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Stabilization and control of the nonlinear plate equation

Cristóbal Loyola

TL;DR

This work analyzes the damped and controlled nonlinear plate equation on a compact Riemannian manifold with hinged boundary conditions and an analytic nonlinearity, focusing on internal damping and internal control localized in a set $\omega$ where the linear Schrödinger equation is observable. The authors build a nonlinear unique continuation result by coupling time-analytic propagation (propagation of analyticity) with linear plate unique continuation, allowing transfer of linear observability to the nonlinear plate. They prove exponential energy decay under a defocusing nonlinearity and, under an asymptotic defocusing condition, semiglobal exact controllability by exploiting the gradient structure and a compact global attractor for the damped system, plus a local controllability result around equilibria. The approach relaxes geometric conditions compared to GCC-based results and leverages the Schrödinger observability transfer and attractor dynamics to achieve stabilization and control, with potential extensions to broader boundary conditions and dimensions.

Abstract

In this article we prove semiglobal stabilization and exact controllability results for nonlinear plate equations with hinged boundary conditions and analytic nonlinearity. These results hold when the damping or control is localized in a region where observability for the linear Schrödinger equation is known to hold. At the core of these results lies a new unique continuation property for the nonlinear plate equation, which significantly relaxes the geometric conditions required for such property to hold. This property is obtained by combining recent results on propagation of analyticity in time and unique continuation for linear plate operators. More broadly, our approach exploits the linear observability of the plate equation to establish both stabilization and control results. First, we prove exponential decay of the nonlinear energy under a defocusing assumption on the nonlinearity. Second, under a weaker asymptotic assumption on the nonlinearity, we prove semiglobal exact control by analyzing control properties inside the compact attractor provided by the dynamics of the damped equation.

Stabilization and control of the nonlinear plate equation

TL;DR

This work analyzes the damped and controlled nonlinear plate equation on a compact Riemannian manifold with hinged boundary conditions and an analytic nonlinearity, focusing on internal damping and internal control localized in a set where the linear Schrödinger equation is observable. The authors build a nonlinear unique continuation result by coupling time-analytic propagation (propagation of analyticity) with linear plate unique continuation, allowing transfer of linear observability to the nonlinear plate. They prove exponential energy decay under a defocusing nonlinearity and, under an asymptotic defocusing condition, semiglobal exact controllability by exploiting the gradient structure and a compact global attractor for the damped system, plus a local controllability result around equilibria. The approach relaxes geometric conditions compared to GCC-based results and leverages the Schrödinger observability transfer and attractor dynamics to achieve stabilization and control, with potential extensions to broader boundary conditions and dimensions.

Abstract

In this article we prove semiglobal stabilization and exact controllability results for nonlinear plate equations with hinged boundary conditions and analytic nonlinearity. These results hold when the damping or control is localized in a region where observability for the linear Schrödinger equation is known to hold. At the core of these results lies a new unique continuation property for the nonlinear plate equation, which significantly relaxes the geometric conditions required for such property to hold. This property is obtained by combining recent results on propagation of analyticity in time and unique continuation for linear plate operators. More broadly, our approach exploits the linear observability of the plate equation to establish both stabilization and control results. First, we prove exponential decay of the nonlinear energy under a defocusing assumption on the nonlinearity. Second, under a weaker asymptotic assumption on the nonlinearity, we prove semiglobal exact control by analyzing control properties inside the compact attractor provided by the dynamics of the damped equation.

Paper Structure

This paper contains 36 sections, 26 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Semiglobal stabilization
  4. Dynamics and controllability
  5. Unique continuation property
  6. Brief literature
  7. Outline of the article
  8. Acknowledgments
  9. Preliminaries
  10. Well-posedness
  11. Nonlinearity properties
  12. Linear observability
  13. About the stabilization strategy
  14. Unique continuation and equilibriums
  15. Stabilization and long time dynamics
  16. ...and 21 more sections

Key Result

Theorem 1.2

Assume that $\omega$ satisfy assumOBS. If $f$ is real analytic in the second variable and it satifies NLP:defdefocusing, then the exponential decay property (ED) holds.

Theorems & Definitions (54)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • ...and 44 more