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

The objective of this paper is to provide a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems for which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce the set-valued return function V and show that the epigraph of V is equal to the viability kernel of a properly chosen closed set for a properly chosen dynamics. 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 is shown to be equal to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [4, 5]. As a result, the epigraph of the approximate set-valued return function converges towards 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 are provided

Set-Valued Return Function and Generalized Solutions for Multiobjective Optimal Control Problems (MOC)

In this paper, we consider a multiobjective optimal control problem where the preference relation in the objective space is defined in terms of a pointed convex cone containing the origin, which defines generalized Pareto optimality. For this problem, we introduce the set-valued return function V and provide a unique characterization for V in terms of contingent derivative and coderivative for set-valued maps, which extends two previously introduced notions of generalized solution to the Hamilton-Jacobi equation for single objective optimal control problems.Comment: 29 pages, submitted to SICO