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

作　　者：Paula Chen Jerome Darbon Tingwei Meng 

机构地区：[1]Division of Applied Mathematics,Brown University,Providence,RI,USA [2]Department of Mathematics,UCLA,Los Angeles,CA,USA

出　　处：《Communications on Applied Mathematics and Computation》2024年第2期1428-1471,共44页应用数学与计算数学学报（英文）

基　　金：supported by the DOE-MMICS SEA-CROGS DE-SC0023191 and the AFOSR MURI FA9550-20-1-0358;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：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) 

分 类 号：O24[理学—计算数学] O29[理学—数学]