最优控制与非平衡弛豫中的有限时间跃迁等价类
Equivalence classes of finite-time transitions in optimal control and non-equilibrium relaxation
- Technische Universität Berlin（柏林工业大学）
- University of Konstanz（康斯坦茨大学）
- Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究建立了最优控制与非平衡弛豫的映射关系，将难采样的有限时间动力学相变转化为可通过最优控制轨迹平均获取的量，并通过光阱胶体粒子实验验证了相关理论。
AI中文摘要：
我们提出了一种针对结构化环境中随机系统最优控制的理论，该环境由代价函数中的惩罚项表征。研究表明，此类控制问题通常会呈现出尖锐的有限时间跃迁，对应于临界时刻控制策略的定性变化。从过阻尼朗之万方程和二次代价函数出发，我们证明所有由此产生的问题可分为三个典型等价类（抛物型、双曲型和椭圆型），其区分依据是控制哈密顿量的行列式符号。对于每个类，我们以闭式形式得到了最优协议、代价函数和临界时间，并证明该跃迁在平均场层面呈现出连续相变的特征。随后，我们建立了最优控制代价与势淬灭后非平衡弛豫所遵循的大偏差率函数之间的映射关系，该映射覆盖抛物型和双曲型类，而椭圆型类没有简单的弛豫对应物。这一对应关系意味着，最近发现的有限时间动力学相变（直接采样需付出指数级代价）可通过对最优控制轨迹的普通平均来获取。为验证理论发现，我们报道了三项基于光阱胶体粒子的实验：平均随机功的控制跃迁，以及自由扩散和谐波弛豫中的有限时间动力学相变。
英文摘要：
We present a theory for the optimal control of stochastic systems in structured environments, represented by penalty terms in the cost functional. We show that such control problems generically feature sharp finite-time transitions associated with a qualitative change in the control strategy at a critical time. Starting from an overdamped Langevin equation and a quadratic cost functional, we show that all resulting problems fall into three canonical equivalence classes (parabolic, hyperbolic, and elliptic), distinguished by the sign of the determinant of the control Hamiltonian. For each class, we obtain the optimal protocol, the cost function, and the critical time in closed form, and show that the transition exhibits features of a continuous phase transition at mean-field level. We then establish a mapping between the optimal control cost and the large-deviation rate function governing non-equilibrium relaxation after a potential quench. The mapping covers the parabolic and hyperbolic classes, while the elliptic class has no simple relaxation counterpart. This correspondence implies that recently discovered finite-time dynamical phase transitions, which are exponentially costly to sample directly, are accessible through ordinary averages over optimally controlled trajectories. To validate our theoretical findings, we report three experiments with optically trapped colloidal particles: a control transition for the mean stochastic work, and the finite-time dynamical phase transitions in free diffusion and in harmonic relaxation.
