Unique Continuation of Static Over-Determined Magnetohydrodynamic Equations

TL;DR

This paper proves the Unique Continuation Property for an overdetermined MHD eigenvalue problem, which is crucial for stabilizing nonlinear MHD systems using localized feedback controls.

Contribution

It establishes the UCP for a specific overdetermined MHD eigenvalue problem, enabling controllability and stabilization of the system.

Findings

UCP established for the MHD eigenvalue problem

Carleman-type estimate used in the proof

Facilitates stabilization of nonlinear MHD systems

Abstract

This paper establishes the Unique Continuation Property (UCP) for a suitably overdetermined Magnetohydrodynamics (MHD) eigenvalue problem, which is equivalent to the Kalman, finite rank, controllability condition for the finite dimensional unstable projection of the linearized dynamic MHD problem. It is the ``ignition key" to obtain uniform stabilization of the dynamic nonlinear MHD system near an unstable equilibrium solution, by means of finitely many, interior, localized feedback controllers of Laseicka et. al 2025. The proof of the UCP result uses a pointwise Carleman-type estimate for the Laplacian following the approach that was introduced in Triggiani 2009 for the Navier-Stokes equations and further extended in Triggiani et. al. 2021 for the Boussinesq system.