Sciences},
title = {Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels},
url = {http://eudml.org/doc/272960},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 1
SP - 95
EP - 115
AB - 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.
LA - eng
KW - multiobjective optimal control; Pareto optimality; viability theory; convergent numerical approximation; dynamic programming; set-valued map; set-valued return function; viability kernel; external stability; recession cone
UR - http://eudml.org/doc/272960
ER -

## References

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