Equivalent Conditions on Periodic Feedback Stabilization for Linear Periodic Evolution Equations

TL;DR

This paper establishes equivalent conditions for the periodic feedback stabilization of linear $T$-periodic evolution equations, linking controllability, spectral properties, and unique continuation, with applications to heat equations with periodic potentials.

Contribution

It provides new equivalent conditions for stabilization of periodic evolution equations, connecting controllability, spectral analysis, and unique continuation properties.

Findings

Equivalent conditions relate stabilization to the attainable subspace and Poincaré map.

Stabilizability is characterized by finite-dimensional subspace conditions.

Applications demonstrate the theory on heat equations with periodic potentials.

Abstract

This paper studies the periodic feedback stabilization for a class of linear $T$-periodic evolution equations.Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related with the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincar$\acute{e}$ map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on different time horizons $[0,T]$ and $[0,n_0T]$ (where $n_0$ is the sum of algebraic multiplicities of distinct unstable eigenvalues of the Poincar$\acute{e}$ map). It is also proved that a $T$-periodic controlled evolution equation is linear $T$-periodic feedback sabilizable if and only if it is linear $T$-periodic feedback sabilizable with respect to a finite dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.