论文标题
非局部连续性方程的最佳控制：数值解决方案
Optimal control of nonlocal continuity equations: numerical solution
论文作者
论文摘要
本文解决了概率度量空间上非局部连续性方程的最佳集合控制问题。我们承认一般的非线性成本功能，以及直接控制驱动向量场非本地术语的选项。对于这个问题，我们根据Pontryagin的最大原理（PMP）设计了一种下降方法。为此，我们通过脱钩的哈密顿系统得出了一种新形式的PMP。具体而言，我们在签署的措施空间上提取线性非局部平衡定律的伴随系统，并证明其适当的性能。作为设计的下降方法的实现，我们提出了一种带回溯的间接确定性数字算法。我们证明了算法的融合，并通过处理涉及相互作用振荡器人群的库拉莫托型模型的简单案例来说明其作案操作。
The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin's maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.
