  This paper presents a theoretical framework for the design and analysis of
gradient descent-based algorithms for coverage control tasks involving robot
swarms. We adopt a multiscale approach to analysis and design to ensure
consistency of the algorithms in the large-scale limit. First, we represent the
macroscopic configuration of the swarm as a probability measure and formulate
the macroscopic coverage task as the minimization of a convex objective
function over probability measures. We then construct a macroscopic dynamics
for swarm coverage, which takes the form of a proximal descent scheme in the
$L^2$-Wasserstein space. Our analysis exploits the generalized geodesic
convexity of the coverage objective function, proving convergence in the
$L^2$-Wasserstein sense to the target probability measure. We then obtain a
consistent gradient descent algorithm in the Euclidean space that is
implementable by a finite collection of agents, via a "variational"
discretization of the macroscopic coverage objective function. We establish the
convergence properties of the gradient descent and its behavior in the
continuous-time and large-scale limits. Furthermore, we establish a connection
with well-known Lloyd-based algorithms, seen as a particular class of
algorithms within our framework, and demonstrate our results via numerical
experiments.
