Skip to main content
SpringerLink
Log in

An Optimal Control Problem for a Nonlinear Hyperbolic Equation with an Infinite Time Horizon

  • Chapter
  • First Online:
Evolution Equations of Hyperbolic and Schrödinger Type

Part of the book series: Progress in Mathematics ((PM,volume 301))

  • 1130 Accesses

Abstract

An optimization control problem for a nonlinear hyperbolic equation with non-smooth nonlinearity and infinite time horizon without global solvability of the boundary problem is considered. This problem is solved using an approximation. The convergence of the approximation is proved. Necessary conditions of optimality are obtained.

Mathematics Subject Classification. 49K20.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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. A. Matveev and V. Jakubovich, Optimal control for a system with distributed parameters. J. Sib. Math. 19 (1978), 5, 1109–1140.

    Google Scholar 

  2. D. Tiba, Optimality conditions for nonlinear distributed control problems. Proc. 22 IEEE Conf. N.Y. Dec. Contr., 1983, 1251–1252.

    Google Scholar 

  3. A.V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications. Providence, Amer. Math. Soc., 1999.

    Google Scholar 

  4. D. Tiba, Optimal control for second-order semilinear hyperbolic equations. Contr. Theory Adv. Techn. 3 (1987), 1, 33–46.

    Google Scholar 

  5. V. Sumin, About relaxation of optimization problems for functional equations in spaces of essential bounded functions. Vestnik Nizhegorod. Univ. 1998, 1, 126–133.

    MATH  Google Scholar 

  6. H. Banks, K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identification and control. SIAM J. Contr. Optim. 20 (1982), 6, 815–849.

    Google Scholar 

  7. H. Kuliev, K. Gasanov, Necessary conditions of optimality for system with distributed parameters and control in the coefficients of high derivatives. J. Differ. Equat. 18 (1982), 6, 1028–1036.

    Google Scholar 

  8. J.L. Lions, Contrô le de Systèmes Distribués Singuliers. Paris, Gauthiers–Villars, 1983.

    Google Scholar 

  9. R. Rockafellar, Convex Analysis. Princeton Univ. Press, 1996.

    Google Scholar 

  10. F.H. Clarke, Optimization and Nonsmooth Analysis. N.Y., Wiley, 1983.

    Google Scholar 

  11. V. Barbu, Boundary control problems with nonlinear state equation. SIAM J. Control and Optim. 20 (1982), 1, 125–143.

    Google Scholar 

  12. S. Serovajsky, Optimal control for the elliptic equation with nonsmooth nonlinearity. J. Differ. Equat. 39 (2003), 10, 1497–1502.

    Google Scholar 

  13. A. Seierstad, Necessary conditions for nonsmooth, infinite–horizon, optimal control problems. J. Optim. Theory Appl. 103 (1999), 1, 201–209.

    Google Scholar 

  14. S. Aseev and A. Kryazhimskiy, The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons. SIAM J. Control Optim. 43 (2004), 3, 1094–1119.

    Google Scholar 

  15. J.L. Lions, Contrô le Optimal de Syst`emes Gouvernès par des Equations aux Dérivées Partielles. Paris, Gauthiers–Villars, 1983.

    Google Scholar 

  16. V. Barbu, Optimal feedback control for a class of nonlinear distributed parameter systems. SIAM J. Control and Optim. 21 (1983), 6, 871–894.

    Google Scholar 

  17. P. Cannarsa, G. Da Prato, Nonlinearity of control with infinite horizon for distributed parameter systems and stationary Hamilton–Jacobi equations. SIAM J. Contr. Optim. 27 (1989), 4, 861–875.

    Google Scholar 

  18. D. Tiba, Optimal control of nonsmooth distributed parameter systems. Lecture Notes in Mathematics. 1459 (1990), Berlin, WJ Springer-Verlag, 159 p.

    Google Scholar 

  19. F.P. Vasiliev, Optimization Metods. Moscow, Factorial, 2002.

    Google Scholar 

  20. T.A. Zolezzi, A characterization of well-posed optimal control systems. SIAM J. Contr. Optim. 19 (1981), 5, 605–616.

    Google Scholar 

  21. S. Serovajsky, Approximate Solution of Singular Optimization Problems. J. Mathematical Notes, 74 (2003), 5, 685–694.

    Google Scholar 

  22. J. Sea, Optimisation. Théorie et algorithmes. Paris, Dunod, 1971.

    Google Scholar 

  23. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Gauthiers–Villars, Paris, 1969.

    Google Scholar 

  24. F.L. Chernous’ko and V.V. Kolmanovsky, Computing and Approximate Optimization Methods. In “Mathematical Analysis. Results of Science and Technique”. 14 (1977), 101–166.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. al-Farabi Kazakh National University, Masanchi str. 39/47, 050012, Almaty, Kazakhstan

    Simon Serovajsky & Kanat Shakenov

Authors
  1. Simon Serovajsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kanat Shakenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simon Serovajsky .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

  3. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel

About this chapter

Cite this chapter

Serovajsky, S., Shakenov, K. (2012). An Optimal Control Problem for a Nonlinear Hyperbolic Equation with an Infinite Time Horizon. In: Ruzhansky, M., Sugimoto, M., Wirth, J. (eds) Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol 301. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0454-7_15

Download citation

Publish with us

Policies and ethics

Search

Navigation