Skip to main content
SpringerLink
Log in

Hilbert Space Treatment of Optimal Control Problems with Infinite Horizon

  • Conference paper
  • First Online:
Modeling, Simulation and Optimization of Complex Processes - HPSC 2012

Abstract

We consider a class of infinite horizon optimal control problems as optimization problems in Hilbert spaces. For typical applications it is demonstrated that the state and control variables belong to a Weighted Sobolev – and Lebesgue space, respectively. In this setting Pontryagin’s Maximum Principle as necessary condition for a strong local minimum is shown. The obtained maximum principle includes transversality conditions as well.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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. Aseev, S.M., Kryazhimskii, A.V., Tarasyev, A.M.: The pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons. Proc. Steklov Inst. Math. 233, 64–80 (2001)

    Google Scholar 

  2. Aseev, S.M., Veliov, V.M.: Maximum principle for infinite-horizon optimal control problems with dominating discount. DCDIS: Dyn. Contin. Discret. Impuls. Syst. Ser. B: Appl. Algorithms 19(1–2), 43–63 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control. Springer, New York/Berlin/Heidelberg (1991)

    Book  MATH  Google Scholar 

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

    Google Scholar 

  5. Elstrodt, J.: MaĂź und Integrationstheorie. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Kufner, A.: Weighted Sobolev Spaces. Wiley, Chichester, etc. (1985)

    MATH  Google Scholar 

  9. Lykina, V.: Beiträge zur Theorie der Optimalsteuerungsprobleme mit unendlichem Zeithorizont. Dissertation. BTU Cottbus (2010)

    Google Scholar 

  10. Magill, M.J.P.: Pricing infinite horizon programs. J. Math. Anal. Appl. 88, 398–421 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Ramsey, F.P.: A mathematical theory of savings. Econ. J. 152(38), 543–559 (1928)

    Article  Google Scholar 

  13. Sethi, S.P., Thompson, G.L.: Optimal Control Theory. Applications to Management Science and Economics, 2nd edn. Kluwer, Boston/Dordrecht/London (1985)

    Google Scholar 

  14. Yosida, K.: Functional Analysis. Springer, New York (1974)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brandenburg University of Technology, Cottbus, 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 .

Editor information

Editors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Hans Georg Bock

  2. Vietnam Academy of Science and Technology(VAST), Hanoi, Vietnam

    Xuan Phu Hoang

  3. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  4. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Heidelberg, Germany

    Johannes P. Schlöder

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Pickenhain, S. (2014). Hilbert Space Treatment of Optimal Control Problems with Infinite Horizon. In: Bock, H., Hoang, X., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes - HPSC 2012. Springer, Cham. https://doi.org/10.1007/978-3-319-09063-4_14

Download citation

Publish with us

Policies and ethics

Search

Navigation