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

Christophe Prieur 1 Emmanuel Trélat 2
1 GIPSA-SYSCO - SYSCO
GIPSA-DA - Département Automatique
2 CaGE - Control And GEometry
Inria de Paris, LJLL - Laboratoire Jacques-Louis Lions
Abstract : 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.
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Submitted on : Thursday, April 18, 2019 - 7:04:09 PM
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Christophe Prieur, Emmanuel Trélat. Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2019, 64 (4), pp.1415-1425. ⟨10.1109/TAC.2018.2849560⟩. ⟨hal-01583199v2⟩

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