Feeble fish in time-dependent waters and homogenization of the G-equation

TL;DR

This paper investigates the ability of a fish with limited speed to reach a destination in time-dependent flows, demonstrating it can do so and applying the results to homogenize the G-equation.

Contribution

It extends previous work by analyzing fish navigation in time-dependent waters and applies findings to the homogenization of the G-equation.

Findings

Fish can reach any destination in finite time despite flow complexity

Time-dependent flows do not prevent controlled navigation

Application to homogenization of the G-equation

Abstract

We study the following control problem. A fish with bounded aquatic locomotion speed swims in fast waters. Can this fish, under reasonable assumptions, get to a desired destination? It can, even if the flow is time-dependent. Moreover, given a prescribed sufficiently large time $t$, it can be there at exactly the time $t$. The major difference from our previous work is the time-dependence of the flow. We also give an application to homogenization of the G-equation.