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）为计算非定常流场中不变流形的时变对应物提供了有力方法。这些流形可用于可视化随流动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，而是根据某种智能轨迹规划或控制律进行驱动；例如，在动态变化的背景流中，常用模型预测控制与强化学习设计高能效轨迹。本研究探讨将FTLE应用于此类受控智能体，以揭示已知非定常流场中导航的最优输运路径。我们发现，这些受控FTLE（cFTLE）相干结构将流场划分为不同的区域，这些区域具有抵达目标位置的相似输运代价。此类分割流形的形态取决于规划算法的超参数（如优化时域与驱动代价）。计算动态流场中主动智能体动力学的不变集与流形，对自主系统的鲁棒运动控制、超参数调优以及安全无碰撞轨迹的确定具有重要价值。此外，这些cFTLE结构为部署具有驱动与能量约束的移动智能体以穿越海洋或大气提供了有效位置选择的依据。