Robin-type boundary control problems for the nonlinear Boussinesq type equations
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.
scientific article; zbMATH DE number 1866577
scientific article