Rapid stabilization of general linear systems with F-equivalence

TL;DR

This paper presents a method for rapidly stabilizing general linear systems with a Riesz basis of eigenvectors using an F-equivalence approach, leading to explicit feedback operators and improved stabilization conditions.

Contribution

It introduces new sufficient conditions for rapid stabilization of linear systems and constructs explicit feedback operators using Fredholm transformations, extending results to non-parabolic operators.

Findings

Achieves exponential stabilization with arbitrarily large decay rate.

Provides explicit feedback operator construction.

Improves existing conditions for non-parabolic systems.

Abstract

We study the rapid stabilization of general linear systems, when the differential operator $\mathcal{A}$ has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an $F$-equivalence approach \textcolor{black}{relying on Fredholm transformation} to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew-adjoint systems.