Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators

¹ Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, 8010 Graz, Austria.
² Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenberger Straße 69, 4040 Linz, Austria.

^* Corresponding author: sergio.rodrigues@ricam.oeaw.ac.at

Received: 29 November 2017
Accepted: 24 September 2018

Abstract

An explicit feedback controller is proposed for stabilization of linear parabolic equations, with a time-dependent reaction–convection operator. The range of the feedback controller is finite-dimensional, and is typically modeled by indicator functions of small subdomains. Its dimension depends polynomially on a suitable norm of the reaction–convection operator. A sufficient condition for stabilizability is given, which involves the asymptotic behavior of the eigenvalues of the (time-independent) diffusion operator, the norm of the reaction–convection operator, and the norm of the nonorthogonal projection onto the controller’s range along a suitable infinite-dimensional (higher-modes) eigenspace. To construct the explicit feedback, the essential step consists in computing the nonorthogonal projection. Numerical simulations are presented, in 1D and 2D, showing the practicability of the controller and its response to measurement errors, where the actuators are indicator functions of suitable small subsets.