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

TL;DR

This paper introduces a convergent numerical method based on Viability Theory to approximate the Pareto optimal set in finite-horizon multiobjective optimal control problems, even when the objective space is non-convex.

Contribution

It develops a novel approach linking viability kernels with set-valued return functions to approximate Pareto sets in complex control problems.

Findings

The method converges to the true Pareto optimal set.

Numerical examples demonstrate the effectiveness of the approach.

The approach handles non-convex objective spaces.

Abstract

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.