Stability of Linear Difference and Differential Inclusions

Abstract

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius ρ(B) arbitrarily close to ρ(A). A difference inclusion $\begin{gathered} x^{k + 1} \in {\text{ }}\mathbb{A}x^k \hfill \\ \mathbb{A} \hfill \\ \end{gathered}$, where $\mathbb{A}$ is a compact set of matrices, is asymptotically stable if and only if $\mathbb{A}$ can be extended to a set $\mathbb{B}$ of nonnegative matrices B with $\left\| B \right\|_{_1 } < {\text{1 }}$ or $\left\| B \right\|_\infty < {\text{1}}$. Similar results are derived for differential inclusions.