Multi-objective Herglotz' variational principle and cooperative Hamilton-Jacobi systems

TL;DR

This paper introduces a deterministic approach to multi-objective Herglotz' variational problems with cooperative coupling, linking them to Hamilton-Jacobi systems and broadening the understanding of their solutions under general conditions.

Contribution

It establishes Euler-Lagrange equations and the relation between value functions and viscosity solutions for cooperative weakly coupled Hamilton-Jacobi systems, extending previous stochastic methods.

Findings

Derived Euler-Lagrange equations for the variational problem

Linked value functions to viscosity solutions of Hamilton-Jacobi systems

Validated approach for general linearly coupled matrices for short time

Abstract

We study a multi-objective variational problem of Herglotz' type with cooperative linear coupling. We established the associated Euler-Lagrange equations and the characteristic system for cooperative weakly coupled systems of Hamilton-Jacobi equations. We also established the relation of the value functions of this variational problem with the viscosity solutions of cooperative weakly coupled systems of Hamilton-Jacobi equations. Comparing to the previous work in stochastic frame, this approach affords a pure deterministic explanation of this problem under more general conditions. We also showed this approach is valid for general linearly coupling matrix for short time.