Positive semigroups and abstract Lyapunov equations

Research paper by Sergiy Koshkin

Indexed on: 04 Mar '14Published on: 04 Mar '14Published in: Positivity

Abstract

We consider abstract equations of the form $$\mathcal {A}x=-z$$ on a locally convex space, where $$\mathcal {A}$$ 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 $$-\mathcal {A}$$ 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.