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.

该论文解决了概率测度空间上非局部连续性方程的最优系综控制问题。我们承认一般非线性成本函数，以及直接控制驱动向量场的非局部项的选项。针对这个问题，我们设计了一种基于庞特里亚金极大值原理（PMP）的下降法。为此，我们推导了一种具有解耦哈密顿系统的 PMP 的新形式。具体来说，我们在有符号测度空间上提取线性非局部平衡律的伴随系统并证明其适定性。作为所设计的下降方法的实现，我们提出了一种带有回溯的间接确定性数值算法。