Set-valued Hamilton-Jacobi-Bellman Equations
Optimization and Control
2025-08-19 v3
Abstract
Building upon the dynamic programming principle for set-valued functions arising from many applications, in this paper we propose a new notion of set-valued PDEs. The key component in the theory is a set-valued It\^{o} formula, characterizing the flows on the surface of the dynamic sets. In the contexts of multivariate control problems, we establish the wellposedness of the set-valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, a moving scalarization for certain time inconsistent problems is constructed by using the classical solution of the set-valued HJB equation.
Cite
@article{arxiv.2311.05727,
title = {Set-valued Hamilton-Jacobi-Bellman Equations},
author = {Melih İşeri and Jianfeng Zhang},
journal= {arXiv preprint arXiv:2311.05727},
year = {2025}
}