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Null controllability of damped nonlinear wave equation

Yan Cui, Peng Lu, Yi Zhou

TL;DR

The paper advances internal null controllability for damped nonlinear wave equations across semilinear,quasi-linear, and fully nonlinear regimes under geometric control assumptions. It combines Galerkin approximations, fixed-point arguments, and iterative contraction schemes with robust observability inequalities to construct controls, often in a constructive, numerically amenable manner. Key contributions include precise regularity requirements, explicit constants and conditions on the nonlinearities, and a time-differentiation strategy for fully nonlinear cases. Collectively, the results extend null controllability to broader nonlinear settings with internal control, under GCC-type conditions, and provide a path toward practical computation of controls.

Abstract

In this paper, we investigate the null controllability of nonlinear wave systems. Initially, we employ a combination of the Galerkin method and a fixed point theorem to establish the null controllability for semi-linear wave equations with nonlinear functions that are dependent on velocities, under the geometric control condition. Subsequently, utilizing a novel iterative method, we demonstrate the null controllability for a class of quasi-linear wave systems in a constructive manner. Lastly, we present a control result for a class of fully nonlinear wave systems, serving as an application.

Null controllability of damped nonlinear wave equation

TL;DR

The paper advances internal null controllability for damped nonlinear wave equations across semilinear,quasi-linear, and fully nonlinear regimes under geometric control assumptions. It combines Galerkin approximations, fixed-point arguments, and iterative contraction schemes with robust observability inequalities to construct controls, often in a constructive, numerically amenable manner. Key contributions include precise regularity requirements, explicit constants and conditions on the nonlinearities, and a time-differentiation strategy for fully nonlinear cases. Collectively, the results extend null controllability to broader nonlinear settings with internal control, under GCC-type conditions, and provide a path toward practical computation of controls.

Abstract

In this paper, we investigate the null controllability of nonlinear wave systems. Initially, we employ a combination of the Galerkin method and a fixed point theorem to establish the null controllability for semi-linear wave equations with nonlinear functions that are dependent on velocities, under the geometric control condition. Subsequently, utilizing a novel iterative method, we demonstrate the null controllability for a class of quasi-linear wave systems in a constructive manner. Lastly, we present a control result for a class of fully nonlinear wave systems, serving as an application.
Paper Structure (21 sections, 10 theorems, 511 equations)

This paper contains 21 sections, 10 theorems, 511 equations.

Table of Contents

  1. Introduction and main results
  2. Semi-linear case
  3. Quasi-linear case
  4. Fully nonlinear case
  5. Organization of this paper
  6. Controllability for linear damped hyperbolic system
  7. Constant case
  8. Various case: controllability in $\mathcal{H}^1\times L^2$
  9. Various case: controllability in $\mathcal{H}^l\times \mathcal{H}^{l-1}$
  10. Proof of Theorem \ref{['thm:semilinear']}
  11. Proof of Theorem \ref{['thm:main1']}
  12. Existence of solutions of system \ref{['system:quasilinear']}
  13. Proof of Lemma \ref{['lem:VZ']}
  14. Basic step 1: The case of $\alpha=1$.
  15. Basic step 2: To prove \ref{['induction4']} and \ref{['induction5']} when $l=0$ for any $\alpha\geq 2$
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume that $a^{ij}$ satisfies sect2:a. Assume that sect2:b is valid. If System system:z-HUM is exactly observable in $\mathcal{H}^2\times \mathcal{H}^1$, then system Damped Wave Eqn with internal control is exactly null controllable.

Theorems & Definitions (49)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • Theorem 2.1
  • Remark 2.2
  • ...and 39 more