Stability and inertia theorems for generalized Lyapunov equations
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.
scientific article; zbMATH DE number 1864747
scientific article