Much attention has recently been given to statement and investigation of new problems for models of continuum mechanics. Control problems for models of hydrodynamics, heat and mass transfer and magnetic hydrodynamics are examples of these problems. The interest to control problems is connected with a variety of important applications in science and engineering such as the crystal growth process, aerodynamic drag reduction, suppression of turbulence and flow separation. In these problems unknown densities of boundary or distributed sources are recovered from additional information on the solution to the boundary value problem. Control and inverse problems for these models are analyzed by applying a unified approach based on the constrained optimization theory in Hilbert or Banach spaces (see [1, 2]).
Our goal is the study of boundary control problems for the Navier-Stokes and Boussinesq equations describing the flow of the viscous incompressible heat conducting fluid in a bounded domain. These problems consist in minimization of certain cost functionals depending on the state and controls. The existence of an optimal solution is based on a priori estimates and standard techniques. Optimality systems describing the first-order necessary optimality conditions are obtained, and, by analysis of their properties,
conditions ensuring the uniqueness and stability of solutions are established.
Two algorithms for the numerical solution of control problems are proposed. The first of them is based on application of the gradient method. This numerical algorithm reduces the solution of the control problem to the solution of direct and adjoint problems at each iterative step. The second algorithm is based on solution of the optimality system applying Newton's method. The influence of the initial guess and parameters of the algorithm on the speed of convergence and accuracy of the obtained numerical solution is investigated. The open source software freeFEM++ (www.freefem.org) is used for the discretization of boundary-value problems by the finite element method. Some results of numerical experiments can be found in [1-3].
The work was supported by the 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 09-I-OMN-03).
1. Alekseev G.V., Tereshko D.A. Analysis and Optimization in Viscous Fluid Hydrodynamics. Dalnaulka. 2008.
2. Alekseev G.V. Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics. Nauchny Mir. 2010.
3. Alekseev G.V., Tereshko D.A. Extremum problems of boundary control for steady equations of thermal convection} // J. Appl. Mech. Tech. Phys. 2010. V. 51. P. 510-520.