Maximal $L^p$-regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems

Abstract

In this paper we present an abstract maximal $L^p$-regularity result up to $T=\infty$, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the $3d$-Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation.

References in corpus (3)

• Uniform stabilization of Navier-Stokes equations in critical $L^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
• Uniform stabilization of Boussinesq systems in critical ${\mathbf{L}}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
• Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory