In this work, we present some results showing the controllability of the linear Schrödinger equation with complex potentials. Firstly we investigate the existence and uniqueness theorem for solution of the considered problem. Then we find the gradient of the cost functional with the help of Hamilton-Pontryagin functions. Finally we state a necessary condition in the form of variational inequality for the optimal solution using this gradient.
Financed by the National Centre for Research and Development under grant No. SP/I/1/77065/10 by the strategic scientific research and experimental development program:
SYNAT - “Interdisciplinary System for Interactive Scientific and Scientific-Technical Information”.