The Hopf-Lax formula for multiobjective costs with non-constant discount via set optimization

TL;DR

This paper develops a set-valued framework for multiobjective optimization with non-constant discounting, deriving Bellman's principle, the Hopf-Lax formula, and a set-valued Hamilton-Jacobi equation to characterize the value function.

Contribution

It introduces a novel set-valued approach to multiobjective costs with non-constant discount, extending classical optimal control results to this new setting.

Findings

Derived Bellman's optimality principle for set-valued problems

Established the Hopf-Lax formula in the set-valued context

Proved the value function solves a set-valued Hamilton-Jacobi equation

Abstract

The minimization of a multiobjective Lagrangian with non-constant discount is studied. The problem is embedded into a set-valued framework and a corresponding definition of the value function is given. Bellman's optimality principle and Hopf-Lax formula are derived. The value function is shown to be a solution of a set-valued Hamilton-Jacobi equation.