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.

我们提出了一个新的公式，用于在维数为1的闭合光滑表面上计算一类Hamilton Hamilton Jacobi Bellman（HJB）方程的解。对于本文中考虑的一类方程，HJB方程的粘度解等于相应的最优控制问题的值函数。在这项工作中，我们将在表面上给出的最佳控制问题扩展到在一维共面的足够细的窄带中定义的等效问题。适当地进行扩展，以使相应的HJB方程在窄带中具有唯一的粘度解，该粘度解与原始最优控制问题的值函数的恒定法向扩展相同。有了这个框架，人们可以轻松地使用在笛卡尔网格上开发的现有（高阶）数值方法来求解曲面上的HJB方程，其计算成本随曲面尺寸而定。该框架还提供了一种系统的方法来求解从表面采样的非结构化点云上的HJB方程。