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

TL;DR

This paper introduces analytical solutions, a Hopf-type formula, and efficient algorithms for high-dimensional optimal control problems with specific cost structures, overcoming the curse of dimensionality and enabling real-time applications.

Contribution

The paper provides a novel Hopf-type representation formula and efficient numerical algorithms tailored for high-dimensional control problems with convex, piecewise affine costs, demonstrating significant speedups.

Findings

Algorithms overcome curse of dimensionality in high dimensions

FPGA implementation achieves 40x speedup over CPU

Numerical examples validate efficiency and scalability

Abstract

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. 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 present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. 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.