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

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.

[1]  Christophe Prieur,et al.  Feedback Stabilization of a 1-D Linear Reaction–Diffusion Equation With Delay Boundary Control , 2019, IEEE Transactions on Automatic Control.

[2]  Yury Orlov,et al.  Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control , 2004 .

[3]  D. Russell.  Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions , 1978 .

[4]  Christophe Prieur,et al.  Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems With Delay Boundary Control , 2021, IEEE Transactions on Automatic Control.

[5]  Christophe Prieur,et al.  PI Regulation of a Reaction–Diffusion Equation With Delayed Boundary Control , 2019, IEEE Transactions on Automatic Control.

[6]  Emmanuel Trélat,et al.  Global Steady-State Controllability of One-Dimensional Semilinear Heat Equations , 2004, SIAM J. Control. Optim..

[7]  Anton Selivanov,et al.  Boundary Delayed Observer-Controller Design for Reaction–Diffusion Systems , 2020, IEEE Transactions on Automatic Control.

[8]  Robert Shorten,et al.  A LMI condition for the Robustness of Linear Predictor Feedback with Respect to Time-Varying Input Delays , 2019, Autom..

[9]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[10]  Yuri V. Orlov,et al.  Discontinuous unit feedback control of uncertain infinite-dimensional systems , 2000, IEEE Trans. Autom. Control..

[11]  Rami Katz,et al.  Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs , 2020, Autom..

[12]  Rami Katz,et al.  Finite-dimensional control of the Kuramoto-Sivashinsky equation under point measurement and actuation , 2020, 2020 59th IEEE Conference on Decision and Control (CDC).

[13]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[14]  Rami Katz,et al.  Delayed finite-dimensional observer-based control of 1-D parabolic PDEs , 2021, Autom..

[15]  Emmanuel Trélat,et al.  GLOBAL STEADY-STATE STABILIZATION AND CONTROLLABILITY OF 1D SEMILINEAR WAVE EQUATIONS , 2006 .

[16]  Robert Shorten,et al.  Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay , 2020, Autom..

[17]  Emilia Fridman,et al.  Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement , 2021, European Journal of Control.

[18]  Yury Orlov,et al.  On General Properties of Eigenvalues and Eigenfunctions of a Sturm–Liouville Operator: Comments on “ISS With Respect to Boundary Disturbances for 1-D Parabolic PDEs” , 2017, IEEE Transactions on Automatic Control.

[19]  Florent Di Meglio,et al.  Late-lumping backstepping control of partial differential equations , 2019, Autom..

[20]  Mark J. Balas,et al.  Finite-dimensional controllers for linear distributed parameter systems: Exponential stability using residual mode filters , 1985 .

[21]  R. Shorten,et al.  An LMI Condition for the Robustness of Constant-Delay Linear Predictor Feedback with Respect to Uncertain Time-Varying Input Delays. , 2019 .

[22]  Joachim Deutscher,et al.  Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers , 2011, Int. J. Control.

[23]  R. Curtain.  Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input , 1982 .

[24]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .