Coverage control and function approximation via semiconcave and semiconvex duality

A min-of-quadratics representation for the cost function underlying a popular approach to classical coverage control is used to motivate and develop connections to function approximation via semiconcave and semiconvex duality. By considering functions that are both semiconcave and semiconvex, upper and lower bounds are developed via their semiconcave and semiconvex relaxations. Min-of-quadratics and max-of-quadratics representations for these bounds are provided by construction, using quadratic basis functions that are synonymous with agents. These representations induce a pair of Voronoi tessellations, whose constituent polytopes delineate regions of space where agents are individually responsible for the upper and lower bounds. The difference in these bounds, weighted by the usual notion of density function, defines a function approximation error that is differentiable via Leibniz. Using the derivatives obtained, a generalized distributed control architecture is developed for minimizing the function approximation error involved. In the special case of spatially invariant function approximation, it is observed that the well-known classical coverage control architecture is recovered.