Analysis of optimal control problems for the 2-D stationary Boussinesq equations
The authors use the system \[ \begin{aligned} - \nu \Delta {\mathbf u} + ({\mathbf u} \cdot \nabla){\mathbf u} &= - \nabla p + \alpha T{\mathbf g} + {\mathbf f}, \\ \nabla \cdot {\mathbf u} &= 0,\\ - \kappa \Delta T + ({\mathbf u} \cdot \nabla)T &= Q \end{aligned} \] in a plane domain \(\Omega\) with boundary \(\Gamma\) as a model of viscous incompressible thermally convected flow. \((({\mathbf u}, p, T) =\) (velocity, pressure, temperature)). The boundary conditions are \[ {\mathbf u} = 0 \;\;\text{on} \;\Gamma, \quad T = h \;\;\text{on} \;\Gamma_D, \quad {\partial T \over \partial {\mathbf n}} = g \;\;\text{on} \;\Gamma_N , \] where \(\Gamma_N\) is an open subset of \(\Gamma\) and \(\Gamma_D = \Gamma \setminus \Gamma_N .\) The controls are \(Q\) and \(g,\) and the authors study minimization of two quadratic functionals, both involving the integrals of \(|Q|^2\) and \(|g|^2.\) In the first functional, the additional term is the integral of \(|\nabla \times {\mathbf u}|^2\) (thus the objective is to minimize vorticity); in the second, the additional term is the integral of \(|T - T_d|^2\) where \(T_d\) is a target temperature field. The authors show existence of minimizers and obtain an optimality system using the method of Lagrange multipliers.
scientific article; zbMATH DE number 1423630
scientific article