A reduced basis method for control problems governed by PDEs.

*(English)* Zbl 0908.93025
Desch, W. (ed.) et al., Control and estimation of distributed parameter systems. International conference in Vorau, Austria, July 14–20, 1996. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 126, 153-168 (1998).

Real time simulations for optimal control of viscous flows is presented as one of the most challenging task in computational engineering and science. The difficulty is mainly due to the nonlinearity of the Navier-Stokes equations (which are the state equations for such systems), where discretizations lead to large scale control problems. The authors discuss a reduction type method called the reduced basis method in the paper. In this method, in contrast to traditional numerical methods (like finite difference or finite elements methods), where grid functions or piecewise polynomials are used as basis, one uses “very few” basis functions closely related to and generated from the problem that is being solved. The authors prove the applicability and feasibility of the reduced basis method to vorticity minimization problems in fluid flows through backward-facing step type channels. Especially attention is paid to two fluid situations. In the first one an electrically conducting fluid (as sea water for instance) under applied magnetic field and controlled by a boundary electric potential is considered. In the second situation, the control of a thermally convective fluid is effected by boundary temperature. In both cases several computational results are presented. A justification of the method is based on an error analysis.

##### MSC:

93B40 | Computational methods in systems theory |

49K20 | Optimal control problems with PDE (optimality conditions) |

49M05 | Numerical methods in calculus of variations based on necessary conditions |

65H10 | Systems of nonlinear equations (numerical methods) |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

80A20 | Heat and mass transfer, heat flow |

76D05 | Navier-Stokes equations (fluid dynamics) |