The Apollonian metric is a generalization of the hyperbolic metric. It is defined in arbitrary domains in ${ℝ}^n$. In this paper, we derive optimal comparison results between this metric and the $j_G$ metric in a large class of domains. These results allow us to prove that Euclidean bilipschitz mappings have small Apollonian bilipschitz constants in a domain $G$ if and only if $G$ is a ball or half-space.

For affine control systems, we study the relationship between an optimal regulation problem on the infinite horizon and stabilizability. We are interested in the case the value function of the optimal regulation problem is not smooth and feedback laws involved in stabilizability may be discontinuous.