Existence and Characterization of an Optimal Control for the Problem of Long Waves in a Shallow-Water Model

Belmiloudi, Aziz

HAL, hal-00648929 / Harvested from HAL

Résumé

In this paper we present a method of optimal control developed in order to calculate the current corresponding to the observed sea level in a fluid domain $\Omega$ and during a time T. The control is the external stress $\f\ $. The cost function measures the distance between the observed and computed sea levels. The equations satisfied by the depth and the depth averaged velocity are of nonlinear shallow-water type. The existence and uniqueness of a solution for the direct problem are studied in the case of Dirichlet nonhomogeneous boundary conditions. We prove, by means of minimizing sequences, the existence of an optimal control $(\f\ ,\U\ )$ in the case of the small data and a very viscous fluid. To characterize it we build a sequence of problems corresponding to a linearization of the direct problem. We obtain the necessary conditions of optimality. The set of equations and the inequality characterizing the optimal control $(\f\ ,\U\ )$ is obtained as the limit of the penalization.

Détail

Publié le : 2001-07-05
Classification:  optimal control,  shallow-water equations,  nonlinear partial differential equations,  minimizing sequences,  penalization,  altimetric measurements,  oceanography,  35Q30, 49J20, 49K20, 65J10, 76B15, 76D05, 76U05, 86A22,  [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC],  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],  [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph],  [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]