Exact controllability for the magnetohydrodynamic equations

Abstract

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.

Local Controllability to Trajectories of the Magnetohydrodynamic Equations

“…The main novelty of the above controllability result is that the two controls are localized in a small subset of the domain. This improves the results of [6,7,31,32] where only the control in the velocity equation is localized. 2.…”

“…The local controllability of magnetohydrodynamic equations has been studied in [6,7,31,32]. However, these quoted works provide controllability results with a non local control of the form P 1 O u 2 in the induction equations where u 2 does not necessarily satisfies (1.4) and where P is the Helmholtz projector related to Ω.…”

“…4. We underline that unlike in [6,7,31,32] we do not treat directly the Dynamo-type equation but we prove a Carleman inequality for the curl of such equation, Vol. 16 (2014) Controllability to Trajectories of MHD Equations 637 which has the advantage of getting rid of the terms ∇κ and μ i g i .…”

On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds

“…Taylor and [8] by R. Dautray and J.-L. Lions. The interested reader may also consult the papers [4], [3] by T. Akiyama, H. Kasai, Y. Shibata and M. Tsutsumi, [5] by V. Barbu, T. Havarneanu, C. Popa and S.S. Sritharan, [10] by E.B. Fabes, J.E.…”

Remarks on the Controllability of Some Parabolic Equations and Systems

“…Again, the local exact controllability of the solutions to the trajectories is established. Let us also mention [2,17], where the controllability of the MHD and other related equations has been analyzed.…”