Abstract
In this paper we prove the local controllability to trajectories of the three dimensional magnetohydrodynamic equations by means of two internal controls, one in the velocity equations and the other in the magnetic field equations and both localized in an arbitrary small subset with not empty interior of the domain. This paper improves the previous results (Barbu et al. in Comm Pure Appl Math 56:732-783, 2003; Barbu et al. in Adv Differ Equ 10:481-504, 2005 ;H avet al. in Adv Differ Equ 11:893-929, 2006 ;H avin SIAM J Control Optim 46:1802-1830, 2007) where the second control is not localized and it allows to deduce the local controllability to trajectories with boundary controls. The proof relies on the Carleman inequality for the Stokes system of Imanuvilov et al. (Carleman estimates for second order nonhomogeneous parabolic equations, preprint) to deal with the velocity equations and on a new Carleman inequality for a Dynamo-type equation to deal with the magnetic field equations. Mathematics Subject Classification. Primary 76W05, 93B05; Secondary 76D55, 35Q35.
Published Version
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