Appl Math Optim 51:333–360 (2005)
2005 Springer Science+Business Media, Inc.
Robust Control Problems Associated with the Multilayer
Quasi-Geostrophic Equations of the Ocean
T. Tachim Medjo
Department of Mathematics, Florida International University, DM413B,
University Park, Miami, FL 33199, USA
Abstract. In this article we study some robust control problems associated with the
multilayer quasi-geostrophic equations of the ocean and related to data assimilation
in oceanography. We prove the existence and uniqueness of solutions using a general
framework given in .
Key Words. Robust control, Quasi-geostrophic, Saddle point.
AMS Classiﬁcation. 65N12, 49K35, 76D55.
With modern geophysical satellites, it is now possible to obtain large and sufﬁciently
detailed ﬁelds of oceanographic observations. This has been very useful to oceanogra-
phers. Unfortunately, these data are contaminated with noise and must be analyzed in
order to study, for instance, the ocean currents or to reﬁne the parameters of oceanic mod-
els. There are several approaches to analyzing data in oceanography including Kalman
ﬁltering and variational methods. In the variational method approach, the problem may
be considered as a deterministic control problem where the forcing error and the initial
error are the control variables. Minimal controls are then sought to steer the solutions
towards the data , .
It is well known that optimal and robust control of a ﬂuid ﬂow are important to many
scientiﬁc applications. Optimal control for ﬂuid ﬂow problems such as those governed
by the Navier–Stokes equations has been the subject of extensive studies in the past years
and much progress has been made mathematically and computationally to understand
the subject, see, e.g., , , , , , –,  and –. The control
of a ﬂuid ﬂow with the aim of minimizing the turbulence inside the ﬂuid is studied in