Monotonicity and Contraction on Polyhedral Cones

In this note, we study monotone dynamical systems with respect to polyhedral cones. Using the half-space representation and the vertex representation, we propose three equivalent conditions to certify monotonicity of a dynamical system with respect to a polyhedral cone. We then introduce the notion of gauge norm associated with a cone and provide closed-from formulas for computing gauge norms associated with polyhedral cones. A key feature of gauge norms is that contractivity of monotone systems with respect to them can be efficiently characterized using simple inequalities. This result generalizes the well-known criteria for Hurwitzness of Metzler matrices and provides a scalable approach to search for Lyapunov functions of monotone systems with respect to polyhedral cones. Finally, we study the applications of our results in transient stability of dynamic flow networks and in scalable control design with safety guarantees.