Year: 2014
Type: preprint
Abstract: We consider abstract equations of the form Ax=-z on a locally convex space, where A generates a positive semigroup and z is a positive element. This is an abstract version of the operator Lyapunov equation A*P+PA=-Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that -A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that z is an order unit, or that the space contains any order units. As an application, we generalize Wonham's theorem on the operator Lyapunov equations with detectable right hand sides to reflexive Banach spaces.
Source: arXiv (Cornell University)
Author: Koshkin, Sergiy
Institutions: University of Houston - Downtown, University of Houston
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951225014