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Equivalent Extensions of Hamilton–Jacobi–Bellman Equations on Hypersurfaces

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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.

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Notes

  1. All code used to produce the numerical simulations can be found at https://github.com/lindsmart/MartinTsaiExtHJB.

  2. The uniform samplings of the sphere were computed using the code provided at https://github.com/AntonSemechko/S2-Sampling-Toolbox.

  3. The point cloud for the torus was generated using the standard parametrization of a torus.

  4. The point cloud for the Stanford bunny is generated from a refinement of the triangulated Stanford bunny from https://casual-effects.com/data/ [26].

  5. The point clouds for the surfaces in Fig. 5 were generated from the triangulations downloaded at https://www.myminifactory.com/scantheworld/.

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Acknowledgements

The authors are supported partially by National Science Foundation Grant DMS-1720171. Tsai also thanks National Center for Theoretical Study, Taipei for hosting his visits, in which some of the ideas presented in this paper originated.

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Lindsay Martin

  2. Department of Mathematics and Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Yen-Hsi Richard Tsai

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Correspondence to Lindsay Martin.

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Martin, L., Tsai, YH.R. Equivalent Extensions of Hamilton–Jacobi–Bellman Equations on Hypersurfaces. J Sci Comput 84, 43 (2020). https://doi.org/10.1007/s10915-020-01292-z

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