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

TL;DR

This paper develops new Carleman estimates for the Korteweg-de Vries equation with discontinuous coefficients, enabling results on controllability, inverse problems, and boundary observability.

Contribution

It introduces a novel two-parameter Carleman estimate for the KdV equation with piecewise constant coefficients, advancing control and inverse problem analysis.

Findings

Established local exact controllability to trajectories.

Proved Lipschitz stability for inverse potential retrieval.

Derived boundary observability remarks.

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{\u\i}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.