Unique Continuation of Static Over-Determined Magnetohydrodynamic Equations

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.