Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Abstract

Solving high-dimensional optimal control problems and their corresponding Hamilton-Jacobi partial differential equations is an important but challenging problem. In particular, handling optimal control problems with state-dependent running costs presents an additional challenge. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We also present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. KeywordsOptimal control • Hamilton-Jacobi partial differential equations • Grid-free numerical methods • High dimensions Research supported by NSF 1820821 and AFOSR MURI FA9550-20-1-0358. P.C. is supported by the SMART Scholarship, which is funded by USD/R&E (The Under Secretary of Defense-Research and Engineering), National Defense Education Program (NDEP) / BA-1, Basic Research. Authors' names are given in last/family name alphabetical order.

SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems

“…In literature, there are some algorithms for solving high-dimensional optimal control problems (or the corresponding Hamilton-Jacobi PDEs), which include optimization methods [18,24,21,23,15,14,87,60,55], max-plus methods [1,2,29,35,39,67,66,68,69], tensor decomposition techniques [28,44,86], sparse grids [10,37,53], polynomial approximation [51,52], model order reduction [4,57], optimistic planning [9], dynamic programming and reinforcement learning [13,11,3,8,89], as well as methods based on neural networks [5,6,27,47,42,45,46,58,73,78,81,84,62,20,…”