Predictor-Based Output Feedback Stabilization of an Input Delayed Parabolic PDE with Boundary Measurement

TL;DR

This paper develops a predictor-based output feedback control method to stabilize 1-D reaction-diffusion PDEs with large input delays, using boundary measurements and finite-dimensional observers, ensuring exponential stability.

Contribution

It introduces a novel control strategy combining a finite-dimensional observer with a predictor to handle arbitrarily large input delays in boundary-controlled PDEs.

Findings

Achieves exponential stabilization for any input delay length.

Works with various boundary conditions and measurement types.

Requires the observer dimension to be sufficiently large.

Abstract

This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input delay that the control strategy achieves the exponential stabilization of the closed-loop system, for system trajectories evaluated in $H^1$ norm (also in $L^2$ norm in the case of a Dirichlet boundary measurement), provided the dimension of the observer is selected large enough. The reported proof of this result requires to perform both control design and stability analysis using simultaneously the (non-homogeneous) original version of the PDE and one of its equivalent homogeneous representations.