Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control

The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equat...

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Bibliographic Details
Published inIEEE transactions on automatic control Vol. 64; no. 4; pp. 1415 - 1425
Main Authors Prieur, Christophe, Trélat, Emmanuel
Format Journal Article
LanguageEnglish
Published Institute of Electrical and Electronics Engineers 01.04.2019
Subjects
Analysis of PDEs
Mathematics
Optimization and Control
Reaction-diffusion equation
Delay control
partial differential equation
Lyapunov function
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Summary:The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method.
ISSN:0018-9286
DOI:10.1109/TAC.2018.2849560