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Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations

Thomas Perrin

Abstract

On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases.

Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations

Abstract

On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases.
Paper Structure (42 sections, 23 theorems, 321 equations, 1 figure)

This paper contains 42 sections, 23 theorems, 321 equations, 1 figure.

Table of Contents

  1. Setting and main results.
  2. Consequences.
  3. Consequences for some focusing nonlinearities.
  4. Consequences for some defocusing nonlinearities.
  5. Connection with existing literature.
  6. Outline of the article.
  7. Notation.
  8. Acknowledgments.
  9. Preliminaries
  10. Strichartz estimates
  11. Case 1:
  12. Case 2:
  13. Case 3:
  14. Case 4:
  15. Basic nonlinear estimates
  16. ...and 27 more sections

Key Result

Theorem 4

Assume that $f$ satisfies (eq_def_nonlinearity_local), and consider $(\mathbf{u}^0, \mathbf{u}^1) \in H_0^1(\Omega) \times L^2(\Omega)$ such that the solution $\mathbf{u}$ of (eq_NLKG) with initial data $(\mathbf{u}^0, \mathbf{u}^1)$ exists on the interval $[0, T]$. We make the following assumptions Then, local controllability around $\mathbf{u}$ at time $T$ holds.

Figures (1)

  • Figure 1: The local admissible exponents $\Lambda_\Omega$, in gray.

Theorems & Definitions (52)

  • Remark 1
  • Definition 2: Local controllability around $\mathbf{u}$ at time $T$
  • Definition 3
  • Theorem 4: Local controllability around a trajectory
  • Definition 5: Null-controllability in a long time
  • Definition 6
  • Definition 7
  • Theorem 8: Null-controllability of scattering solutions
  • Remark 9
  • Remark 10
  • ...and 42 more