Matrix Measures in the Qualitative Analysis of Parametric Uncertain Systems

The paper considers parametric uncertain systems of the form ẋ t Mx t ,M ∈ M,M ⊂ Rn×n, where M is either a convex hull, or a positive cone of matrices, generated by the set of vertices V {M1,M2, . . . ,MK} ⊂ Rn×n. Denote by μ‖ ‖ the matrix measure corresponding to a vector norm ‖ ‖. When M is a convex hull, the condition μ‖ ‖ Mk ≤ r < 0, 1 ≤ k ≤ K, is necessary and sufficient for the existence of common strong Lyapunov functions and exponentially contractive invariant sets with respect to the trajectories of the uncertain system. When M is a positive cone, the condition μ‖ ‖ Mk ≤ 0, 1 ≤ k ≤ K, is necessary and sufficient for the existence of common weak Lyapunov functions and constant invariant sets with respect to the trajectories of the uncertain system. Both Lyapunov functions and invariant sets are described in terms of the vector norm ‖ ‖ used for defining the matrix measure μ‖ ‖. Numerical examples illustrate the applicability of our results.