Interval systems - construction and exploitation of diagonal Lyapunov functions

Abstract

Our paper proposes a framework that allows the construction of several types of diagonal Lyapunov functions for discrete- and continuous-time interval systems. The Lyapunov function candidates are defined as weighted vector norms, where the weighting matrices are diagonal and positive definite. For the existence of such Lyapunov functions, we provide necessary and sufficient conditions in the case of 1 and ℝ norms, as well as two sufficient conditions in the case of 2-norm. The results offer construction procedures focusing on diagonal Lyapunov functions with the fastest decreasing rate, fact that also ensures the identification of the fastest contractive invariant sets shaped in accordance with the aforementioned norms. The exposition covers the discrete- and continuous-time dynamics, in parallel, aiming to emphasize both similarities and differences between the two types of results. The applicability of the theoretical developments is illustrated by a numerical example.