Feedback stabilization of magnetohydrodynamic equations

Abstract

We prove the local exponential stabilizability for the MHD system, with internally distributed feedback controllers. These controllers take values in a finite dimensional space which is the unstable manifold of the elliptic part of the linearized operator. The stabilization of the linear system is derived using a unique continuation property for systems of parabolic and elliptic equations, as well as the equivalence between controllability and feedback stabilizability in the case of finite dimensional systems. The feedback that stabilizes the linearized system is also stabilizing the nonlinear system in the domain of a fractional power of the elliptic operator.