On Riccati equations in Banach algebras

TL;DR

This paper investigates the existence of solutions to Riccati equations within Banach algebras, providing counterexamples, sufficient conditions, and applications to distributed systems.

Contribution

It offers new insights into Riccati equations in Banach algebras, including counterexamples to prior results and conditions for solutions.

Findings

Counterexample to previous Riccati equation results

Sufficient conditions for solution existence in Banach algebras

Applications to spatially distributed systems

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 $$ PBB^\star P-PA-A^\star P-C^\star C=0 $$ 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.