  We present theoretical and numerical results concerning the problem to find
the path that minimizes the time to navigate between two given points in a
complex fluid under realistic navigation constraints. We contrast deterministic
Optimal Navigation (ON) control with stochastic policies obtained by
Reinforcement Learning (RL) algorithms. We show that Actor-Critic RL algorithms
are able to find quasi-optimal solutions in the presence of either
time-independent or chaotically evolving flow configurations. For our
application, ON solutions develop unstable behavior within the typical duration
of the navigation process, and are therefore not useful in practice. We first
explore navigation of turbulent flow using a constant propulsion speed. Based
on a discretized phase-space, the propulsion direction is adjusted with the aim
to minimize the time spent to reach the target. Further, we explore a case
where additional control is obtained by allowing the engine to power off.
Exploiting advection of the underlying flow, allows the target to be reached
with less energy consumption. In this case, we optimize a linear combination
between the total navigation time and the total time the engine is switched
off. Our approach can be generalized to other setups, for example, navigation
under imperfect environmental forecast or with different models for the moving
vessel.
