Skip to main content
SpringerLink
Log in

Optimal Control of Self-Consistent Classical and Quantum Particle Systems

  • Chapter
  • First Online:
Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

  • 1764 Accesses

Abstract

We study optimal control problems for self-consistent interacting classical and quantum particle systems from the analytical and numerical point of view. This involves microscopic as well as macroscopic quantum models, which have two main features in common: The control enters in a bilinear manner into the partial differential equations and in all models particle interaction takes place via a self-consistent electrostatic potential. This special model structure appears in many different types of applications, like quantum semiconductor devices, optimal quantum control or biomedical applications and it is used to construct fast optimization algorithms.

The authors acknowledge support from the DFG via SPP 1253/2.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N.B. Abdallah, A. Unterreiter, On the stationary quantum drift diffusion model. Z. Angew. Math. Phys. 49, 251–275 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Arnold, Mathematical properties of quantum evolution equations, in Quantum Transport, ed. by N. Ben Abdallah, G. Frosali. Lecture Notes in Mathematics (Springer, Berlin/Heidelberg, 2008)

    Google Scholar 

  3. E. Brown, H. Rabitz, Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena. J. Math. Chem. 31, 17–63 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Burger, R. Pinnau, Fast optimal design of semiconductor devices. SIAM J. Appl. Math 64, 108–126 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Fouego, Optimal control of transient drift-diffusion and nonlinear schrödinger-poisson problems. PhD thesis, WWU Münster, 2013

    Google Scholar 

  6. M. Fouego, M. Burger, Optimal dopant doping profiling with tv penalty. PAMM 12(1), 679–680 (2012)

    Article  Google Scholar 

  7. M. Hintermüller, I. Kopacka, Mathematical programs with complementarity constraints in function space: C-and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Hinze, R. Pinnau, An optimal control approach to semiconductor design. Math. Models Methods Appl. Sc. 12(1), 89–107 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE constraints (Springer, New York, 2009)

    MATH  Google Scholar 

  10. A. Jüngel, A steady-state quantum euler-poisson system for potential flows. Commun. Math. Phys. 194 463–479, (1998)

    Article  MATH  Google Scholar 

  11. E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a trap: a rigorous derivation of the gross-pitaevskii energy functional, in The Stability of Matter: From Atoms to Stars (Springer, Berlin, 2005), pp. 759–771

    Google Scholar 

  12. M. Burger, M. Fouego, D. Mahrahrens, Optimal design of ground states in nonlinear schrödinger-poisson systems. Preprint, WWU Münster, 2013.

    Google Scholar 

  13. M. Burger, M. Fouego, R. Pinnau, Optimal control of transient drift-diffusion models. Preprint, WWU Münster, 2013

    Google Scholar 

  14. F. Pacard, A. Unterreiter, A variational analysis of the thermal equilibrium state of charged quantum fluids. Commun. Partial Differ. Equ. 20(5-6), 885–900 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Pinnau, A. Unterreiter, The stationary current-voltage characteristics of the quantum drift diffusion model. SIAM J. Numer. Anal. 37(1), 211–245 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Pinnau, S. Rau, F. Schneider, Optimal quantum semiconductor design based on the quantum Euler-Poisson model (2012, submitted)

    Google Scholar 

  17. S. Rau, Optimal control of interacting quantum particle systems, PhD thesis, TU Kaiserslautern, 2013

    Google Scholar 

  18. F. Schneider, Optimal design of quantum semiconductor devices. Master’s thesis, University of Kaiserslautern, 2011

    Google Scholar 

  19. A. Unterreiter, S. Volkwein, Optimal control of the stationary quantum drift-diffusion model. Commun. Math. Sci. 5, 85–111 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62, 48149, Münster, Germany

    Martin Burger & Marcisse Fouego

  2. Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653, Kaiserslautern, Germany

    René Pinnau & Sebastian Rau

Authors
  1. Martin Burger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. René Pinnau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcisse Fouego
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sebastian Rau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to René Pinnau .

Editor information

Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Burger, M., Pinnau, R., Fouego, M., Rau, S. (2014). Optimal Control of Self-Consistent Classical and Quantum Particle Systems. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_29

Download citation

Publish with us

Policies and ethics

Search

Navigation