Finite-time Lyapunov exponents (FTLEs) provide a powerful approach to compute time-varying analogs of invariant manifolds in unsteady fluid flow fields. These manifolds are useful to visualize the transport mechanisms of passive tracers advecting with the flow. However, many vehicles and mobile sensors are not passive, but are instead actuated according to some intelligent trajectory planning or control law; for example, model predictive control and reinforcement learning are often used to design energy-efficient trajectories in a dynamically changing background flow. In this work, we investigate the use of FTLE on such controlled agents to gain insight into optimal transport routes for navigation in known unsteady flows. We find that these controlled FTLE (cFTLE) coherent structures separate the flow field into different regions with similar costs of transport to the goal location. These separatrices are functions of the planning algorithm's hyper-parameters, such as the optimization time horizon and the cost of actuation. Computing the invariant sets and manifolds of active agent dynamics in dynamic flow fields is useful in the context of robust motion control, hyperparameter tuning, and determining safe and collision-free trajectories for autonomous systems. Moreover, these cFTLE structures provide insight into effective deployment locations for mobile agents with actuation and energy constraints to traverse the ocean or atmosphere.

翻译：有限时间李雅普诺夫指数（FTLE）提供了一种强有力的方法，用于计算非定常流场中不变流形的时变类似物。这些流形有助于可视化随流输运的被动示踪剂的传输机制。然而，许多运载器与移动传感器并非被动，而是依据某种智能轨迹规划或控制律进行驱动；例如，模型预测控制与强化学习常用于在动态变化背景流中设计节能轨迹。本文探究了将FTLE应用于此类受控智能体，以获取在已知非定常流中导航的最优输运路径的洞见。我们发现，这些受控FTLE（cFTLE）相干结构将流场划分为不同区域，每个区域具有到达目标位置相近的输运成本。这些分界线是规划算法超参数（如优化时域与驱动成本）的函数。在动态流场中计算主动智能体动力学的不变集与流形，对于自主系统鲁棒运动控制、超参数调优以及安全无碰撞轨迹确定具有重要意义。此外，这些cFTLE结构为具有驱动与能量约束的移动智能体穿越海洋或大气时，如何选择有效部署位置提供了洞见。