On some properties of diagonally stable polytopic systems

For polytopic matrices and polytopic systems we introduce the concept of diagonal stability with respect to a p-norm, 1 ≤ p ≤ ∞. We address both the discrete-time and the continuous-time cases. We prove the equivalence between this algebraic concept and the following properties of uncertain linear dynamics described by polytopic models: · the existence of diagonal Lyapunov functions defined by weighted p-norm; · the existence of contractive sets shaped by weighted p-norm, which are invariant with respect to the system trajectories. We also discuss the problem of the best rate for the Lyapunov function decrease and, respectively, for the invariant set contraction. A numerical procedure has been developed to support this exploration.