Piecewise optimal distributed controls for 2D Boussinesq equations.

*(English)*Zbl 0951.35108The authors consider the velocity and temperature tracking problem for viscous incompressible thermally convected flows in two dimensions (which is similar to the velocity tracking problem for viscous incompressible flows). This problem is formulated as a piecewise (in time) distributed optimal control problem for the Boussinesq equations. The semidiscrete (in time) approximation of the piecewise optimal control problem is also studied. The main theorem asserts that the \(L^2\) and \(H^1\) norms of the difference between the controlled state and the desired state both decay to zero exponentially in time. A similar result is obtained for the semidiscrete approximation of the piecewise optimal control problem.

Reviewer: C.Popa (Iaşi)

##### MSC:

35Q30 | Navier-Stokes equations |

35B37 | PDE in connection with control problems (MSC2000) |

35B40 | Asymptotic behavior of solutions to PDEs |

49M25 | Discrete approximations in optimal control |

##### Keywords:

velocity and temperature tracking problem for viscous incompressible thermally convected flows; Boussinesq equations; piecewise optimal control problem; semidiscrete approximation
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\textit{H.-C. Lee} and \textit{B. C. Shin}, Math. Methods Appl. Sci. 23, No. 3, 227--254 (2000; Zbl 0951.35108)

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##### References:

[1] | Sobolev Spaces, Academic, New York, 1975. · Zbl 0314.46030 |

[2] | Abergel, Theoret. Comput. Fluid Dyn. 1 pp 303– (1990) · Zbl 0708.76106 |

[3] | and Navier-Stokes Equations, University of Chicago, Chicago, 1988. |

[4] | and Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986. · Zbl 0585.65077 |

[5] | Finite Element Methods for Viscous Incompressible Flows, Academic Press, London, 1989. · Zbl 0697.76031 |

[6] | Hou, SIAM J. Control Optim. 35 pp 654– (1997) · Zbl 0871.49008 |

[7] | Hou, SIAM J. Control Optim. 35 pp 1847– (1997) · Zbl 0889.35074 |

[8] | and ?Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media?, to appear in SIAM J. Control Optim. · Zbl 0968.49007 |

[9] | and ?Dynamics for controlled Boussinesq equations with distributed control?, preprint. |

[10] | Lions, Pub. de IHES. 19 pp 5– (1964) · Zbl 0148.11403 |

[11] | Navier-Stokes Equations, Theory and Numerical Methods, North-Holland, Amsterdam, 1983. |

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