Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains

Abstract

We study the exact controllability for the magnetohydrodynamic equations in multi-connected bounded domains. We show that the value of any solution of the magnetohydrodynamic system at any given time is locally exactly controllable provided that this solution is smooth enough. This means that the chosen value of such a solution can be reached by starting from initial states which are sufficiently close to the initial value of the solution and by acting with controls distributed in a given small subdomain. So, a previous controllability result for the magnetohydrodynamic equations is improved in several directions. Our treatment reduces the local controllability for the magnetohydrodynamic equations to the global controllability for their linearizations by means of an infinite-dimensional variant of the local inversion theorem. The proof of the global controllability relies on a Carleman-type estimate for the adjoint linearized equations.