Skip to main content
SpringerLink
Log in

Bang-bang optimal control for the dam problem

  • Published:
Applied Mathematics and Optimization Aims and scope Submit manuscript

Abstract

The dam problem with general geometry is considered. Fluid is drawn from the bottomS 1 at a ratek where 0 ≤k ≤ N, ∫ S 1 k ≤ M; the objective is to minimize the “total pressure” of the fluid in the dam. A bang-bang principle is established for any optimal controlk 0, that is,k 0 = 0 on a setA andk 0 =N on the complement setS 1 ∖A. In the case of a rectangular dam the structure ofA is determined and the uniqueness of the minimizerk 0 is established.

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. Alt, HW (1977) A free boundary problem associated with the flow of ground water. Arch Rational Mech Anal 64:111–126

    Google Scholar 

  2. Alt HW (1979) Stromungen durch inhomogene porose medien mit freien rand. J Reine Angew Math 305:89–115

    Google Scholar 

  3. Baiocchi C (1972) Su un problema a frontiera libera connesso a questione di idraulica. Ann. Mat Pura Appl 92:107–127

    Google Scholar 

  4. Barbu V (1984) Optimal Control of Variational Inequalities. Pitman, London.

    Google Scholar 

  5. Brezis H, Kinderlehrer D, Stampacchia G (1978) Sur une nouvelle formulation de probleme de l'ecoulement a traverse une digue. C R Acad Sci Paris 287:711–714

    Google Scholar 

  6. Carrillo-Menendez J, Chipot M (1982) On the dam problem. J Differential Equations 45:234–271

    Google Scholar 

  7. Friedman A (1976) A problem in hydraulics with non-monotone free boundary. Indiana Univ Math J 25:577–592

    Google Scholar 

  8. Friedman A (1982) Variational Principles and Free Boundary Problems. Wiley, New York

    Google Scholar 

  9. Friedman A, Optimal control for parabolic variational inequalities. SIAM J Control Optim (to appear)

  10. Friedman A, Yaniro D (1985) Optimal control for the dam problem. Appl Math Optim 13:59–78

    Google Scholar 

  11. Gilbarg D, Trudinger NS (1977) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Heidelberg

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Purdue University, 47907, West Lafayette, IN, USA

    Avner Friedman & Jiongmin Yong

  2. Peking University, Peking, People's Republic of China

    Shaoyun Huang

Authors
  1. Avner Friedman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaoyun Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiongmin Yong
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work is partially supported by National Science Foundation Grants DMS-8501397 and DMS-8420896.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Friedman, A., Huang, S. & Yong, J. Bang-bang optimal control for the dam problem. Appl Math Optim 15, 65–85 (1987). https://doi.org/10.1007/BF01442646

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01442646

Keywords

Search

Navigation