We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents, as the number of interactions grows O(N2) for N agents. For reducing the computational cost to O(N), we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. First, the considered time interval is divided into a number of subintervals. In each subinterval, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics in the L2-expectation norm as the length of subintervals goes to zero. For this approximated dynamics, the corresponding optimal control can be computed efficiently using a classical gradient descent. The resulting control is not optimal for the original system, but for a reduced RBM model. We therefore adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. This leads to a semi-feedback control strategy, where the control is applied only for a short time interval to the original system, and then compute the optimal control for the next time interval with the state of the (controlled) original dynamics. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics.

中文翻译：

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使用随机批处理方法对指导问题进行模型预测控制

我们建模、模拟和控制在令人厌恶的驾驶员作用下的一群逃避者的引导问题。该问题是在一个最优控制框架中制定的，其中驱动程序（控制）旨在将逃避者（状态）引导到欧几里得空间的所需区域。随着交互次数的增加，这种模型的数值模拟对于大量交互代理来说很快变得不可行○(ñ2)为了ñ代理。为了降低计算成本○(ñ)，我们使用随机批量方法（RBM），它提供了计算上可行的动力学近似。首先，将考虑的时间间隔划分为多个子间隔。在每个子区间中，RBM ​​将粒子集随机划分为小的子集（批次），仅考虑每个批次内的相互作用。由于平均效应，RBM ​​近似收敛到精确的动态大号2-期望范数随着子区间的长度变为零。对于这种近似动态，可以使用经典梯度下降有效地计算相应的最优控制。由此产生的控制对于原始系统不是最佳的，而是对于简化的 RBM 模型。因此，我们采用模型预测控制 (MPC) 策略来处理动态误差。这导致了一种半反馈控制策略，其中控制仅应用于原始系统的短时间间隔，然后使用（受控）原始动态的状态计算下一个时间间隔的最优控制。通过数值实验，我们表明 RBM 和 MPC 的结合可以显着降低计算成本，保持控制整体动力学的能力。