Unique continuation of static overdetermined 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 non-linear MHD system near an unstable equilibrium solution, by means of finitely many, interior, localized feedback controllers [Res. Math. Sci. 12 (2025), article no. 7]. The proof of the UCP result uses a pointwise Carleman-type estimate for the Laplacian following the approach that was introduced in [Nonlinear Anal. 71 (2009), 4967–4976] for the Navier–Stokes equations and further extended in [Appl. Math. Optim. 84 (2021), 2099–2146] for the Boussinesq system.