# Positive semigroups and abstract Lyapunov equations

Positive semigroups and abstract Lyapunov equations We consider abstract equations of the form $$\mathcal {A}x=-z$$ A x = - z on a locally convex space, where $$\mathcal {A}$$ A generates a positive semigroup and $$z$$ z is a positive element. This is an abstract version of the operator Lyapunov equation $$A^*P+PA=-Q$$ A ∗ P + P A = - Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that $$-\mathcal {A}$$ - A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that $$z$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive semigroups and abstract Lyapunov equations

, Volume 19 (1) – Mar 4, 2014
21 pages

1

/lp/springer_journal/positive-semigroups-and-abstract-lyapunov-equations-DJf0qNAnFM
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0279-3
Publisher site
See Article on Publisher Site

### Abstract

We consider abstract equations of the form $$\mathcal {A}x=-z$$ A x = - z on a locally convex space, where $$\mathcal {A}$$ A generates a positive semigroup and $$z$$ z is a positive element. This is an abstract version of the operator Lyapunov equation $$A^*P+PA=-Q$$ A ∗ P + P A = - Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that $$-\mathcal {A}$$ - A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that $$z$$ 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.

### Journal

PositivitySpringer Journals

Published: Mar 4, 2014

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