The optimal control related to Riemannian manifolds and the viscosity solutions to H-J-B equations

TL;DR

This paper explores the connection between optimal control problems on Riemannian manifolds and viscosity solutions to Hamilton-Jacobi-Bellman equations, establishing the uniqueness of solutions via the Dynamic Programming Principle.

Contribution

It extends the Dynamic Programming Principle to stochastic differential equations on Riemannian manifolds and proves the uniqueness of viscosity solutions to the associated PDEs.

Findings

DPP holds for SDEs on Riemannian manifolds

Cost function is the unique viscosity solution to the PDEs

Establishes a link between control problems and PDE solutions on manifolds

Abstract

This paper is concerned with the Dynamic Programming Principle (DPP in short) with SDEs on Riemannian manifolds. Moreover, through the DPP, we conclude that the cost function is the unique viscosity solution to the related PDEs on manifolds.