In this paper we consider mean-field optimal control problems with selective action of the control, where the constraint is a continuity equation involving a non-local term and diffusion. First order optimality conditions are formally derived in a general framework, accounting for boundary conditions. Hence, the optimality system is used to construct a reduced gradient method, where we introduce a novel algorithm for the numerical realization of the forward and the backward equations, based on exponential integrators. We illustrate extensive numerical experiments on different control problems for collective motion in the context of opinion formation and pedestrian dynamics.

翻译：本文考虑具有控制选择性作用的平均场最优控制问题，其约束条件涉及非局部项和扩散的连续性方程。我们在包含边界条件的通用框架下形式化推导了一阶最优性条件。进而利用最优性系统构建了简化梯度方法，其中我们提出了一种基于指数积分器的新型算法，用于实现前向和反向方程的数值求解。通过意见形成与行人动力学背景下的集体运动控制问题，我们展示了大量的数值实验。