Diagonal stability of interval matrices and applications
An interval matrix \({\mathcal A}^I= \{A\in\mathbb{R}^{n\times n}\mid A^-\leq A\leq A^+\}\) is called Schur stable, respectively, Hurwitz stable if the eigenvalues of all matrices \(A\in{\mathcal A}^I\) lie in the open unit disc of the complex plane, respectively, in its open left half plane. It is called Schur, respectively, Hurwitz diagonally stable relative to a Hölder \(p\)-norm \(\|\cdot\|_p\) \((1\leq p\leq\infty)\) if there exists a positive definite diagonal matrix \(D\) such that \[ \| A\|^D_p:=\| D^{-1} AD\|_p< 1, \] respectively \[ m^D_p(A):= \lim_{h\downarrow 0}\,h^{-1}(\| I+ hA\|^D_p- 1)< 0 \] holds for all matrices \(A\in{\mathcal A}^I\). The first part of the paper provides criteria for these latter types of stability. It presents methods for finding \(D\), analyses the robustness and investigates the connection with the standard concept of Schur and Hurwitz stability for interval matrices. The second part considers an equivalence of Schur respectively Hurwitz diagonal stability of \({\mathcal A}^I\) with some properties of a discrete- or continuous-time dynamical interval system whose motion is described by \({\mathcal A}^I\).
(diagonally invariant) exponential stability
scientific article; zbMATH DE number 5804418
scientific article