连续时间线性二次规划的重叠 Schwarz 方案
Overlapping Schwarz Scheme for Linear-Quadratic Programs in Continuous Time
AI总结：
提出先优化后离散的框架，基于庞特里亚金最小值原理和重叠 Schwarz 分解求解连续时间线性二次最优控制问题，并证明灵敏度指数衰减性质在连续时间下成立，数值实验验证了其适用性。
摘要：
我们提出了一种先优化后离散的框架，用于求解由时间非齐次常微分方程（ODE）控制的线性二次最优控制问题（OCP）。我们的方法采用基于庞特里亚金最小值原理的修正重叠 Schwarz 分解，将时间域划分为重叠区间，并在连续时间内独立求解哈密顿系统。我们证明，通过适当更新各个哈密顿动力学的边界条件，可以确保收敛性。我们分析的核心在于证明离散时间 OCP 中展现的灵敏度指数衰减（EDS）性质能够延续到连续时间设置中。与先离散后优化的方法不同，我们的方法可以灵活地采用不同的数值积分方法来求解由此产生的哈密顿两点边值子问题，包括自适应时间积分器。在一个线性二次 OCP 上的数值实验说明了我们方法在广泛科学应用中的实用性。
英文摘要：
We present an optimize-then-discretize framework for solving linear-quadratic optimal control problems (OCP) governed by time-inhomogeneous ordinary differential equations (ODEs). Our method employs a modified overlapping Schwarz decomposition based on the Pontryagin Minimum Principle, partitioning the temporal domain into overlapping intervals and independently solving Hamiltonian systems in continuous time. We demonstrate that the convergence is ensured by appropriately updating the boundary conditions of the individual Hamiltonian dynamics. The cornerstone of our analysis is to prove that the exponential decay of sensitivity (EDS) exhibited in discrete-time OCPs carries over to the continuous-time setting. Unlike the discretize-then-optimize approach, our method can flexibly incorporate different numerical integration methods for solving the resulting Hamiltonian two-point boundary-value subproblems, including adaptive-time integrators. A numerical experiment on a linear-quadratic OCP illustrates the practicality of our approach in broad scientific applications.
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