A Convergent Approximation of the Pareto Optimal Set for Finite Horizon Multiobjective Optimal Control Problems (MOC) Using Viability Theory

摘要

The objective of this paper is to provide a convergent numericalapproximation of the Pareto optimal set for finite-horizon multiobjectiveoptimal control problems for which the objective space is not necessarilyconvex. Our approach is based on Viability Theory. We first introduce theset-valued return function V and show that the epigraph of V is equal to theviability kernel of a properly chosen closed set for a properly chosendynamics. We then introduce an approximate set-valued return function withfinite set-values as the solution of a multiobjective dynamic programmingequation. The epigraph of this approximate set-valued return function is shownto be equal to the finite discrete viability kernel resulting from theconvergent numerical approximation of the viability kernel proposed in [4, 5].As a result, the epigraph of the approximate set-valued return functionconverges towards the epigraph of V. The approximate set-valued return functionfinally provides the proposed numerical approximation of the Pareto optimal setfor every initial time and state. Several numerical examples are provided.