P0-property and linear inequalities in positive systems analysis

A matrix M ∊ R^n×n is a P0-matrix if every principal minor of M is nonnegative. We use this concept in a generalized form, called row (column)-P0-property, which refers to a finite set of matrices M = {M¹,…, M^N} ⊂ R^n×n. In the current work, the set M collects matrices of the form M^θ = A^θ − rI, θ = 1,…, N, with A^θ ∊ R^n×n essentially nonnegative and Hurwitz stable, and r < 0. Relying on the P0-property of M, we investigate the existence of nonnegative vectors v ∊Rⁿ (depending on r<0) that solve the inequalities v^T A^θ ≤ rv^T, θ = 1,…, N. The exploration of such inequalities is strongly related to the construction of Lyapunov functions for switching positive systems.