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

TL;DR

This paper establishes maximal $L^p$-regularity results for linear parabolic PDEs with boundary feedback controls, enabling advanced analysis and stabilization of complex systems like Navier-Stokes and Boussinesq equations.

Contribution

It provides a novel abstract maximal $L^p$-regularity framework applicable to PDEs with boundary feedback controls, including complex fluid and thermal systems.

Findings

Maximal $L^p$-regularity up to infinite time for controlled PDEs.

Application to stabilization of Navier-Stokes and Boussinesq systems.

Framework facilitates analysis of 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.