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

TL;DR

This paper introduces a set-valued return function for multiobjective optimal control problems with generalized Pareto optimality, providing a unique characterization using advanced derivative concepts, extending solutions of Hamilton-Jacobi equations.

Contribution

It extends the notion of generalized solutions for Hamilton-Jacobi equations to multiobjective control problems using set-valued analysis.

Findings

Characterization of the set-valued return function V.

Extension of generalized solution concepts to multiobjective problems.

Unique representation of V via contingent derivatives.

Abstract

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.