TL;DR
This paper explores the fundamental role of Lyapunov equations in nonlinear control theory, linking stability and observability through fixed-point theorems, providing a new perspective on their solvability.
Contribution
It presents a novel fixed-point perspective on Lyapunov equations, connecting stability, observability, and solvability via Brouwer's fixed-point theorem.
Findings
Lyapunov equations relate to stability and observability.
Fixed-point theorems can be used to prove solvability.
New insights into nonlinear control theory.
Abstract
The Lyapunov equation is the gateway drug of nonlinear control theory. In these notes we revisit an elegant statement connecting the concepts of asymptotic stability and observability, to the solvability of Lyapunov equations, and discuss how this statement can be proved using the Brouwer fixed-point theorem.
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