# American Institute of Mathematical Sciences

December  2011, 15(3): 769-788. doi: 10.3934/dcdsb.2011.15.769

## Robust control problems for primitive equations of the ocean

 1 Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  February 2009 Revised  April 2010 Published  February 2011

In this article, we study some robust control problems associated with the primitive equations of the ocean and related to data assimilation in oceanography. We prove the existence and uniqueness of solutions to these control problems.
Citation: T. Tachim Medjo. Robust control problems for primitive equations of the ocean. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 769-788. doi: 10.3934/dcdsb.2011.15.769
##### 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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##### 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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