A review from the PDE viewpoint of Hamilton-Jacobi-Bellman Equations Arising in Optimal Control with Vectorial Cost

TL;DR

This review explores the derivation of Hamilton-Jacobi-Bellman equations for optimal control problems with vector-valued costs, emphasizing the use of viscosity solutions to handle the resulting parametric PDE families.

Contribution

It introduces a novel approach to derive single PDEs for vectorial optimal control problems, avoiding systems of equations and enabling viscosity solution techniques.

Findings

Derivation of parametric families of HJB equations for vector-valued costs

Use of Pareto minimality to define vectorial value functions

Application of viscosity solutions to these PDEs

Abstract

This paper is a review of results on Optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation is considered with respect to a partial ordering generated by a given cone. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. Our main objective is the derivation of the model PDEs which turn out to be parametric families of HJB single equations instead of systems of PDEs. However, this allows the use of the theory of Viscosity Solutions.