AI 中文总结
该研究针对受随机扰动的惯性蜂拥，基于薛定谔桥理论，提出嵌套不动点方案求解最优控制，实现有限时间内将蜂拥群体引导至规定端点分布的目标。
AI 中文摘要
范式交互模型解释了复杂系统中，集体行为如何由组成智能体间的交互涌现产生。然而在受生物启发的蜂拥行为中，仅靠交互可能无法让群体在规定时间范围内达到期望的整体构型，这一需求存在于从靶向治疗到集体运输、紧急疏散等各类应用场景中。本研究针对受随机扰动的惯性蜂拥，考虑有限时间范围的最小能量集体控制问题，聚焦于由Cucker--Smale对齐或Morse吸引-排斥交互驱动的多智能体系统的平均场表示。我们的目标是通过状态反馈控制，将蜂拥群体在规定的端点分布之间进行引导，其中端点规范可以是完整的相空间分布（位置和速度）或仅位置边际分布。我们的形式体系基于薛定谔桥理论，该理论已推动了统计推断、生物建模、随机控制和生成式学习等领域的当代发展。在桥框架内，不受控的交互随机动力学被视为先验模型，最优控制则被视为实现规定分布所需的最小能量校正漂移。我们推导了具有时间对称结构的非线性耦合最优性必要系统，其结构与经典薛定谔桥类似，并提出了嵌套不动点方案对其进行数值求解。数值示例表明，所获得的最优控制（校正漂移）可根据交互力对引导任务是有利还是不利，动态地利用或抵消这些交互力。
英文摘要
Paradigmatic interaction models explain how collective behaviors can emerge in complex systems from interactions among the constituent agents. In bio-inspired swarms, however, interactions alone may not suffice to bring the population to a desired aggregate configuration within a prescribed time horizon, as needed in applications ranging from targeted therapy to collective transport and emergency evacuation. In the present work, we consider finite-horizon minimum-energy collective steering for inertial swarms that are subject to stochastic disturbances. We focus on the mean-field representations of these multi-agent systems driven by Cucker--Smale alignment or Morse attraction--repulsion interactions. Our objective is to steer the swarm between prescribed endpoint distributions using a state-feedback control, where the endpoint specifications can be full phase-space distributions (positions and velocities) or position marginals alone. Our formalism is rooted in the theory of Schrödinger bridges, which has inspired contemporary developments spanning statistical inference, biological modeling, stochastic control, and generative learning. Within the bridges framework, the uncontrolled interacting stochastic dynamics are viewed as a prior model, and the optimal control as the minimum-energy corrective drift needed to realize the prescribed distributions. We derive nonlinear, coupled necessary optimality systems with a time-symmetric structure reminiscent of classical Schrödinger bridges, and propose nested fixed-point schemes to numerically solve them. Numerical examples show that the obtained optimal control (corrective drift) can dynamically exploit or counteract the interaction forces, depending on whether the latter are favorable or adversarial to the steering task.
