Gradient methods in an optimal control problem for a nonlinear elliptic system
The author considers the nonlinear elliptic equation \[ Ay+| y|^\rho y= f_\Omega+ v_\Omega,\quad x\in\Omega\subset \mathbb{R}^n, \] with the boundary condition \[ {\partial y\over\partial\nu_A}= f_S+ v_S,\quad x\in S=\partial\Omega, \] where \(\rho>0\), \(f_\Omega\) and \(f_S\) are known functions, \(v_\Omega\) and \(v_S\) are controls, \(\partial/\partial\nu_A\) stands for the derivative in the direction of the conormal determined by the elliptic operator \(A\). He considers also the space \(Y= H'(\Omega)\cap L_q(\Omega)\), \(q=\rho+2\). For a given closed bounded convex subset \(U\) of \(Y'\) and a lower semicontinuous functional \({\mathcal J}:Y'\times Y\to\mathbb{R}\) bounded from below, he defines the objective functional \[ I(v)= {\mathcal J}[v, y(v)] \quad\forall v\in Y' \] and studies the following variational problems: find a control \(u\in Y'\) minimizing \(I\) on \(Y'\); find a control \(u\in U\) minimizing \(I\) on \(U\).
scientific article; zbMATH DE number 1023915