Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions
The problem consists in minimizing the functionals \[ \begin{aligned}{{\mathcal J}_f}({\mathbf u}, f) &= {1 \over 2} \int_\Omega | {\mathbf u} - {\mathbf U}| ^2 dx + {\delta \over 2} \int_\Omega | f| ^2 dx,\\ {{\mathcal J}_{\mathbf g}}({\mathbf u}, {\mathbf g}) &= {1 \over 2} \int_\Omega | {\mathbf u} - {\mathbf U}| ^2 dx + {\delta \over 2} \int_\Omega | {\mathbf g}| ^2 dx\end{aligned} \] in a 3-dimensional domain \(\Omega,\) the scalar function \(f\) and the vector functions \({\mathbf u}, {\mathbf g}\) satisfying \[ \nabla \cdot {\mathbf u} = f, \quad \nabla \times {\mathbf u} = {\mathbf g} \qquad \text{in} \;\Omega \] with either of the boundary conditions \({\mathbf u} \cdot {\mathbf n} = 0\) or \({\mathbf u} \times {\mathbf n} = 0\) on the boundary \(\Gamma.\) There are in addition compatibility conditions for \(f\) and \({\mathbf g}\) depending on the boundary condition used; in optimal control problems, these compatibility conditions play the role of constraints on the control variables. The authors show existence and uniqueness of solutions to the minimization problem and existence of Lagrange multipliers, and derive a method to compute optimal controls and states from an \textit{optimality system} of partial differential equations. The solution of this system is approximated with finite elements.
scientific article; zbMATH DE number 1311093