2014/01/28 by Sergiy Koshkin, Koshkin, Sergiy
Computer Science · Engineering · Mathematics · #34D20 #47B65 #47D06 #49K27 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1401.7059
openalex publication_date 2014/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.