Robin-type boundary control problems for the nonlinear Boussinesq type equations
type: Item
instance of: scholarly article
title: Robin-type boundary control problems for the nonlinear Boussinesq type equations
zbMATH Open document ID: 1032.76017
DOI: 10.1016/S0022-247X(02)00252-4
author: A. Belmiloudi
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The paper studies the existence and uniqueness of solutions and an optimal control problem for the nonlinear Boussinesq equation arising in oceanographic and atmospheric problems. The equation is described in \(\Omega\times(0,T)\), and has the state \(({\mathbf u},{\mathcal T},{\mathcal S},p)\), where \({\mathbf u}\) denotes the velocity, \({\mathcal T}\) the temperature, \({\mathcal S}\) the salinity, \(p\) the pressure, and \(\Omega\) the tropical oceanic domain extending on both sides of the equator and of constant depth \(H\). The curvature of the Earth is neglected. The boundary condition is of mixed type, including periodicity condition in \(x\)-direction and linear or nonlinear Robin-type condition on the surface \(z=0\). The optimal control problem \(({\mathcal P})\) is to minimize the cost functional \(J({\mathbf u},{\mathbf f})\), where \({\mathbf f}\) denotes the surface stress considered as the control. To characterize the optimal control, a family of penalized control problems \(\{( {\mathcal P}_\varepsilon)\}\) is introduced, and the convergence of the corresponding optimal solutions is studied. The problem is weakly formulated via bilinear forms, considered first in the case of nonlinear Robin-type boundary, and then for linear Robin-type boundary.
Mathematics Subject Classification ID: 76D55
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 76U05
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 86A05
zbMATH DE Number: 1866577
zbMATH Keywords: tropical oceanography
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: nonliner Boussinesq equation
zbMATH Keywords: nonlinear Robin-type condition
zbMATH Keywords: cost functional
reviewed by: Takao Nambu
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00252-4
OpenAlex ID: W2005915865
description: scientific article; zbMATH DE number 1866577