# Investigation of optimal control problems governed by a time-dependent Kohn-Sham model

## TL;DR

This paper explores optimal control problems within the time-dependent Kohn-Sham model of TDDFT, proposing a numerical framework validated by experiments to control multi-electron quantum systems efficiently.

## Contribution

It develops a theoretical and numerical framework for optimal control of Kohn-Sham TDDFT models, including existence proofs and a computational scheme.

## Key findings

- Numerical experiments demonstrate the effectiveness of the control approach.
- The proposed discretization and conjugate gradient method successfully solve control problems.
- The framework can handle various control objectives in quantum systems.

## Abstract

Many application models in quantum physics and chemistry require to control multi-electron systems to achieve a desired target configuration. This challenging task appears possible in the framework of time-dependent density functional theory (TDDFT) that allows to describe these systems while avoiding the high dimensionality resulting from the multi-particle Schr\"{o}dinger equation. For this purpose, the theory and numerical solution of optimal control problems governed by a Kohn-Sham TDDFT model are investigated, considering different objectives and a bilinear control mechanism. Existence of optimal control solutions and their characterization as solutions to Kohn-Sham TDDFT optimality systems are discussed. To validate this control framework, a time-splitting discretization of the optimality systems and a nonlinear conjugate gradient scheme are implemented. Results of numerical experiments demonstrate the computational capability of the proposed control approach.