Approximation and regularization of the optimal control problem for a Schrödinger type equation
The authors discuss the problem \[ J(u)\equiv \int_{\Omega}| E(x,y,Z)-g(x,y)|^ 2 dx dy\to \min, \] \[ (1)\quad \partial E/\partial Z+i(\partial^ 2/\partial x^ 2+\partial^ 2/\partial y^ 2)E+if(x,y)E=0,\quad (x,y,z)\in \Omega \times (O,Z), \] \[ E|_{\partial \Omega \times (O,Z)}=0,\quad E|_{z=0}=u,\quad u\in U, \] where the set of controls U is a bounded convex set in \({\mathcal H}\equiv H^ 1_ 0(\Omega)\cap H^ 2(\Omega)\), \(\Omega\) is a bounded domain with regular boundary and f,g are given functions. A discretization of the problem with piecewise constant functions \(E_ h:[O,Z]\to {\mathcal H}_ h\subset {\mathcal H}\) and \(u_ h\in {\mathcal H}_ h\) is considered. Estimates for the convergence of approximate solutions of (1) and values of the functional J are given. It is shown that Tikhonov's regularization method is applicable.
scientific article; zbMATH DE number 4009195