This talk concerns a question that frequently occurs in various applications: Is any diffusive coupling of stable linear systems, also stable? Although it has been known for a long time that this is not the case, we shall identify a reasonably diverse class of systems for which it is true.
In this final part of the talk, I will focus on the question of when a finite collection of linear systems with a proper invariant cone, have a common linear Lyapunov function on the cone. The goal is to provide geometric conditions in terms of the system matrices, and the cone. We shall give a sufficient condition in case of arbitrary proper cones, and necessary and sufficient conditions in case of finitely generated proper cones. This proof leads to somewhat surprising observations such as: "The sum of two closed cones need not be closed", and "The image of a closed cone under a linear map need not be a closed."