On Path-Complete Lyapunov Functions: Geometry and Comparison

We study optimization-based criteria for the stability of switching systems, known as path-complete Lyapunov functions, and ask the question “can we decide algorithmically when a criterion is less conservative than another?”. Our contribution is twofold. First, we show that a path-complete Lyapunov function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each path-complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path-complete Lyapunov functions.