Solvability of the $H^\infty$ algebraic Riccati equation in Banach algebras

TL;DR

This paper establishes conditions for solving the $H^$ algebraic Riccati equation within Banach algebras, with implications for spatially distributed systems, expanding the theoretical framework for control in infinite-dimensional settings.

Contribution

It provides new sufficient conditions for the existence of stabilizing solutions to the $H^$ Riccati equation in Banach algebra contexts, extending control theory.

Findings

Sufficient conditions for Riccati equation solvability in Banach algebras.

Application to spatially distributed systems.

Enhanced understanding of control in infinite-dimensional spaces.

Abstract

Let $R$ be a commutative complex unital semisimple Banach algebra with the involution $\cdot ^\star$. Sufficient conditions are given for the existence of a stabilizing solution to the $H^\infty$ Riccati equation when the matricial data has entries from $R$. Applications to spatially distributed systems are discussed.