Temporal Parallelization of the HJB Equation and Continuous-Time Linear Quadratic Control

• 成果类型: Article
• 署名作者: Sarkka, Simo; Garcia-Fernandez, Angel F.
• 署名单位: Aalto University; University of Liverpool
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3518309
• 发表日期: 2025
• 页码: 3755-3770
• 关键词: Optimal control; mathematical models; trajectory; dynamic programming; viscosity; time complexity; Symmetric matrices; Graphics processing units; COSTS; reviews; Continuous-time control; graphics processing unit (GPU); Hamilton-Jacobi--Bellman (HJB) equation; linear quadratic tracking; multicore; parallel computing

摘要

This article presents a mathematical formulation to perform temporal parallelization of continuous-time optimal control problems, which can be solved via the Hamilton-Jacobi-Bellman (HJB) equation. We divide the time interval of the control problem into subintervals, and define a control problem in each subinterval, conditioned on the start and end states, leading to conditional value functions for the subintervals. By defining an associative operator as the minimization of the sum of conditional value functions, we obtain the elements and associative operators for a parallel associative scan operation. This allows for solving the optimal control problem on the whole time interval in parallel in logarithmic time complexity in the number of subintervals. We derive the HJB-type of backward and forward equations for the conditional value functions and solve them in closed form for linear quadratic problems. We also discuss numerical methods for computing the conditional value functions. The computational advantages of the proposed parallel methods are demonstrated via simulations run on a multicore central processing unit and a graphics processing unit.