Skip to main content
SpringerLink
Log in

Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

In this paper a class of linear-quadratic infinite horizon optimal control problems is considered. Problems of this type are not only of practical interest. They also appear as an approximation of nonlinear problems. The key idea is to introduce weighted Sobolev spaces as state space and weighted Lebesgue spaces as control spaces into the problem setting. We investigate the question of existence of an optimal solution in these spaces and establish a Pontryagin type Maximum Principle as a necessary optimality condition including transversality conditions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aseev, S.M., Kryazhimskii, A.V.: The Pontryagin Maximum Principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257, 1–255

  2. Aseev, S.M., Veliovm, V.M: Maximum principle for problems with dominating discount. Dynamics of Continuous, Discrete and Impulsive Systems, Series B 19 (1-2b), 43–63 (2012)

    MATH  Google Scholar 

  3. Balder, E.J.: An existence result for optimal economic growth problems. J. Math. Anal. Appl. 95, 195–213 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bähr, M., Burtchen, A. Pseudospectralmethoden zur Lösung von Optimalsteuerungsaufgaben mit unendlichen Zeithorizont, Masterthesis, BTU Cottbus-Senftenberg, http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/ http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/ (2013)

  5. Carlson, D.A., Haurie, A.B.: Infinite Horizon Optimal Control. Springer-Verlag, New York, Berlin, Heidelberg (1991)

    Book  MATH  Google Scholar 

  6. Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York, etc (1988)

    MATH  Google Scholar 

  7. Elstrodt, J.: Maß und Integrationstheorie. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  8. Göpfert, A: A. Mathematische Optimierung in allgemeinen Vektorräumen. Teubner (1973)

  9. Garg, D., Hager, W.W., Rao, A.V.: Pseudospectral methods for solving infinite-horizon optimal control 47, 829–837 (2011)

  10. Halkin, H.: Necessary conditions for optimal control problems with infinite horizons. Econometrica 42, 267–272 (1979)

    Article  MathSciNet  Google Scholar 

  11. Ioffe, A.D., Tichomirow, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)

    MATH  Google Scholar 

  12. Kalman, R.E.: Contribution to the theory of optimal control. Bol. Soc. Matem. Mex, 5 (1960)

  13. Kufner, A.: Weighted Sobolev Spaces. John Wiley & Sons, Chichester, etc (1985)

    MATH  Google Scholar 

  14. Letov, A.M.: Analytic controller design I, II. Autom. Remote Contr. 21, 303–306 (1960)

    MATH  Google Scholar 

  15. Lykina, V. Beiträge zur Theorie der Optimalsteuerungsprobleme mit unendlichem Zeithorizont, Dissertation. BTU Cottbus, http://opus.kobv.de/btu/volltexte/2010/1861/pdf/dissertationLykina.pdf (2010)

  16. Lykina, V., Pickenhain, S., Wagner, M.: On a resource allocation model with infinite horizon, vol. 204, pp 595–601, Appl. Math. Comput. (2008)

  17. Pickenhain, S., Lykina, V.: Sufficiency conditions for infinite horizon optimal control problems. In Recent Advances in Optimization. In: Seeger, A. (ed.) Lecture Notes in Economics and Mathematical Systems, vol. 563, pp 217–232. Springer, Berlin, etc (2006)

    Google Scholar 

  18. Pickenhain, S.: On adequate transversality conditions for infinite horizon optimal control problems – a famous example of Halkin. In: Crespo Cuaresma, J., Palokangas, T., Tarasyev, A (eds.) Dynamic Systems, Economic Growth, and the Environment. Dynamic Modeling and Econometrics in Economics and Finance, vol. 12, pp 3–22. Springer, Berlin etc (2010)

    Google Scholar 

  19. Pickenhain, S. Hilbert Space Treatment of Optimal Control Problems with Infinite Horizon. Preprint Reihe Mathematik M-01/2012, BTU Cottbus. (accepted in Modelling, Simulation and Optimization of Complex Processes, Springer). http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/preprint/pickenh/M_01_2012.pdf (2012)

  20. Yosida, K.: Functional Analysis. Springer-Verlag, New York (1974)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brandenburg University of Technology, Cottbus-Senftenberg, Germany

    Sabine Pickenhain

Authors
  1. Sabine Pickenhain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sabine Pickenhain.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pickenhain, S. Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces. Set-Valued Var. Anal 23, 169–189 (2015). https://doi.org/10.1007/s11228-014-0304-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-014-0304-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation