Journal of Dynamical and Control Systems

, Volume 24, Issue 4, pp 657–679 | Cite as

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

  • M. Sprengel
  • G. Ciaramella
  • A. BorzìEmail author


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.


Kohn-Sham model Optimal control Time-dependent density functional theory 

Mathematics Subject Classification (2010)

35Q40 49J20 81Q93 



We are very grateful to Fredi Tröltzsch for many helpful remarks and to Andrei Fursikov for continued support of this work.


  1. 1.
    Andrews B, Hopper C. The Ricci flow in Riemannian geometry. Lecture notes in mathematics. Berlin: Springer; 2011.CrossRefzbMATHGoogle Scholar
  2. 2.
    Attaccalite C, Moroni S, Gori-Giorgi P, Bachelet GB. Correlation energy and spin polarization in the 2D electron gas. Phys Rev Lett 2002;88:256,601.CrossRefGoogle Scholar
  3. 3.
    Baudouin L, Kavian O, Puel JP. Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control. J Differ Equ 2005; 216(1):188–222.CrossRefzbMATHGoogle Scholar
  4. 4.
    Borzì A, Ciaramella G, Sprengel M. Formulation and numerical solution of quantum control problems. Philadelphia: Society for Industrial and Applied Mathematics; 2017.CrossRefzbMATHGoogle Scholar
  5. 5.
    Borzì A, Schulz V. Computational optimization of systems governed by partial differential equations. Philadelphia: Society for Industrial and Applied Mathematics; 2012.zbMATHGoogle Scholar
  6. 6.
    Cancès E, le Bris C, Pilot M. Contrôle optimal bilinéaire d’une équation de schrödinger. Comptes Rendus de l’Acadé,mie des Sciences - Series I - Mathematics 2000;330(7):567–571.zbMATHGoogle Scholar
  7. 7.
    Cancès E, Le bris C. On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics. Math Models Methods Appl Sci 1999;9(7):963–990.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castro A, Appel H, Oliveira M, Rozzi CA, Andrade X, Lorenzen F, Marques MAL, Gross EKU, Rubio A. Octopus: a tool for the application of time-dependent density functional theory. Phys Status Solidi (B) 2006;243(11):2465–88.CrossRefGoogle Scholar
  9. 9.
    Castro A, Werschnik J, Gross EKU. Controlling the dynamics of many-electron systems from first principles: a combination of optimal control and time-dependent density-functional theory. Phys Rev Lett 2012;109:153,603.CrossRefGoogle Scholar
  10. 10.
    Ciarlet PG. Linear and nonlinear functional analysis with applications. Philadelphia: Society for Industrial and Applied Mathematics; 2013.zbMATHGoogle Scholar
  11. 11.
    Constantin LA. Dimensional crossover of the exchange-correlation energy at the semilocal level. Phys Rev B 2008;78:155,106.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Engel E, Dreizler RM. Density functional theory, an advanced course. Heidelberg: Springer; 2011.CrossRefzbMATHGoogle Scholar
  13. 13.
    Evans LC. Partial differential equations, Graduate Studies in Mathematics, vol 19, 2nd edn. Providence: American Mathematical Society; 2010.Google Scholar
  14. 14.
    Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys Rev 1964;136: B864–71.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jerome JW. Time dependent closed quantum systems: nonlinear Kohn-Sham potential operators and weak solutions. J Math Anal Appl 2015;429(2):995–1006.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys Rev 1965;140:A1133–8.MathSciNetCrossRefGoogle Scholar
  17. 17.
    van Leeuwen R. Mapping from densities to potentials in time-dependent density-functional theory. Phys Rev Lett 1999;82:3863–6.CrossRefGoogle Scholar
  18. 18.
    Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars: Dunod; 1969.zbMATHGoogle Scholar
  19. 19.
    Maday Y, Salomon J, Turinici G. Monotonic time-discretized schemes in quantum control. Numer Math 2006;103(2):323–338.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Marques MAL, Oliveira MJT, Burnus T. Libxc: a library of exchange and correlation functionals for density functional theory. Comput Phys Commun 2012; 183(10):2272–81.CrossRefGoogle Scholar
  21. 21.
    Marques MAL, Ullrich CA, Nogueira F, Rubio A, Burke K, Gross EKU. Time-dependent density functional theory, Lecture notes in physics, vol. 706. Berlin: Springer; 2006.CrossRefGoogle Scholar
  22. 22.
    Parr RG, Yang W. Density-functional theory of atoms and molecules. Oxford: Oxford University Press; 1989.Google Scholar
  23. 23.
    Remmert E. Theory of complex functions. Heidelberg: Springer; 1991.CrossRefzbMATHGoogle Scholar
  24. 24.
    Ruggenthaler M, Penz M, Van Leeuwen R. Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory. J Phys: Condens Matter 2015;27(20):203,202.Google Scholar
  25. 25.
    Runge E, Gross EKU. Density-functional theory for time-dependent systems. Phys Rev Lett 1984;52:997–1000.CrossRefGoogle Scholar
  26. 26.
    Salomon J. Limit points of the monotonic schemes in quantum control. Proceedings of the 44th IEEE Conference on Decision and Control Sevilla; 2005.Google Scholar
  27. 27.
    Sprengel M, Ciaramella G, Borzì A. A COKOSNUT code for the control of the time-dependent Kohn–Sham model. Comput Phys Commun 2017;214:231–8.CrossRefzbMATHGoogle Scholar
  28. 28.
    Sprengel M, Ciaramella G, Borzì A. A theoretical investigation of time-dependent Kohn–Sham equations. SIAM J Math Anal 2017;49(3):1681–1704.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Stone MH. On one-parameter unitary groups in Hilbert space. Annals of Mathematics. Second Series 1932;33(3):643–8.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tröltzsch F. Optimal control of partial differential equations, 1st ed. Providence: American Mathematical Society; 2010.zbMATHGoogle Scholar
  31. 31.
    Tröltzsch F, Valli A. 2014. Optimal control of low-frequency electromagnetic fields in multiply connected conductors. DFG-Research Center Matheon, Matheon preprint.Google Scholar
  32. 32.
    von Winckel G, Borzì A. Computational techniques for a quantum control problem with H 1-cost. Inverse Problems 2008;24(3):034,007,23.zbMATHGoogle Scholar
  33. 33.
    von Winckel G, Borzì A, Volkwein S. A globalized Newton method for the accurate solution of a dipole quantum control problem. SIAM J Sci Comput 2009/10;31(6):4176–4203.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yserentant H. Regularity and approximability of electronic wave functions, Lecture notes in mathematics, vol 2000. Berlin: Springer; 2010.zbMATHGoogle Scholar
  35. 35.
    Zowe J, Kurcyusz S. Regularity and stability for the mathematical programming problem in Banach spaces. Appl Math Optim 1979;5(1):49–62.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Institut für MathematikUniversität KonstanzKonstanzGermany

Personalised recommendations