Skip to main content
SpringerLink
Log in

Relaxation problems in control theory

  • Conference paper
  • First Online:
Calculus of Variations and Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1340))

  • 933 Accesses

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 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight 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. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Ambrosio, Nuovi risultati sulla semicontinuità inferiore di certi funzionali integrali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (to appear).

    Google Scholar 

  3. H. Attouch, Variational Convergence of Functionals and Operators, Pitman, Appl. Math. Ser., Boston 1984.

    Google Scholar 

  4. E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim. 4 (1984), 570–598.

    Article  MathSciNet  MATH  Google Scholar 

  5. L.D. Berkowitz, Lower semicontinuity of integral functionals, Trans. Amer. Math. Soc. 192 (1974), 51–57.

    Article  MathSciNet  Google Scholar 

  6. G. Buttazzo, Problemi di semicontinuità e rilassamento in calcolo delle variazioni, Proceedings “Equazioni Differenziali e Calcolo delle Variazioni”, Pisa 1985, Edited by L. Modica, ETS, Pisa (1985), 23–36.

    Google Scholar 

  7. G. Buttazzo, Some relaxation problems in optimal control theory, J. Math. Anal. Appl., (to appear).

    Google Scholar 

  8. G. Buttazzo and E. Cavazzuti, Paper in preparation.

    Google Scholar 

  9. G. Buttazzo and G. Dal Maso, Γ-limits of integral functionals, J. Analyse Math. 37 (1980), 145–185.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Buttazzo and G. Dal Maso, Γ-convergence and optimal control problems, J. Optim. Theory Appl. 38 (1982), 385–407.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl. 122 (1979), 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Lecture Notes in Math. 580, Berlin 1977.

    Google Scholar 

  13. L. Cesari, Semicontinuità e convessità nel calcolo delle variazioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964), 389–423.

    MathSciNet  MATH  Google Scholar 

  14. L. Cesari, Optimization-Theory and Applications, Springer-Verlag, New York 1983.

    Book  MATH  Google Scholar 

  15. F.H. Clarke, Admissible relaxation in variational and control problems, J. Math. Anal. Appl. 51 (1975), 557–576.

    Article  MathSciNet  MATH  Google Scholar 

  16. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York 1983.

    MATH  Google Scholar 

  17. G. Dal Maso and L. Modica, A general theory of variational integrals, “Topics in Functional Analysis 1980–81”, Scuola Normale Superiore, Pisa (1982), 149–221.

    Google Scholar 

  18. E. De Giorgi, Convergence problems for functionals and operators, Proceedings “Recent Methods in Nonlinear Analysis”, Rome 1978, Edited by E. De Giorgi, E. Magenes and U. Mosco, Pitagora, Bologna (1979), 131–188.

    Google Scholar 

  19. E. De Giorgi, Some semicontinuity and relaxation problems, Proceedings “Ennio De Giorgi Colloquium”, Paris 1983, Edited by P. Kree, Pitman, Res. Notes in Math. 125, Boston (1985), 1–11.

    MathSciNet  Google Scholar 

  20. E. De Giorgi and G. Dal Maso, Γ-convergence and calculus of variations, Proceedings “Mathematical Theories of Optimization”, S. Margherita Ligure 1981, Edited by J.P. Cecconi and T. Zolezzi, Springer-Verlag, Lecture Notes in Math. 979, Berlin (1983), 121–143.

    Article  MathSciNet  Google Scholar 

  21. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842–850.

    MathSciNet  MATH  Google Scholar 

  22. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam 1976.

    MATH  Google Scholar 

  23. C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178.

    Article  MathSciNet  MATH  Google Scholar 

  24. A.D. Ioffe, On lower semicontinuity of integral functionals I, SIAM J. Control Optim. 15 (1977), 521–538.

    Article  MathSciNet  MATH  Google Scholar 

  25. A.D. Ioffe, On lower semicontinuity of integral functionals II, SIAM J. Control Optim. 15 (1977), 991–1000.

    Article  MathSciNet  MATH  Google Scholar 

  26. A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems, North-Holland, Amsterdam (1979).

    MATH  Google Scholar 

  27. E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley and Sons, London 1968.

    Google Scholar 

  28. P. Marcellini and C. Sbordone, Dualità e perturbazione di funzionali integrali, Ricerche Mat. 26 (1977), 383–421.

    MathSciNet  MATH  Google Scholar 

  29. P. Marcellini and C. Sbordone, Semicontinuity problems in the calculus of variations, Nonlinear Anal. 4 (1980), 241–257.

    Article  MathSciNet  MATH  Google Scholar 

  30. E.J. McShane, Relaxed controls and variational problems, SIAM J. Control Optim. 5 (1967), 438–485.

    Article  MathSciNet  MATH  Google Scholar 

  31. C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin 1966.

    MATH  Google Scholar 

  32. F. Murat, Contre-exemples pour divers problèmes où le controle intervient dans les coefficients, Ann. Mat. Pura Appl. 112 (1977), 49–68.

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Olech, Weak lower semicontinuity of integral functionals, J. Optim. Theory Appl. 19 (1976), 3–16.

    Article  MathSciNet  MATH  Google Scholar 

  34. Y. Reshetniak, General theorems on semicontinuity and on convergence with a functional, Siberian Math. J. 8 (1967), 801–816.

    Article  Google Scholar 

  35. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1972.

    MATH  Google Scholar 

  36. J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139–167.

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Tonelli, Fondamenti di Calcolo delle Variazioni, Vols. 1,2, Zanichelli, Bologna 1921–23.

    Google Scholar 

  38. J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 111–128.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Warga, Necessary conditions for minimum in relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 129–145.

    Article  MathSciNet  MATH  Google Scholar 

  40. L.C. Young, Lectures on the Calculus of Variations, W.B. Saunders, Philadelphia 1969.

    MATH  Google Scholar 

  41. T. Zolezzi, Book in preparation.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56100, Pisa, Italy

    Giuseppe Buttazzo

Authors
  1. Giuseppe Buttazzo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stefan Hildebrandt David Kinderlehrer Mario Miranda

Additional information

Dedicated to Hans Lewy

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Buttazzo, G. (1988). Relaxation problems in control theory. In: Hildebrandt, S., Kinderlehrer, D., Miranda, M. (eds) Calculus of Variations and Partial Differential Equations. Lecture Notes in Mathematics, vol 1340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082883

Download citation

  • DOI: https://doi.org/10.1007/BFb0082883

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50119-0

  • Online ISBN: 978-3-540-45932-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation