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中非局部连续性方程的最优集控制问题。我们允许使用一般非线性代价泛函，并具备直接控制驱动向量场中非局部项的可选功能。针对该问题，我们基于庞特里亚金最大值原理（PMP）设计了一种下降方法。为此，我们推导了具有解耦哈密顿系统的新型PMP形式。具体来说，我们在有符号测度空间上提取了线性非局部平衡律的伴随系统，并证明其适定性。作为所设计下降方法的实现，我们提出了一种带有回溯机制的间接确定性数值算法。我们证明了该算法的收敛性，并通过处理一个涉及库拉莫托型相互作用振子种群模型的简单案例，阐明了其运行机制。