Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

doi:10.1007/s42967-024-00371-4

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (2): 1428-1471.doi: 10.1007/s42967-024-00371-4

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Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Paula Chen¹, Jér?me Darbon¹, Tingwei Meng²

1. Division of Applied Mathematics, Brown University, Providence, RI, USA;
2. Department of Mathematics, UCLA, Los Angeles, CA, USA

收稿日期:2023-03-13 修回日期:2023-10-03 接受日期:2023-12-21 出版日期:2024-04-29 发布日期:2024-04-29
通讯作者: Jér?me Darbon,E-mail:jerome_darbon@brown.edu;Paula Chen,E-mail:paula_chen@alumni.brown.edu;Tingwei Meng,E-mail:tingwei@math.ucla.edu E-mail:jerome_darbon@brown.edu;paula_chen@alumni.brown.edu;tingwei@math.ucla.edu
基金资助:
This research is supported by the DOE-MMICS SEA-CROGS DE-SC0023191 and the AFOSR MURI FA9550-20-1-0358. P.C. is supported by the SMART Scholarship, which is funded by the USD/R&E (The Under Secretary of Defense-Research and Engineering), National Defense Education Program (NDEP)/BA-1, Basic Research.

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Paula Chen¹, Jér?me Darbon¹, Tingwei Meng²

1. Division of Applied Mathematics, Brown University, Providence, RI, USA;
2. Department of Mathematics, UCLA, Los Angeles, CA, USA

Received:2023-03-13 Revised:2023-10-03 Accepted:2023-12-21 Online:2024-04-29 Published:2024-04-29
Contact: Jér?me Darbon,E-mail:jerome_darbon@brown.edu;Paula Chen,E-mail:paula_chen@alumni.brown.edu;Tingwei Meng,E-mail:tingwei@math.ucla.edu E-mail:jerome_darbon@brown.edu;paula_chen@alumni.brown.edu;tingwei@math.ucla.edu
Supported by:
This research is supported by the DOE-MMICS SEA-CROGS DE-SC0023191 and the AFOSR MURI FA9550-20-1-0358. P.C. is supported by the SMART Scholarship, which is funded by the USD/R&E (The Under Secretary of Defense-Research and Engineering), National Defense Education Program (NDEP)/BA-1, Basic Research.

摘要/Abstract

摘要： Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions. In this paper, we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians. Additionally, we present an efficient, grid-free numerical solver based on our representation formulas, which is shown to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit (CPU) and a field-programmable gate array (FPGA). In several cases, our FPGA implementation obtains over a 10 times speedup compared to the CPU, which demonstrates the promising performance boosts FPGAs can achieve. Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

关键词: Optimal control, Hamilton-Jacobi partial differential equations, Grid-free numerical methods, High dimensions, Field-programmable gate arrays(FPGAs)

Abstract: Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions. In this paper, we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians. Additionally, we present an efficient, grid-free numerical solver based on our representation formulas, which is shown to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit (CPU) and a field-programmable gate array (FPGA). In several cases, our FPGA implementation obtains over a 10 times speedup compared to the CPU, which demonstrates the promising performance boosts FPGAs can achieve. Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

Key words: Optimal control, Hamilton-Jacobi partial differential equations, Grid-free numerical methods, High dimensions, Field-programmable gate arrays(FPGAs)

Paula Chen, Jér?me Darbon, Tingwei Meng. Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1428-1471.

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Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

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[1]	Frederic Y. M. Wan. Growth of RB Population in the Conversion Phase of Chlamydia Life Cycle[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 90-112.
[2]	Hong Xiong, Maoning Tang, Qingxin Meng. Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1386-1415.