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.

翻译：本文研究了概率测度空间上非局部连续性方程的最优整体控制问题。我们允许一般的非线性代价泛函，并支持直接控制驱动向量场的非局部项。针对该问题，我们设计了一种基于庞特里亚金最大值原理的下降法。为此，我们推导出具有解耦哈密顿系统的庞特里亚金最大值原理新形式。具体而言，我们在带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。作为所设计下降法的实现，我们提出了一种带有回溯的间接确定性数值算法。我们证明了该算法的收敛性，并通过处理涉及相互作用的振荡器群体的Kuramoto型模型的简单案例，阐释了其运行机制。