Diagonally weighted Hölder norms as Lyapunov functions for arbitrary switching linear systems — Discrete-time case

The paper focuses on the class of Lyapunov functions defined by diagonally weighted Holder vector p-norms for arbitrary switching linear systems with discrete-time dynamics. Our work is motivated by the simple forms of such Lyapunov function candidates, as well as by the mathematical perspective of getting a generalized result able to incorporate, as particular cases, several previous results, individually reported in literature. The presentation is organized in three parts. The first part provides a global characterization of the considered class of Lyapunov functions, which relies on the properties of induced p-norms for finite-dimensional operators, with 1 ≤ p ≤ ∞. The second part provides characterizations corresponding to p ϵ {1,2, ∞}, which exploit the theory of representatives associated with a family of nonnegative matrices built from the matrices of the constituent switched subsystems. The third part applies these characterization procedures to a numerical case study, and comments on the similarities / differences between the two distinct types of approaches.