Diagonal stability of interval matrices and applications

Abstract

Let A I = { A ∈ R n × n | A - ⩽ A ⩽ A + } be an interval matrix and 1 ⩽ p ⩽ ∞ . We introduce the concept of Schur and Hurwitz diagonal stability, relative to the Hölder p-norm, of A I , abbreviated as SDS p and HDS p , respectively. This concept is formulated in terms of a matrix inequality using the p-norm, which must be satisfied by the same positive definite diagonal matrix for all A ∈ A I . The inequality form is different for SDS p and HDS p . The particular case of p = 2 is equivalent to the condition of quadratic stability of A I . The SDS 2 inequality is equivalent to the Stein inequality ∀ A ∈ A I : A T PA - P ≺ 0 , and the HDS 2 inequality is equivalent to the Lyapunov inequality ∀ A ∈ A I : A T P + PA ≺ 0 ; in both cases P is a positive definite diagonal matrix and the notation “ ≺ 0 ” means negative definite. The first part of the paper • provides SDS p and HDS p criteria, • presents methods for finding the positive definite diagonal matrix requested by the definition of SDS p and HDS p , • analyzes the robustness of SDS p and HDS p and • explores the connection with the Schur and Hurwitz stability of A I . The second part shows that the SDS p or HDS p of A I is equivalent to the following properties of a discrete- or continuous-time dynamical interval system whose motion is described by A I : • the existence of a strong Lyapunov function defined by the p-norm and • the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system’s trajectories.