[Paper Review] Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with a either Dirichlet or Neumann boundary measurement

Hugo Lhachemi, Christophe Prieur|arXiv (Cornell University)|Aug 24, 2021

Stability and Controllability of Differential EquationsEngineering24 references48 citations

TL;DR

The paper develops finite-dimensional observer-based boundary control for reaction-diffusion PDEs with either Dirichlet or Neumann boundary measurements, providing LMI-based design conditions that guarantee H1-stability when the observer order is large enough.

ABSTRACT

This paper investigates the output feedback boundary control of reaction-diffusion equations with either distributed or boundary measurement by means of a finite-dimensional observer. A constructive method dealing with the design of finite-dimensional observers for the feedback stabilization of reaction-diffusion equations was reported in a recent paper in the case where either the control or the observation operator is bounded and also satisfies certain regularity assumptions. In this paper, we go beyond by demonstrating that a finite-dimensional state-feedback combined with a finite-dimensional observer can always be successfully designed in order to achieve the Dirichlet boundary stabilization of reaction-diffusion PDEs with a either Dirichlet or Neumann boundary measurement.

Motivation & Objective

• Motivate and enable output feedback boundary stabilization for parabolic PDEs using finite-dimensional observers.
• Extend prior finite-dimensional observer designs to boundary measurements (Dirichlet or Neumann).
• Provide constructive LMI-based conditions guaranteeing closed-loop stability in the H1 norm.
• Show that stability can be achieved by increasing the number of observed modes and using a scaling approach.

Proposed method

• Use spectral (modal) reduction to express the PDE in terms of its eigenmodes.
• Design a finite-dimensional observer for the first N modes and a state-feedback on the first N0 modes.
• Formulate Lyapunov-based LMIs (Theta1, Theta2) to certify exponential stability with an observer–controller loop.
• Introduce a scaling of outputs when measurements are boundary-based to keep LMIs feasible as N grows.
• Provide Dirichlet-control with Dirichlet or Neumann measurements (Sections 3–5) and prove stability under stated LMI conditions.
• Apply an auxiliary integral action to handle boundary control and ensure convergence in H1.

Experimental results

Research questions

• RQ1 How can finite-dimensional observers be designed to stabilize reaction-diffusion PDEs under boundary measurements?
• RQ2 Can a unified LMI-based design be developed that works for both Dirichlet and Neumann boundary observations?
• RQ3 What conditions on the observer order and LMIs ensure exponential stability in the H1 norm?
• RQ4 How does scaling of outputs aid feasibility of the LMIs as the number of observed modes increases?
• RQ5 Are the proposed designs constructive and applicable to both Dirichlet and Neumann observation scenarios?

Key findings

• A constructive method to design finite-dimensional observers for boundary-stabilized reaction-diffusion PDEs is provided.
• Stability is guaranteed under LMIs Theta1 and Theta2 with sufficiently large observer order N, for both Dirichlet and Neumann observations.
• A scaling procedure ensures that the LMIs remain feasible as N grows, enabling stabilization by observing a large number of modes.
• Exponential stability in the H1 norm is achieved for the closed-loop system when the stated conditions hold.
• The approach extends prior results to the more challenging case of boundary measurements, including Dirichlet and Neumann cases.