Steering large-scale swarms in only a few control updates is challenging because real systems operate in sampled-data form: control inputs are updated intermittently and applied over finite intervals. In this regime, the natural object is not an instantaneous velocity field, but a finite-window control quantity that captures the system response over each sampling interval. Inspired by MeanFlow, we introduce a control-space learning framework for swarm steering under linear time-invariant dynamics. The learned object is the coefficient that parameterizes the finite-horizon minimum-energy control over each interval. We show that this coefficient admits both an integral representation and a local differential identity along bridge trajectories, which leads to a simple stop-gradient training objective. At implementation time, the learned coefficient is used directly in sampled-data updates, so the prescribed dynamics and actuation map are respected by construction. The resulting framework provides a scalable approach to few-step swarm steering that is consistent with the sampled-data structure of real control systems.

翻译：在仅有少量控制更新的情况下引导大规模集群具有挑战性，因为实际系统以采样数据形式运行：控制输入间歇性更新并在有限时间间隔内施加。在此情况下，自然的研究对象并非瞬时速度场，而是一个有限窗口控制量，用于捕获每个采样间隔内的系统响应。受MeanFlow启发，我们提出了一种面向线性时不变动力学下集群引导的控制空间学习框架。该学习对象是参数化每个间隔内有限时域最小能量控制的系数。我们证明该系数同时具有沿桥接轨迹的积分表示与局部微分恒等式，从而导出一个简单的停止梯度训练目标。在实现时，学习到的系数直接用于采样数据更新，因此预设的动力学和驱动映射在构造上得到遵守。该框架为与真实控制系统采样数据结构一致的少步集群引导提供了可扩展方法。