[Paper Review] Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control

Christophe Prieur, Emmanuel Trélat|arXiv (Cornell University)|Sep 7, 2017

Stability and Controllability of Differential Equations22 references114 citations

TL;DR

The paper designs a finite-dimensional delay controller for the unstable part of a 1D reaction-diffusion PDE with delayed Dirichlet boundary control and proves exponential stabilization of the full PDE. A constructive Artstein-based approach and Lyapunov analysis are used.

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.

Motivation & Objective

• Motivate and address the stabilization of a 1D reaction-diffusion equation with boundary input delay.
• Develop a constructive stabilization method that handles an unbounded boundary control operator.
• Split the infinite-dimensional system into unstable finite-dimensional and stable infinite-dimensional parts.
• Design a delayed finite-dimensional controller for the unstable part and prove it stabilizes the whole PDE.
• Provide numerical illustration of the proposed design method.

Proposed method

• Transform the delayed boundary control problem into a coupled PDE-transport system to model the delay.
• Perform spectral reduction to decompose the state into an unstable finite-dimensional part and a stable infinite-dimensional part.
• Apply Artstein model reduction to the finite-dimensional part to remove the input delay and use pole-shifting to design a stabilizing gain.
• Invert the Artstein transform to express the feedback as a function of the original state and derive a Lyapunov function for the closed-loop system.
• Construct a Lyapunov functional V_D combining the finite-dimensional Lyapunov term and a dissipative term for the infinite-dimensional part, and show exponential decay.
• Provide a numerical illustration of the design method.

Experimental results

Research questions

• RQ1 Can a boundary control with constant delay stabilize a 1D reaction-diffusion equation that is unstable without control?
• RQ2 Is it sufficient to stabilize only the finite-dimensional unstable part to guarantee stabilization of the entire infinite-dimensional system?
• RQ3 How can Artstein transformation be used to design an explicit, implementable delayed feedback for the unstable modes?
• RQ4 Does the proposed Lyapunov functional ensure exponential stability of the full cascaded PDE-transport system?
• RQ5 How does the delay D affect the design and stability, and can the method accommodate arbitrary delays?

Key findings

• There exists a stabilizing boundary feedback for the delayed Dirichlet boundary control of the 1D reaction-diffusion equation for any D ≥ 0.
• A finite-dimensional autonomous linear controller with delayed input can be designed via Artstein reduction and pole-shifting to stabilize the unstable finite-dimensional part.
• The designed feedback, when inverted back to the original variables, yields a retarded control that stabilizes the entire PDE system exponentially.
• A Lyapunov function combining a finite-dimensional quadratic term and a dissipative infinite-dimensional term proves exponential stability of the closed-loop system.
• The approach does not require any smallness assumption on the delay D and includes a constructive numerical illustration.