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 & 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. 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.

[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. 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. 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. 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. doi: 10.1515/rnam.2000.15.5.419.

[8]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", Series Classics in Applied Mathematics, (1999).

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology,", John Wiley and Sons, (1980).

[10]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption,, Nonlinear Anal., 61 (2005), 425. 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. 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.

[13]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. 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.

[15]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. 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.

[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.

[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.

[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. 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. 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., (2010).

[22]

T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean,, in, (2007), 107.

[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. doi: 10.1016/j.na.2008.02.044.

[24]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987).

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate,", American Institute of Physics, (1992).

[26]

O. Talagrand, On the mathematics of data assimilation,, Tellus, 33 (1981), 321. 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. 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. 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, (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. 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. 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.

[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. 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. 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. 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. doi: 10.1515/rnam.2000.15.5.419.

[8]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", Series Classics in Applied Mathematics, (1999).

[9]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology,", John Wiley and Sons, (1980).

[10]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption,, Nonlinear Anal., 61 (2005), 425. 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. 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.

[13]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. 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.

[15]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. 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.

[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.

[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.

[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. 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. 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., (2010).

[22]

T. Tachim Medjo and R. Temam, A small eddy correction algorithm for the primitive equations of the ocean,, in, (2007), 107.

[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. doi: 10.1016/j.na.2008.02.044.

[24]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer-Verlag, (1987).

[25]

J. P. Peixoto and A. H. Oort, "Physics of Climate,", American Institute of Physics, (1992).

[26]

O. Talagrand, On the mathematics of data assimilation,, Tellus, 33 (1981), 321. 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. 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. 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, (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. doi: 10.1175/1520-0493(1997)125<2274:AGWCAT>2.0.CO;2.

[1]

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

[2]

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

[3]

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

[4]

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 - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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 - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152

[11]

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

[12]

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

[13]

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

[14]

T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201

[15]

Magdi S. Mahmoud, Omar Al-Buraiki. Robust control design of autonomous bicycle kinematics. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 181-191. doi: 10.3934/naco.2014.4.181

[16]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717

[17]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[18]

Monica Lazzo, Paul G. Schmidt. Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 566-575. doi: 10.3934/proc.2005.2005.566

[19]

M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365

[20]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]