We study the local exact controllability of the steady state solutions of the magnetohydrodynamic equations. The main result of the paper asserts that the steady state solutions of these equations are locally controllable if they are smooth enough. We reduce the local exact controllability of the steady state solutions of the magnetohydrodynamic equations to the global exact controllability of the null solution of the linearized magnetohydrodynamic system via a fixed-point argument. The treatment of the reduced problem relies on two Carleman-type inequalities for the backward adjoint system.
In this paper we establish the local exact internal controllability for the two-dimensional magnetohydrodynamic equations. The needed Carleman estimate for the adjoint linearized magnetohydrodynamic equations is also obtained here.
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