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A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances

Jun Zheng, Guchuan Zhu

Abstract

This paper addresses input-to-state stability (ISS) properties with respect to boundary and in-domain disturbances for a class of semi-linear partial differential equations (PDEs) subject to Dirichlet boundary conditions. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS in $L^\infty$-norm for a 1-$D$ transport equation and the ISS in $L^2$-norm for Burgers' equation have been established using this method. As an application of the main result for the 1-D transport equation, a study on ISS properties in $L^\infty$-norm of a 1-D transport equation with anti-stable term under boundary feedback control is carried out. This is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and be applied to a wider class of nonlinear parabolic PDEs.

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances

Abstract

This paper addresses input-to-state stability (ISS) properties with respect to boundary and in-domain disturbances for a class of semi-linear partial differential equations (PDEs) subject to Dirichlet boundary conditions. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS in -norm for a 1- transport equation and the ISS in -norm for Burgers' equation have been established using this method. As an application of the main result for the 1-D transport equation, a study on ISS properties in -norm of a 1-D transport equation with anti-stable term under boundary feedback control is carried out. This is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and be applied to a wider class of nonlinear parabolic PDEs.

Paper Structure

This paper contains 13 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. De Giorgi iteration
  4. Preliminary inequalities
  5. Problem Formulation and Main Results
  6. Problem formulation and well-posedness analysis
  7. Main results
  8. ISS Properties for 1-$D$ Transport PDE
  9. ISS properties for the split subsystems
  10. Proof of ISS properties for 1-$D$ transport PDE (Theorem \ref{['Theorem 6']})
  11. ISS of a 1-$D$ transport equation with anti-stable term under boundary feedback control
  12. ISS Properties for Burgers' Equation
  13. Concluding Remarks

Key Result

Lemma 1

Suppose that $\varphi$ is a non-negative decreasing function on $[k_0,+\infty)$ satisfying where $\alpha>0,\beta>1,M\in \mathbb{R}$ are constants. Then there holds with $d=2^{\frac{\beta}{\beta-1}}M(\varphi(k_0))^{\frac{\beta-1}{\alpha}}$.

Theorems & Definitions (23)

  • Lemma 1: Wu2006
  • Lemma 2: Young's inequality
  • Lemma 3: Poincaré's inequality Zheng:2017
  • Lemma 4
  • Remark 1
  • Lemma 5: Gronwall's inequality Zheng:2017
  • Proposition 6
  • Definition 1
  • Definition 2
  • Theorem 7
  • ...and 13 more