On Riccati equations in Banach algebras

Abstract

Let $R$ be a commutative complex Banach algebra with the involution ${\cdot }^{\star }$ and suppose that $A\in R^{n\times n}$, $B\in R^{n\times m}$, $C\in R^{p\times n}$. The question of when the Riccati equation $$PB{B}^{\star }P-PA-A^{\star }P-C^{\star }C=0$$ has a solution $P\in R^{n\times n}$ is investigated. A counterexample to a previous result in the literature on this subject is given, followed by sufficient conditions on the data guaranteeing the existence of such a $P$. Finally, applications to spatially distributed systems are discussed.

arXiv commentSubmitted to a journal on 20 August