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

ESAIM: Control, Optimisation and Calculus of Variations (2014)

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

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topGuigue, 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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