Skip to main content
SpringerLink
Log in

Recent Advances in the Analysis of State-constrained Elliptic Optimal Control Problems

  • Conference paper
  • First Online:
Book cover Optimal Control of Coupled Systems of Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 158))

  • 1272 Accesses

Abstract

Pointwise state-constrained control problems associated with semilinear elliptic equations are studied. Theoretical results are derived, which are necessary to carry out the numerical analysis of the control problem. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.

The first author was partially supported by the Spanish Ministry of Education and Science under project MTM2005-06817 and “Ingenio Mathematica (i-MATH)” CSD2006-00032 (Consolider Ingenio 2010).

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Casas, Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM:COCV, (to appear).

    Google Scholar 

  2. E. Casas, J. de los Reyes, and F. Tröltzsch, Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., (to appear).

    Google Scholar 

  3. E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim., 40 (2002), pp. 1431–1454.

    Article  MathSciNet  Google Scholar 

  4. E. Casas and F. Tröltzsch, Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations, App. Math. Optim., 39 (1999), pp. 211–227.

    Article  MathSciNet  Google Scholar 

  5. E. Casas, F. Tröltzsch, and A. Unger, Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, SIAM J. Control Optim., 38 (2000), pp. 1369–1391.

    Article  MathSciNet  Google Scholar 

  6. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

    Book  Google Scholar 

  7. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

    MATH  Google Scholar 

  8. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), pp. 161–219.

    Article  MathSciNet  Google Scholar 

  9. M. Mateos, Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado, PhD thesis, University of Cantabria, 2000.

    Google Scholar 

  10. H. Maurer and J. Zowe, First-and second-order conditions in infinite-dimensional programming problems, Math. Programming, 16 (1979), pp. 98–110.

    Article  MathSciNet  Google Scholar 

  11. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), pp. 189–258.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departmento de Matemática Aplicada y Ciencias de la Computación E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005, Santander, Spain

    Eduardo Casas

  2. Institut für Mathematik, Technische Universität Berlin, D-10623, Berlin, Germany

    Fredi Tröltzsch

Authors
  1. Eduardo Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fredi Tröltzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Karl Kunisch

  2. Abt. Partielle Differentialgleichung Weierstraß-Institut für Angewandte Analysis und Stochastik, Humboldt-Universität zu Berlin, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

  3. LS Angewandte Mathematik II Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, Martensstrasse 3, 91058, Erlangen, Germany

    Günter Leugering

  4. Institut für Mathematik Fak. II Mathematik & Naturwissenschaften, TU Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Fredi Tröltzsch

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Casas, E., Tröltzsch, F. (2009). Recent Advances in the Analysis of State-constrained Elliptic Optimal Control Problems. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Optimal Control of Coupled Systems of Partial Differential Equations. International Series of Numerical Mathematics, vol 158. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8923-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation