Exact Internal Controllability for the Two-Dimensional Magnetohydrodynamic Equations

Havârneanu, Teodor; Popa, Cătălin; Sritharan, S. S.

doi:10.1137/040611884

Cited by 14 publications

(23 citation statements)

References 10 publications

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“…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.…”

Section: Introductionsupporting

confidence: 51%

“…2. The regularity assumptions (1.8) are weaker than those made in previous work on the subject, see in particular [31,32]. Moreover, the initial regularity assumption…”

Section: Introductionmentioning

confidence: 74%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

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Local Controllability to Trajectories of the Magnetohydrodynamic Equations

Badra

2014

J. Math. Fluid Mech.

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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; Havârneanu et al. in Adv Differ Equ 11:893-929, 2006; Havârneanu, in 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.

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Section: Introductionsupporting

confidence: 51%

“…2. The regularity assumptions (1.8) are weaker than those made in previous work on the subject, see in particular [31,32]. Moreover, the initial regularity assumption…”

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local Controllability to Trajectories of the Magnetohydrodynamic Equations

Badra

2014

J. Math. Fluid Mech.

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“…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.…”

Section: Other More Realistic Nonlinear Equations and Systemsmentioning

confidence: 99%

Remarks on the Controllability of Some Parabolic Equations and Systems

Fernández-Cara

2009

Computational Methods in Applied Sciences

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239[3] M. Mateos, Problemas de controlóptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado. cara@us.es AbstractWe present a review of recent results concerning the null controllability of parabolic systems. Among others, we will consider the heat equation, the Burgers, Navier-Stokes and Ginzburg-Landau equations, etc. We will also indicate some open questions.

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Existence of Controls Insensitizing the Rotational of the Solution of the Navier–Stokes System Having a Vanishing Component

Prada

2023

Appl Math Optim

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Exact Internal Controllability for the Two-Dimensional Magnetohydrodynamic Equations

Abstract: 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.

Cited by 14 publications

References 10 publications

Local Controllability to Trajectories of the Magnetohydrodynamic Equations

Local Controllability to Trajectories of the Magnetohydrodynamic Equations

Remarks on the Controllability of Some Parabolic Equations and Systems

Existence of Controls Insensitizing the Rotational of the Solution of the Navier–Stokes System Having a Vanishing Component

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