Properties of eventually positive linear input–output systems

In this study, the author considers systems with trajectories originating in the non-negative orthant and becoming non-negative after some finite time transient. These systems are called eventually positive and the results are based on recent theoretical developments in linear algebra. The author considers dynamical systems (i.e. fully observable systems with no inputs), for which they compute forward-invariant cones and Lyapunov functions. They then extend the notion of eventually positive systems to the input–output system case. The extension is performed in such a manner, that some valuable properties of classical internally positive input–output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have non-negative derivatives. The author illustrates the theoretical results on numerical examples.