Analysis and computational methods of Dirichlet boundary optimal control problems for 2D Boussinesq equations

Advances in Computational Mathematics

Volumn 19, Issue 1-3, 2003, Pages 255-275

  Lee, Hyung Chun ^a  

^a Ajou University   (South Korea)

Author keywords

Boussinesq equations; Navier Stokes equation; Optimal control

Indexed keywords

EID: 0037243410     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022872602498     Document Type: Conference Paper

References (31)

1. F. Abergal and R. Temmam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics 1 (1990) 303-325.
2. R. Adams, Sobolev Spaces (Academic Press, New York, 1975).
3. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 17 (1964) 35-92.
4. F. Brezzi, J. Rappaz and P.A. Raviart, Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math. 36 (1980) 1-25.
5. P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).
6. P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization (Cambridge, 1989).
7. P. Cuvelier, Optimal control of a system governed by the Navier-Stokes equations coupled with the heat equations, in: New Developments in Differential Equations, ed. W. Eckhaus (North-Holland, Amsterdam, 1976) pp. 81-98.
8. H.O. Fattorini and S.S. Sritharan, Optimal controls for viscous flow problems, Proc. Roy. Soc. London Ser. A 439 (1992) 81-102.
9. A.V. Fursikov and O.Yu. Imanuvilov, Local exect boundary controllability of the Boussinesq equations, SIAM J. Control Optim. 36 (1998) 391-421.
10. V. Girault and P.A. Raviart, Finite Element Method for Navier-Stokes Equations (Springer, Berlin, 1986).
11. M. Gunzburger, Finite Element Methods for Incompressible Viscous Flows: A Quide to Theory, Practice and Algorithms (Academic Press, Boston, 1989).
12. M. Gunzburger, L. Hou and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, Modél. Math. Anal. Numér. 25 (1991) 711-748.
13. M. Gunzburger, L. Hou and T. Svobodney, Treating inhomogenous essential boundary conditions in finite element methods, SIAM J. Numer. Anal. 29(2) (1992) 390-424.
14. M. Gunzburger, L. Hou and T. Svobodney, Heating and cooling control of temperature distributions along boundaries of flow domains, J. Math. Systems Estim. Control 3(2) (1993) 147-172.
15. M. Gunzburger and H.-C. Lee, Analysis, approximation, and comutation of a copled solid/fluid temperature control problem, Comput. Math. Appl. Engrg. 118 (1994) 133-152.
16. M.D. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with bounded distributed controls, SIAM J. Control Optim. 37(6) (1999).
17. M.D. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with boundary control, SIAM J. Control Optim. 39(2) (2000).
18. M.D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM J. Numer. Anal. 37(5) (2000).
19. L. Hou, Analysis and finite element approximation of some optimal control problem associated with the Navier-Stokes equations, Ph.D. thesis, Carnegie Mellon University, Pittsburgh (1989).
20. L.S. Hou and Y. Yan, Dynamics for controlled Navier-Stokes systems with distributed controls, SIAM J. Control Optim. 35(2) (1997) 654-677.
21. L.S. Hou and Y. Yan, Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls, SIAM J. Control Optim. 35(6) (1997) 1847-1885.
22. K. Ito and S.S. Ravindran, Optimal control of thermally convected fluid flows, SIAM J. Sci. Comput. 9(6) (1998) 1847-1869.
23. H.-C. Lee and O.Y. Imanuvilov, Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media, SIAM J. Control Optim. 39(2) (2000) 457-477.
24. H.-C. Lee and O.Y. Imanuvilov, Analysis of optimal control problems for the 2-D stationary Boussinesq equations, J. Math. Anal. Appl. 242 (2000) 191-211.
25. H.-C. Lee and B.C. Shin, Piecewise optimal distributed controls for 2D Boussinesq equations, Math. Methods Appl. Sci. 23 (2000) 227-254.
26. H.-C. Lee and B.C. Shin, Dynamics for linear feedback controlled Bénard equations with distributed controls, Appl. Math. Optim., to appear.
27. J.L. Lions and E. Magnes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1 (Springer, New York, 1972).
28. S.S. Ravindran, Numerical solutions of optimal control for thermally convective flows, Internat. J. Numer. Methods Fluids 25 (1997) 205-227.
29. G. Savare, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations 22 (1997) 869-899.
30. R. Temam, Navier-Stokes Equations (North-Holland, Amsterdam, 1979).
31. V. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems (Wiley, Chichester, 1982).