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Robust control problems for primitive equations of the ocean
1. | Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States |
References:
[1] |
F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325.
doi: 10.1007/BF00271794. |
[2] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation, Comput. Math. Math. Phys., 37 (1997), 348-358. |
[3] |
A. Belmiloudi and F. Broissier, A control method for assimilation of surface data in a linearized Navier-Stokes-type problem related to oceanography, SIAM. J. Control Optim., 35 (1997), 2183-2197.
doi: 10.1137/S0363012995286137. |
[4] |
A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994. |
[5] |
T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392.
doi: 10.1016/S0167-2789(99)00206-7. |
[6] |
C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[7] |
F. X. Le Dimet and V. P. Shutyaev, On Newton methods in data assimilation, Russ. J. Numer. Anal. Math. Modelling, 15 (2000), 419-434.
doi: 10.1515/rnam.2000.15.5.419. |
[8] |
I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. |
[9] |
G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. |
[10] |
C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460.
doi: 10.1016/j.na.2004.12.005. |
[11] |
C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. B, 23 (2002), 1-16.
doi: 10.1142/S025295990200002X. |
[12] |
C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. |
[13] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[14] |
F. X. Le Dimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. Math. Modelling, 16 (2001), 247-259. |
[15] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
[16] |
J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54. |
[17] |
J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120. |
[18] |
J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163. |
[19] |
J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A: Theory Methods, 40 (2000), 439-482.
doi: 10.1016/S0362-546X(00)85026-9. |
[20] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean, Nonlinear Anal., 68 (2008), 3550-3564
doi: 10.1016/j.na.2007.03.046. |
[21] |
T. Tachim Medjo, Optimal control of the primitive equations of the ocean with state constraints, Accepted in Nonlinear Anal., Ser. A: Theory Methods, 2010. |
[22] |
T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean, in "Mathematical Modeling, Simulation, Visualization and E-Learning: Proceedings of the Bellagio International Conference" (D. Konate, editor), Springer, (2007), 107-150. |
[23] |
T. Tachim Medjo and R. Temam, A two-grid finite difference method for the primitive equations of the ocean, Nonlinear Anal., 69 (2008), 1034-1056.
doi: 10.1016/j.na.2008.02.044. |
[24] |
J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New-York, second edition, 1987. |
[25] |
J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. |
[26] |
O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339.
doi: 10.1111/j.2153-3490.1981.tb01755.x. |
[27] |
O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328.
doi: 10.1256/smsqj.47811. |
[28] |
E. Tziperman and W. C. Thacker, An optimal-control/adjoint approach to studying the oceanic general circulation, Journal of Physical Oceanography, 19 (1989), 1471-1485.
doi: 10.1175/1520-0485(1989)019<1471:AOCEAT>2.0.CO;2. |
[29] |
W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. |
[30] |
C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996.
doi: 10.1017/CBO9780511629570. |
[31] |
D. Zupanski, A general weak constraint applicable to operational 4D-var data assimilation system, Mon. Weather Rev., 125 (1993), 2274-2292.
doi: 10.1175/1520-0493(1997)125<2274:AGWCAT>2.0.CO;2. |
show all references
References:
[1] |
F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325.
doi: 10.1007/BF00271794. |
[2] |
V. I. Agoshkov and V. M. Ipatova, Solvability of the altimeter data assimilation problem in the quasi-geostrophic multi-layer model of ocean circulation, Comput. Math. Math. Phys., 37 (1997), 348-358. |
[3] |
A. Belmiloudi and F. Broissier, A control method for assimilation of surface data in a linearized Navier-Stokes-type problem related to oceanography, SIAM. J. Control Optim., 35 (1997), 2183-2197.
doi: 10.1137/S0363012995286137. |
[4] |
A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994. |
[5] |
T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392.
doi: 10.1016/S0167-2789(99)00206-7. |
[6] |
C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[7] |
F. X. Le Dimet and V. P. Shutyaev, On Newton methods in data assimilation, Russ. J. Numer. Anal. Math. Modelling, 15 (2000), 419-434.
doi: 10.1515/rnam.2000.15.5.419. |
[8] |
I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," Series Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. |
[9] |
G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. |
[10] |
C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal., 61 (2005), 425-460.
doi: 10.1016/j.na.2004.12.005. |
[11] |
C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chin. Ann. of Math. B, 23 (2002), 1-16.
doi: 10.1142/S025295990200002X. |
[12] |
C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst., 9 (2003), 97-131. |
[13] |
N. Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179.
doi: 10.3934/dcds.2007.17.159. |
[14] |
F. X. Le Dimet and V. Shutyaev, On data assimilation for quasilinear parabolic problems, Russian J. Numer. Anal. Math. Modelling, 16 (2001), 247-259. |
[15] |
J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
[16] |
J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I), Computational Mechanics Advance, 1 (1993), 3-54. |
[17] |
J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Computational Mechanics Advance, 1 (1993), 55-120. |
[18] |
J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII), Math. Pures et Appl., 73 (1995), 105-163. |
[19] |
J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday, Nonlinear Anal. Ser. A: Theory Methods, 40 (2000), 439-482.
doi: 10.1016/S0362-546X(00)85026-9. |
[20] |
T. Tachim Medjo, On strong solutions of the multi-layer quasi-geostrophic equations of the ocean, Nonlinear Anal., 68 (2008), 3550-3564
doi: 10.1016/j.na.2007.03.046. |
[21] |
T. Tachim Medjo, Optimal control of the primitive equations of the ocean with state constraints, Accepted in Nonlinear Anal., Ser. A: Theory Methods, 2010. |
[22] |
T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean, in "Mathematical Modeling, Simulation, Visualization and E-Learning: Proceedings of the Bellagio International Conference" (D. Konate, editor), Springer, (2007), 107-150. |
[23] |
T. Tachim Medjo and R. Temam, A two-grid finite difference method for the primitive equations of the ocean, Nonlinear Anal., 69 (2008), 1034-1056.
doi: 10.1016/j.na.2008.02.044. |
[24] |
J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New-York, second edition, 1987. |
[25] |
J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. |
[26] |
O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339.
doi: 10.1111/j.2153-3490.1981.tb01755.x. |
[27] |
O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equations i: Theory, Q. J. R. Meteorol. Soc., 113 (1987), 1311-1328.
doi: 10.1256/smsqj.47811. |
[28] |
E. Tziperman and W. C. Thacker, An optimal-control/adjoint approach to studying the oceanic general circulation, Journal of Physical Oceanography, 19 (1989), 1471-1485.
doi: 10.1175/1520-0485(1989)019<1471:AOCEAT>2.0.CO;2. |
[29] |
W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. |
[30] |
C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996.
doi: 10.1017/CBO9780511629570. |
[31] |
D. Zupanski, A general weak constraint applicable to operational 4D-var data assimilation system, Mon. Weather Rev., 125 (1993), 2274-2292.
doi: 10.1175/1520-0493(1997)125<2274:AGWCAT>2.0.CO;2. |
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