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Feedback Stabilization and Finite Element Error Analysis of Viscous Burgers Equation around Non-Constant Steady State

Wasim Akram

TL;DR

The work addresses stabilizing the viscous Burgers equation around a non-constant steady state using interior controls and provides a comprehensive FE-based error analysis for the stabilized system. By linearizing about the steady state and solving an algebraic Riccati equation, the authors obtain a stabilizing feedback with exponential decay $-\omega$ for the linear problem, and show that a conforming FE discretization yields a uniformly stabilizable discrete system with decay $-\omega+\epsilon$. They extend the analysis to the nonlinear problem via a Banach fixed-point argument, obtaining local stabilization and quadratic convergence, and derive rigorous error estimates for both stabilized states and stabilizing controls, including discrete Riccati operators. Numerical implementations validate the theoretical results, demonstrate convergence rates, and illustrate stabilization in practice. The results advance the understanding of Riccati-based stabilization for nonlinear parabolic systems around non-constant steady states and provide a solid FE framework for error analysis and computation.

Abstract

In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay $-ω<0$ for any $ω>0$. The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay $-ω+ε$ for any $ε>0$. The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.

Feedback Stabilization and Finite Element Error Analysis of Viscous Burgers Equation around Non-Constant Steady State

TL;DR

The work addresses stabilizing the viscous Burgers equation around a non-constant steady state using interior controls and provides a comprehensive FE-based error analysis for the stabilized system. By linearizing about the steady state and solving an algebraic Riccati equation, the authors obtain a stabilizing feedback with exponential decay for the linear problem, and show that a conforming FE discretization yields a uniformly stabilizable discrete system with decay . They extend the analysis to the nonlinear problem via a Banach fixed-point argument, obtaining local stabilization and quadratic convergence, and derive rigorous error estimates for both stabilized states and stabilizing controls, including discrete Riccati operators. Numerical implementations validate the theoretical results, demonstrate convergence rates, and illustrate stabilization in practice. The results advance the understanding of Riccati-based stabilization for nonlinear parabolic systems around non-constant steady states and provide a solid FE framework for error analysis and computation.

Abstract

In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay for any . The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay for any . The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.
Paper Structure (22 sections, 32 theorems, 188 equations, 8 figures, 5 tables)

This paper contains 22 sections, 32 theorems, 188 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Literature survey
  4. Contributions
  5. Organization
  6. Preliminaries
  7. Linear system
  8. Analytic semigroup and spectral analysis
  9. Stabilizability
  10. Finite element approximation
  11. Approximated linear system
  12. Implementation
  13. Non-linear system
  14. Stabilizability of non-linear system
  15. Error estimates
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $f\in L^p(\Omega),$$g\in L^q(\Omega),$ and $h\in L^r(\Omega),$ where $1\le p, q ,r<\infty$ are such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1.$ Then $fgh\in L^1(\Omega)$ and

Figures (8)

  • Figure 1: $\normalfont\Sigma(-\widehat{\nu}; \theta_0)$ and $\color{red}{\Gamma}=\Gamma_+\cup \Gamma_-\cup\Gamma_0$
  • Figure 2: (Example 1.) (a) Evolution of computed solution $y_h$ in $L^2$-norm with time $t,$ (b) on $\log$-scale, and (c) $\log-\log$ plot of errors with discretization parameter $h$.
  • Figure 3: (Example 1.) (a) Evolution of stabilized solution, (b) on $\log$-scale, (c) evolution of stabilizing control and (d) error plot in $\log-\log$ scale.
  • Figure 4: (Example 2.) Log-log plots of errors against discretization parameter $h$.
  • Figure 5: (Example 3.) (a) Evolution of solution without control, (b) on log scale.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Proposition 2.1: Generalized Hölder's inequality
  • Lemma 2.2: Sobolev embedding
  • Lemma 2.3: Agmon's inequality Agmon10
  • Remark 3.1
  • Theorem 3.2: analytic semigroup
  • Remark 3.3: regularity
  • Proposition 3.4: adjoint semigroup
  • Proposition 3.5: properties of spectrum of $\mathcal{A}$
  • Lemma 3.6
  • Proposition 3.7: open loop stabilizability of $(\mathcal{A}_\omega,\mathcal{B})$
  • ...and 37 more