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Diagonal stability of interval matrices and applications. (English) Zbl 1204.15028
An interval matrix ${\Cal A}^I= \{A\in\bbfR^{n\times n}\mid A^-\le A\le A^+\}$ is called Schur stable, respectively, Hurwitz stable if the eigenvalues of all matrices $A\in{\Cal A}^I$ 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 $\Vert\cdot\Vert_p$ $(1\le p\le\infty)$ if there exists a positive definite diagonal matrix $D$ such that $$\Vert A\Vert^D_p:=\Vert D^{-1} AD\Vert_p< 1,$$ respectively $$m^D_p(A):= \lim_{h\downarrow 0}\,h^{-1}(\Vert I+ hA\Vert^D_p- 1)< 0$$ holds for all matrices $A\in{\Cal A}^I$. 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 ${\Cal A}^I$ with some properties of a discrete- or continuous-time dynamical interval system whose motion is described by ${\Cal A}^I$.

15A42Inequalities involving eigenvalues and eigenvectors
15A60Applications of functional analysis to matrix theory
15B48Positive matrices and their generalizations; cones of matrices
34A30Linear ODE and systems, general
34C14Symmetries, invariants (ODE)
34D20Stability of ODE
93C05Linear control systems
93C41Control problems with incomplete information
93D20Asymptotic stability of control systems
65G30Interval and finite arithmetic
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