I. Introduction
We consider the general problem of finding and patrolling an underwater perimeter with UUVs that use a scalar sensor. We defme a perimeter as a curve of constant concentration, i.e. a curve that satisfies where is a two-dimensional scalar field, is the curve parameter, and is the perimeter concentration. Starting from an initial configuration, a team of UUVs must find the perimeter and maintain a lock on it This problem is of practical importance for thin layer monitoring [1], harmful algae bloom monitoring [2], tactical oceanography [3]–[6], harbor protection [7], and possibly many other problems.