Brizitskii R.  

Stability estimates of solutions of control problems for stationary equations of magnetohydrodynamics

In recent years, much attention has been given to optimal control problems
for flows of viscous electrically conducting fluids. The study of these
problems was motivated by the necessity of the most effective control mechanisms
for hydrodynamic processes in such fluids. A rigorous theoretical study
of these problems can be found, for example, in [1-4].

Along with optimal control problems, an important role is played by identification
problems for MHD models. In the latter problems, the unknown coefficients involved
in the differential equations or in the boundary conditions for the model in question
are determined from additional data on the solution. Note that identification
problems can be reduced to optimization problems with a suitable choice
of the minimized cost functional. In [4] this approach was used to analyze the
solvability, uniqueness and stability of solutions to identification
problems for models of magnetohydrodynamics of viscous incompressible fluid.

In this paper  identification problems for the stationary magnetohydrodynamic (MHD)
model governing a flow of a viscous electrically-conducting fluid are stated and analyzed.
This model consists of the Navier-Stokes equations, the generalized Ohm law,
and the stationary Maxwell equations without displacement currents,
considered under Dirichlet boundary condition for the velocity
and mixed boundary conditions for the electromagnetic field.
The solvability of the problem is proved,
an optimality system is derived, and sufficient conditions on the initial data
are established that ensure the uniqueness and stability of the solution.

The work was supported by the
Council about grants of the President of the Russian Federation for the state support
of Young Russian scientists - candidates of science (MK-3311.2011.1),
Russian Foundation for Basic Research
(project no. 10-01-00219-a) and the Far East
Branch of the Russian Academy of Sciences (projects no. 09-I-P29-01 and


1. Alekseev G.V. Solvability of the control problems for the
stationary magnetohydrodynamic equations of viscous fluid // Sib. Math. J. 2004. V. 45. N 2 P. 243-262.

2. Alekseev G.V., Brizitskii R.V. Control problems for stationary
magnetohydrodynamic equations of viscous heat-conducting fluid under
mixed boundary conditions  // J. Comp. Math. and Math. Physics. 2005. V. 45. N 12. P. 2049-2065

3. Alekseev G.V., Tereshko D.A.  Analysis and Optimization in Viscous Fluid Hydrodynamics. Dalnaulka. 2008.

4.  Alekseev G.V. Optimization in Stationary Problems of Heat and Mass
Transfer and Magnetic Hydrodynamics. Moscow. Nauchny Mir. 2010.

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