Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels. (6th February 2014)

• Record Type: Journal Article
• Authors: Guigue, Alexis
• Is Part Of: ESAIM. Volume 20:Number 1(2014:Jan.)
• Page Start: 95
• Page End: 115
• Publication Date: 2014-02-06
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013056
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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 <italic>V</italic> and show that the epigraph of <italic>V</italic> 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, <italic>Set-Valued Analysis </italic><bold>8 </bold>(2000) 111–126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of <italic>V</italic>. 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.</p> </abstract>