2019/03/26 by Lindsay Martin, Martin, Lindsay, Richard Tzong‐Han Tsai +1
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.48550/arxiv.1903.11173
We present a new formulation for the computation of solutions of a class of\nHamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of\nco-dimension one. For the class of equations considered in this paper, the\nviscosity solution of the HJB equation is equivalent to the value function of a\ncorresponding optimal control problem. In this work, we extend the optimal\ncontrol problem given on the surface to an equivalent one defined in a\nsufficiently thin narrow band of the co-dimensional one surface. The extension\nis done appropriately so that the corresponding HJB equation, in the narrow\nband, has a unique viscosity solution which is identical to the constant normal\nextension of the value function of the original optimal control problem. With\nthis framework, one can easily use existing (high order) numerical methods\ndeveloped on Cartesian grids to solve HJB equations on surfaces, with a\ncomputational cost that scales with the dimension of the surfaces. This\nframework also provides a systematic way for solving HJB equations on the\nunstructured point clouds that are sampled from the surface.\n