Investigation of Optimal Control Problems Governed by a Time-Dependent Kohn-Sham Model

A viable way to develop optimal control strategies for multi-particle quantum systems is to consider the framework of time-dependent density functional theory (TDDFT), where low-dimensional nonlinear Schrödinger models are developed to compute the electronic density of related high-dimensional linear Schrödinger equations. Among these models, the Kohn-Sham TDDFT system allows to accommodate control mechanisms in the same potentials that appear in the original multi-dimensional Schrödinger equations, thus allowing a physical interpretation and a laboratory implementation. The purpose of this paper is the mathematical analysis of optimal control problems governed by the time-dependent Kohn-Sham (TDKS) equations including a control potential that has the purpose to drive the evolution of the electron density to perform given tasks. For the resulting optimal control problems, existence of optimal solutions is proved and their characterization as solutions of TDKS optimality systems is investigated.