Diagonal stability of interval matrices and applications

Let AI={A∈Rn×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 AI, abbreviated as SDSp and HDSp, 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∈AI. The inequality form is different for SDSp and HDSp. The particular case of p=2 is equivalent to the condition of quadratic stability of AI. The SDS2 inequality is equivalent to the Stein inequality ∀A∈AI:ATPA-P≺0, and the HDS2 inequality is equivalent to the Lyapunov inequality ∀A∈AI:ATP+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 SDSp and HDSp criteria,
• presents methods for finding the positive definite diagonal matrix requested by the definition of SDSp and HDSp,
• analyzes the robustness of SDSp and HDSp and
• explores the connection with the Schur and Hurwitz stability of AI. The second part shows that the SDSp or HDSp of AI is equivalent to the following properties of a discrete- or continuous-time dynamical interval system whose motion is described by AI:
• 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.