# Equivalent extensions of Hamilton-Jacobi-Bellman equations on hypersurfaces

## TL;DR

This paper introduces a new method to solve Hamilton-Jacobi-Bellman equations on surfaces by extending the problem to a narrow band around the surface, enabling the use of Cartesian grid methods for efficient computation.

## Contribution

The authors develop an equivalent formulation of HJB equations on surfaces by extending the control problem to a narrow band, facilitating the use of existing numerical methods.

## Key findings

- Unique viscosity solutions are obtained in the narrow band.
- The method allows efficient computation using Cartesian grid techniques.
- Applicable to unstructured point clouds sampled from surfaces.

## Abstract

We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. In this work, we extend the optimal control problem given on the surface to an equivalent one defined in a sufficiently thin narrow band of the co-dimensional one surface. The extension is done appropriately so that the corresponding HJB equation, in the narrow band, has a unique viscosity solution which is identical to the constant normal extension of the value function of the original optimal control problem. With this framework, one can easily use existing (high order) numerical methods developed on Cartesian grids to solve HJB equations on surfaces, with a computational cost that scales with the dimension of the surfaces. This framework also provides a systematic way for solving HJB equations on the unstructured point clouds that are sampled from the surface.