首页|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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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 con-trol 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.Addition-ally,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 demon-strates 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.

Optimal controlHamilton-Jacobi partial differential equationsGrid-free numerical methodsHigh dimensionsField-programmable gate arrays(FPGAs)

Paula Chen、Jér?me Darbon、Tingwei Meng

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Division of Applied Mathematics,Brown University,Providence,RI,USA

Department of Mathematics,UCLA,Los Angeles,CA,USA

DOE-MMICS SEA-CROGS DE-SC0023191AFOSR MURI FA9550-20-1-0358SMART ScholarshipUSD/R&E(The Under Secretary of Defense-Research and Engineering)National Defense Education Program(NDEP)National Defense Education Program(NDEP)

BA-1Basic Research

2024

10.1007/s42967-024-00371-4

应用数学与计算数学学报
上海大学

应用数学与计算数学学报

影响因子：0.165
ISSN：1006-6330
年,卷(期)：2024.6(2)