发表机构
Czech Technical University in Prague; CNRS; LAAS; Université de Toulouse（布拉格捷克理工大学; 法国国家科学研究中心；图卢兹大学拉特兰应用数学与计算机科学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种统一框架，结合径向紧化、时钟变量与折扣占用测度，通过可达性问题和二分法计算无限时间范围内动力系统轨迹的极值，并证明叠加原理以统一处理瞬态与渐近行为。
AI 中文摘要
这项工作研究了在无限时间范围内动力系统轨迹上可观测量的极值计算问题。该设定本质上是全局的：轨迹可能表现出大的瞬态偏移，仅当时间趋于无穷时才渐近地达到其最大值，或者在有限时间内逃逸至无穷。所提出的框架以统一的方式处理这些情景，使得所得的极值同时反映动力学的瞬态和渐近行为。主要困难在于时间间隔和状态空间均不假设为紧致。我们通过将状态空间的径向紧化与紧化的时钟变量和折扣占用测度相结合来解决这一问题。这通过占用测度及相关的Liouville方程，在紧集上给出了轨迹系的描述。极值随后通过可观测量水平集的一族可达性问题来表征，这些问题可以通过二分法求解。在理论方面，我们证明了具有时间分布终端测度的无限水平Liouville方程的叠加原理。该结果表明，测度公式表示停止的允许轨迹的系综，包括从不停止的轨迹，并建立了轨迹级与测度级描述之间的等价性。
英文摘要
This work addresses the computation of extreme values of observables along trajectories of dynamical systems over an \ks{infinite time horizon}. The setting is intrinsically global: trajectories may exhibit large transient excursions, approach their largest values only asymptotically as time tends to infinity, or escape to infinity in finite time. The proposed framework treats these \mk{ scenarios} in a unified way, so that the resulting extreme values reflect both the transient and the asymptotic behavior of the dynamics. The main difficulty is that neither the time interval nor the state space is assumed to be compact. We address this by combining a radial compactification of the state space with a compactified clock variable and discounted occupation measures. This yields a description of trajectory ensembles on compact sets via occupation measures and the associated Liouville's equation. Extreme values are then characterized through a family of reachability problems for level sets of the observable, which can be solved by a bisection procedure. On the theoretical side, we prove a superposition principle for an infinite-horizon Liouville's equation with a time-distributed terminal measure. This result shows that the measure formulation represents ensembles of stopped admissible trajectories, including trajectories \ks{that never stop}, and establishes equivalence between the trajectory-level and measure-level descriptions.
