Sufficient criteria for stabilization properties in Banach spaces

TL;DR

This paper establishes general criteria for the stabilization of linear control systems in Banach spaces, extending previous results and applying to fractional elliptic operators with thick control sets.

Contribution

It provides unified sufficient conditions for stabilizability in Banach spaces, generalizing earlier Hilbert space results and applicable to fractional elliptic operators.

Findings

Criteria depend on a single spectral parameter

Constants are independent of growth rate

Applicable to fractional elliptic operators in $L_p$ spaces

Abstract

We study abstract sufficient criteria for open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in $L_p(\mathbb{R}^d)$ for $p\in [1,\infty)$ and thick control sets.