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Feedback Stabilizability of a generalized Burgers-Huxley equation with kernel around a non-constant steady state

Wasim Akram, Manika Bag, Manil T. Mohan

TL;DR

The paper analyzes a generalized Burgers–Huxley equation with a fading memory term on bounded domains in $\mathbb{R}^d$ ($d\in\{1,2,3\}$), establishing local well-posedness via Banach fixed point and global existence through energy estimates. It then develops a Riccati-based feedback framework for stabilizing the linear principal system and transfers these results to the nonlinear GBHE, achieving stabilization around both zero and non-constant steady states through interior control, with finite- and infinite-dimensional feedback operators. The authors provide a detailed spectral analysis of the principal operator, proving an analytic semigroup generation and a Riesz-basis of eigenfunctions, which underpins the stabilization results. Numerical simulations using finite elements validate the theoretical findings for zero and non-constant steady states, illustrating exponential decay of the stabilized solutions and the efficacy of the feedback control. Overall, the work extends stabilization theory to nonlinear parabolic equations with memory around non-constant equilibria and offers a versatile approach applicable to related memory-driven evolution equations.

Abstract

In this article, we investigate a generalized Burgers-Huxley equation with a smooth kernel defined in a bounded domain $Ω\subset\mathbb{R}^d$, $d\in\{1,2,3\}$, focusing on feedback stabilizability around a non-constant steady state. Initially, employing the Banach fixed point theorem, we establish the local existence and uniqueness of a strong solution, which is subsequently extended globally using an energy estimate. To analyze stabilizability, we linearize the model around a non-constant steady state and examine the stabilizability of the principal system. For the principal system, we develop a feedback control operator by solving an appropriate algebraic Riccati equation. This allows for the construction of both finite and infinite-dimensional feedback operators. By applying this feedback operator and establishing necessary regularity results, we utilize the Banach fixed point theorem to demonstrate the stabilizability of the entire system. Furthermore, we also explore the stabilizability of the model problem around the zero steady state and validate our findings through numerical simulations using the finite element method for both zero and non-constant steady states.

Feedback Stabilizability of a generalized Burgers-Huxley equation with kernel around a non-constant steady state

TL;DR

The paper analyzes a generalized Burgers–Huxley equation with a fading memory term on bounded domains in (), establishing local well-posedness via Banach fixed point and global existence through energy estimates. It then develops a Riccati-based feedback framework for stabilizing the linear principal system and transfers these results to the nonlinear GBHE, achieving stabilization around both zero and non-constant steady states through interior control, with finite- and infinite-dimensional feedback operators. The authors provide a detailed spectral analysis of the principal operator, proving an analytic semigroup generation and a Riesz-basis of eigenfunctions, which underpins the stabilization results. Numerical simulations using finite elements validate the theoretical findings for zero and non-constant steady states, illustrating exponential decay of the stabilized solutions and the efficacy of the feedback control. Overall, the work extends stabilization theory to nonlinear parabolic equations with memory around non-constant equilibria and offers a versatile approach applicable to related memory-driven evolution equations.

Abstract

In this article, we investigate a generalized Burgers-Huxley equation with a smooth kernel defined in a bounded domain , , focusing on feedback stabilizability around a non-constant steady state. Initially, employing the Banach fixed point theorem, we establish the local existence and uniqueness of a strong solution, which is subsequently extended globally using an energy estimate. To analyze stabilizability, we linearize the model around a non-constant steady state and examine the stabilizability of the principal system. For the principal system, we develop a feedback control operator by solving an appropriate algebraic Riccati equation. This allows for the construction of both finite and infinite-dimensional feedback operators. By applying this feedback operator and establishing necessary regularity results, we utilize the Banach fixed point theorem to demonstrate the stabilizability of the entire system. Furthermore, we also explore the stabilizability of the model problem around the zero steady state and validate our findings through numerical simulations using the finite element method for both zero and non-constant steady states.

Paper Structure

This paper contains 16 sections, 22 theorems, 187 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Literature survey
  4. Methodology, contributions and novelty
  5. Preliminaries and notations
  6. Organization
  7. Principal operator
  8. Strong solution
  9. Stabilizability around the zero state
  10. Stabilizability of the linear system
  11. Stabilizability of non-linear system
  12. Stabilizability around a non-constant steady state
  13. Numerical simulations
  14. Simulation for zero steady state case
  15. Simulation around non-constant steady state
  16. ...and 1 more sections

Key Result

Proposition 1.1

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

Figures (4)

  • Figure 1: Eigenvalues of the shifted principal operator $A+\nu I$ and after stabilization $A+\nu I +BK$
  • Figure 2: Plot of norm of solution of linear system before and after stabilization
  • Figure 3: Plot of norm of solution of full non-linear system before and after stabilization
  • Figure 4: Plot of norm of solution of full non-linear system before and after stabilization

Theorems & Definitions (35)

  • Proposition 1.1: Generalized Hölder's inequality
  • Lemma 1.2: Agmons' inequality Agmon10
  • Lemma 1.3: Sobolev embedding
  • Theorem 1.4: Gagliardo-Nirenberg inequality Nirenberg1959
  • Lemma 1.5: General Gronwall inequality Canon99
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Theorem 2.2: Spectral analysis
  • Theorem 2.3
  • ...and 25 more