Distributed coverage while not being covered

We consider the problem of cooperatively covering a group of static targets while simultaneously minimizing exposure from a different set of static locations. Starting from the work of Bullo et al. [8] who formalized the problem of cooperative sensor coverage as a multicenter optimization problem, we show that also this problem can be formulated and solved using related concepts. However, we evidence that the resulting function to be optimized loses some desirable properties and requires a more sophisticated controller. After having identified the peculiar aspects of this new function to be optimized and having studied its critical points, we provide a controller and show that it will drive the system to a stable local optimizer. The control strategy is distributed in the sense of Voronoi, and it implements a distributed version of the gradient projection method known in the optimization literature. Numerical simulations illustrate how the controller works, and we conclude sketching some theoretical properties.