Study of control problems for the stationary MHD equations

Abstract

Abstract.The optimal control problems for the stationary magnetohydrodynamic equations under inhomogeneous mixed boundary conditions for a magnetic field are considered. The role of control in control xs under study is played by normal component of the magnetic field on the part of the boundary. In the capacity of cost fucntionals quadratic tracking-type functionals for a velocity, magnetic field or pressure are taken. Introduction. Statement of the boundary value problemGreat attention has been paid recently to the optimal control problems for the models of magnetohydrodynamics (MHD) for a viscous conducting incompressible fluid (see [1][2][3][4][5][6]).In this paper we study control problems for the new MHD model in which stationary equations of magnetohydrodynamic are considered under mixed boundary conditions for a magnetic field and under Dirichlet boundary condition for a velocity with tracking-type functionals. The role of controls in control problems under study is played by a normal component of the magnetic field on the part of the boundary.One of the features of this work consist in proving of the solvability of optimal control problem under minimum requirements on a normal component of the magnetic field given on the part of boundary and playing the role of control. This result, in contrast to [3], allows to use a simple L 2 -norm for the normal component of the magnetic field instead of the H 1/2 -norm on the part of the boundary by Tikhonov regularization of considered incorrect control problem. This regularization is needed to prove stability and uniqueness of the control problem's solution.Let Ω be a bounded domain of space R 3 with boundary ∂Ω consisting of two parts Σ ν and Σ τ . In this paper we study control problems for the stationary magnetohydrodynamic equations of viscous incompressible fluid