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<p>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 Bukhgeĭm–Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.</p>

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Carleman Estimates for the Korteweg–de Vries Equation with Piecewise Constant Main Coefficient

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • Cristóbal Loyola

摘要

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 Bukhgeĭm–Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.