Stability and inertia theorems for generalized Lyapunov equations
type: Item
instance of: scholarly article
title: Stability and inertia theorems for generalized Lyapunov equations
zbMATH Open document ID: 1016.15010
DOI: 10.1016/S0024-3795(02)00354-3
author: Tatjana Stykel
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: The author considers the continuous-time Lyapunov equation \(E^*XA+A^*XE=-G\) and its discrete-time analog \(A^*XA-E^*XE=-G\) with given matrices \(E\), \(A\), \(G\) and unknown matrix \(X\). Such equations arise in control problems, stability theory for differential and difference equations and in problems of spectral dichotomy. Generalizations of Lyapunov stability theorems are presented for such equations as well as some matrix inertia theorems for matrix pencils \(\lambda E-A\) (w.r.t. the imaginary axis and w.r.t. the unit circle). The author discusses applications of generalized Lyapunov equations with special right-hand sides in stability theory and control problems for descriptor systems.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 1864747
zbMATH Keywords: generalized Lyapunov equations
zbMATH Keywords: descriptor systems
zbMATH Keywords: inertia
zbMATH Keywords: controllability
zbMATH Keywords: observability
zbMATH Keywords: continuous-time Lyapunov equation
zbMATH Keywords: stability
zbMATH Keywords: differential equation
zbMATH Keywords: difference equation
zbMATH Keywords: spectral dichotomy
zbMATH Keywords: matrix pencils
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1864747