On special quadratic Lyapunov functions for linear dynamical systems with an invariant cone

TL;DR

This paper proves that asymptotically stable linear systems with certain invariant cones admit specially structured quadratic Lyapunov functions, with implications for systems like positive and ice-cream cone systems.

Contribution

It establishes the existence of structured quadratic Lyapunov functions for systems with invariant cones, including self-dual, homogeneous, and specific cone cases.

Findings

Existence of special quadratic Lyapunov functions for systems with invariant cones.

Linear complexity of the Lyapunov function relative to system dimension.

Derived a new quadratic Lyapunov function for ice-cream cone invariant systems.

Abstract

We consider a continuous-time linear time-invariant dynamical system that admits an invariant cone. For the case of a self-dual and homogeneous cone we show that if the system is asymptotically stable then it admits a quadratic Lyapunov function with a special structure. The complexity of this Lyapuonv function scales linearly with the dimension of the dynamical system. In the particular case when the cone is the nonnegative orthant this reduces to the well-known and important result that a positive system admits a diagonal Lyapunov function. We demonstrate our theoretical results by deriving a new special quadratic Lyapunov function for systems that admit the ice-cream cone as an invariant set.