American Institute of Mathematical Sciences

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

Robust control problems for primitive equations of the ocean

T. Tachim Medjo 1,

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.
Keywords: saddle point., quasi-geostrophic, Robust control.
Mathematics Subject Classification: 65N12, 49K35, 76D5.
Citation: T. Tachim Medjo. Robust control problems for primitive equations of the ocean. Discrete & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New-York, second edition, 1987. Google Scholar

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. Google Scholar

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339. doi: 10.1111/j.2153-3490.1981.tb01755.x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. Google Scholar

[30]

C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996. doi: 10.1017/CBO9780511629570.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

A. Bennett, "Inverse Methods in Physical Oceanography," Cambridge University Press, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology," John Wiley and Sons, New York, 1980. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, New-York, second edition, 1987. Google Scholar

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate," American Institute of Physics, New-York, 1992. Google Scholar

[26]

O. Talagrand, On the mathematics of data assimilation, Tellus, 33 (1981), 321-339. doi: 10.1111/j.2153-3490.1981.tb01755.x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling," Oxford University Press, Oxford, 1986. Google Scholar

[30]

C. Wunsch, "The Ocean Circulation Inverse Problem," Cambridge University Press, 1996. doi: 10.1017/CBO9780511629570.  Google Scholar

[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.  Google Scholar

[1]

Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023
[2]

T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171
[3]

May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179
[4]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025
[5]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336
[6]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023
[7]

Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525
[8]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[9]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[10]

Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010
[11]

Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277
[12]

T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119
[13]

Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133
[14]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[15]

Eleftherios Gkioulekas, Ka Kit Tung. Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 293-314. doi: 10.3934/dcdsb.2007.7.293
[16]

Tongtong Liang, Yejuan Wang. Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4697-4726. doi: 10.3934/dcdsb.2020309
[17]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093
[18]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[19]

Xiao-Fei Peng, Wen Li. A new Bramble-Pasciak-like preconditioner for saddle point problems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 823-838. doi: 10.3934/naco.2012.2.823
[20]

T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy