The quest for the optimal navigation strategy in a complex environment is at the heart of microswimmer applications like cargo carriage or drug targeting to cancer cells. In this talk we formulate a variational Fermat's principle which determines the optimal path allowing a self-propelled active particle (which can freely steer but moves with a preferred speed) to reach a target fastest. For piecewise constant force or flow fields the principle leads to Snell's law from geometrical optics, showing that the optimal path to the target is piecewise linear, as for light rays, but with a generalized refraction law. For complex environments, like general 1D-, shear- or vortex-fields, we obtain exact analytical expressions for the optimal path, showing, for example, that microswimmers sometimes have to temporarily navigate away from their target to reach it fastest.
In the second half of this talk, we focus on microswimmers, which in contrast to dry active particles, create a characteristic flow field. This flow field induces hydrodynamic interactions with remote walls or obstacles which oblige the swimmers to take significant detours to reach their target fastest, even in the absence of external fields. Such strategic detours are particularly useful in
the presence of fluctuations: they effectively protect microswimmers against fluctuations and allow them to reach a target (e.g. a food source) up to twice faster than when greedily heading straight towards it.