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Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient

Cristóbal Loyola

TL;DR

This work addresses observability and control for the Korteweg–de Vries equation with a piecewise constant main coefficient by deriving a two-parameter Carleman estimate with internal observation under a monotonicity assumption on the coefficient jumps. The authors employ a piecewise-constructed weight and a global Carleman framework to obtain an observability inequality, which they then use to prove local controllability to trajectories for the nonlinear system via a duality argument and a local inversion approach. They also establish Lipschitz stability for the inverse problem of recovering an unknown potential using the Bukhgeĭm–Klibanov method under symmetry/interface hypotheses. The results advance the theory of control and inverse problems for dispersive PDEs in heterogeneous media and illuminate the role of interface conditions in boundary and internal observability analyses.

Abstract

In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate for the linear equation with internal observation, assuming a monotonicity condition on the main coefficient. As a primary application, we establish the local exact controllability to the trajectories by employing a duality argument for the linear case and a local inversion theorem for the nonlinear equation. Secondly, we establish the Lipschitz-stability of the inverse problem of retrieving an unknown potential using the Bukhge{\uı}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.

Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient

TL;DR

This work addresses observability and control for the Korteweg–de Vries equation with a piecewise constant main coefficient by deriving a two-parameter Carleman estimate with internal observation under a monotonicity assumption on the coefficient jumps. The authors employ a piecewise-constructed weight and a global Carleman framework to obtain an observability inequality, which they then use to prove local controllability to trajectories for the nonlinear system via a duality argument and a local inversion approach. They also establish Lipschitz stability for the inverse problem of recovering an unknown potential using the Bukhgeĭm–Klibanov method under symmetry/interface hypotheses. The results advance the theory of control and inverse problems for dispersive PDEs in heterogeneous media and illuminate the role of interface conditions in boundary and internal observability analyses.

Abstract

In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate for the linear equation with internal observation, assuming a monotonicity condition on the main coefficient. As a primary application, we establish the local exact controllability to the trajectories by employing a duality argument for the linear case and a local inversion theorem for the nonlinear equation. Secondly, we establish the Lipschitz-stability of the inverse problem of retrieving an unknown potential using the Bukhge{\uı}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.
Paper Structure (35 sections, 19 theorems, 213 equations)

This paper contains 35 sections, 19 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Carleman estimate
  4. Local controllability to the trajectories
  5. Retrieving a potential term
  6. Some comments on the literature
  7. Outline of the paper
  8. Notation
  9. Well-posedness results
  10. Linear Cauchy problem
  11. Nonlinear system
  12. Adjoint system
  13. A global Carleman estimate
  14. Step 1. Piecewise affine on the left.
  15. Step 2. Quartic polynomial on the observation zone.
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $(\omega, p)$ satisfy Hypothesis assumM and let $\omega_0\Subset \omega$ be non-empty and open. There exist $s_0>0$, $\lambda_0>0$ and a constant $C>0$ depending on $\omega$, $\Gamma$, $L$, $T$, $p$, $\lVert\beta\rVert_{C^3([0, L]\setminus\Gamma)}$, $s_0$ and $\lambda_0$ such that for all $u\in for any $s\geq s_0$ and $\lambda\geq \lambda_0.$

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 31 more