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

Guigue, Alexis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 95-115.

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.

Classification : 49M2, 49L20, 54C60, 90C29
Mots clés : multiobjective optimal control, Pareto optimality, viability theory, convergent numerical approximation, dynamic programming

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