Optimal control of the primitive equations of the ocean with lagrangian observations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 400-419.

We consider an optimal control problem for the three-dimensional non-linear Primitive Equations of the ocean in a vertically bounded and horizontally periodic domain. We aim to reconstruct the initial state of the ocean from lagrangian observations. This inverse problem is formulated as an optimal control problem which consists in minimizing a cost function representing the least square error between lagrangian observations and their model counterpart, plus a regularization term. This paper proves the existence of an optimal control for the regularized problem. To this end, we also prove new energy estimates for the Primitive Equations, thanks to well-chosen functional spaces, which distinguish the vertical dimension from the horizontal ones. We illustrate the result with a numerical experiment.

DOI: 10.1051/cocv/2009003
Classification: 35Q35,  49J20,  65K10
Keywords: optimal control, partial differential equations, primitive equations, numerical simulation
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Nodet, Maëlle. Optimal control of the primitive equations of the ocean with lagrangian observations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 400-419. doi : 10.1051/cocv/2009003. http://archive.numdam.org/articles/10.1051/cocv/2009003/

[1] P. Courtier, J.N. Thepaut and A. Hollingsworth, A strategy for operational implementation of 4D-Var, using an incremental approach. Q. J. Roy. Meteor. Soc. 120 (1994) 1367-1387.

[2] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics 749. Springer-Verlag, Berlin (1979). | Zbl

[3] C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations. Chinese Ann. Math. Ser. B 23 (2002) 277-292. Dedicated to the memory of Jacques-Louis Lions. | Zbl

[4] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean. Nonlinearity 20 (2007) 2739-2753. | Zbl

[5] F.-X. Le Dimet, A general formalism of variational analysis. Technical Report 73091 22, CIMMS, Norman, Oklahoma, USA (1982).

[6] F.-X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus Series A 38 (1986) 97.

[7] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968). | Zbl

[8] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | Zbl

[9] J.-L. Lions, R. Temam and S.H. Wang, New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5 (1992) 237-288. | Zbl

[10] G. Madec, P. Delecluse, M. Imbard and C. Lévy, OPA 8.1 Ocean General Circulation Model reference manual. Note du Pôle de Modélisation, Institut Pierre Simon Laplace, France (1999).

[11] M. Nodet, Variational assimilation of Lagrangian data in oceanography. Inverse Problems 22 (2006) 245-263. | Zbl

[12] R. Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications 2. North-Holland Publishing Co., Amsterdam (1977). | Zbl

[13] R. Temam, Navier-Stokes equations, Theory and numerical analysis. Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI (2001). | Zbl

[14] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of mathematical fluid dynamics III, North-Holland, Amsterdam (2004) 535-657.

[15] E. Titi and C. Cao, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Annals Math. 166 (2007) 245-267. | Zbl

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