论文标题
HJB方程和连续时间线性二次控制的时间平行
Temporal Parallelisation of the HJB Equation and Continuous-Time Linear Quadratic Control
论文作者
论文摘要
本文提出了一种数学公式，以执行连续时间最佳控制问题的时间平行，可以通过汉密尔顿 - 雅各布 - 贝尔曼（HJB）方程来解决。我们将控制问题的时间间隔划分为子间隔，并在每个子间隔中定义一个控制问题，以起始和最终状态为条件，从而导致子互动的条件价值函数。通过将关联运算符定义为条件值函数之和的最小化，我们为并行的关联扫描操作获得了元素和关联操作员。这允许在整个时间间隔中以对数时间复杂性在子互相数量中并行解决最佳控制问题。我们为条件值函数得出了向后和正方程的HJB型，并以封闭形式解决线性二次问题。我们还讨论用于计算条件值函数的数值方法。通过在多核中央处理单元和图形处理单元上运行的模拟来证明所提出的并行方法的计算优势。
This paper presents a mathematical formulation to perform temporal parallelisation 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 sub-intervals, and define a control problem in each sub-interval, conditioned on the start and end states, leading to conditional value functions for the sub-intervals. By defining an associative operator as the minimisation 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 sub-intervals. 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 multi-core central processing unit and a graphics processing unit.
