Local Controllability to Trajectories of the Magnetohydrodynamic Equations

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; 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.

Local exact controllability to trajectories of the magneto-micropolar fluid equations

“…This form of the control v has been also considered in recent works on the local exact controllability of the MHD system [4,5,18,19]. There is only a recent result on the controllability of MHD system [3] in which the control acting on the magnetic field has support in an arbitrarily small open subset of the spatial domain, i.e., the control has the form 1 O P O (v1 O ), where P O is the classical Helmholtz projector related to O (i.e., the orthogonal projection operator from L 2 (O) d onto the completion of the set {v ∈ C ∞ 0 (O) d | ∇ • v = 0 in O} in the norm of L 2 (O) d . However, since the boundary conditions on the magnetic field in our system is different from that in [3], so here we cannot use ideas in [3] to establish our Carleman estimate for the component C of the adjoint system respectively to the magnetic field.…”

“…However, since the boundary conditions on the magnetic field in our system is different from that in [3], so here we cannot use ideas in [3] to establish our Carleman estimate for the component C of the adjoint system respectively to the magnetic field. Hence, we are not able to get an estimate of the right-hand side of the component C having the form O e −2sα ξ 3 |P O C| 2 dxdt as in [3]. So we only obtain the controllability of (1) with the control function acting on the magneto field has the form (2).…”

“…[10,12,26] and references therein. In recent years, the local exact controllability of the MHD system was also studied by a number of authors in [3,4,5,18,19], and that of the micropolar fluid equation was studied in [9,17].…”

“…Now, we proceed to define the spaces where (26)-(27) will be solved. The main space will be E = ( y, p, ω, B, u, w, v) : e sβ y, e sβ γ −1/2 ω, e sβ B ∈ L 2 (Q) 3 ,…”

“…Moreover, we can see from (36) that (ŷ,ω,B) ∈ 3 . On the other hand, from (34) and 35, we see that (ŷ,ω,B) together with some pressurep is the unique solution of (26) which is defined by the transposition with u =û, w =ŵ, v =v.…”