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Diagonal stability of interval matrices and applications. (English) Zbl 1204.15028

An interval matrix ${𝒜}^{I}=\left\{A\in {ℝ}^{n×n}\mid {A}^{-}\le A\le {A}^{+}\right\}$ is called Schur stable, respectively, Hurwitz stable if the eigenvalues of all matrices $A\in {𝒜}^{I}$ 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 ${\parallel ·\parallel }_{p}$ $\left(1\le p\le \infty \right)$ if there exists a positive definite diagonal matrix $D$ such that

${\parallel A\parallel }_{p}^{D}:={\parallel {D}^{-1}AD\parallel }_{p}<1,$

respectively

${m}_{p}^{D}\left(A\right):=\underset{h↓0}{lim}\phantom{\rule{0.166667em}{0ex}}{h}^{-1}{\left(\parallel I+hA\parallel }_{p}^{D}-1\right)<0$

holds for all matrices $A\in {𝒜}^{I}$.

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 ${𝒜}^{I}$ with some properties of a discrete- or continuous-time dynamical interval system whose motion is described by ${𝒜}^{I}$.

##### MSC:
 15A42 Inequalities involving eigenvalues and eigenvectors 15A60 Applications of functional analysis to matrix theory 15B48 Positive matrices and their generalizations; cones of matrices 34A30 Linear ODE and systems, general 34C14 Symmetries, invariants (ODE) 34D20 Stability of ODE 93C05 Linear control systems 93C41 Control problems with incomplete information 93D20 Asymptotic stability of control systems 65G30 Interval and finite arithmetic