  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.
