Relaxation problems in control theory

  • Giuseppe Buttazzo
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1340)


Control Problem Optimal Control Problem Variational Problem Lower Semicontinuity Differential Inclusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [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. [3]
    H. Attouch, Variational Convergence of Functionals and Operators, Pitman, Appl. Math. Ser., Boston 1984.Google Scholar
  4. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    L.D. Berkowitz, Lower semicontinuity of integral functionals, Trans. Amer. Math. Soc. 192 (1974), 51–57.MathSciNetCrossRefGoogle Scholar
  6. [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. [7]
    G. Buttazzo, Some relaxation problems in optimal control theory, J. Math. Anal. Appl., (to appear).Google Scholar
  8. [8]
    G. Buttazzo and E. Cavazzuti, Paper in preparation.Google Scholar
  9. [9]
    G. Buttazzo and G. Dal Maso, Γ-limits of integral functionals, J. Analyse Math. 37 (1980), 145–185.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    G. Buttazzo and G. Dal Maso, Γ-convergence and optimal control problems, J. Optim. Theory Appl. 38 (1982), 385–407.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals, Ann. Mat. Pura Appl. 122 (1979), 1–60.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Lecture Notes in Math. 580, Berlin 1977.Google Scholar
  13. [13]
    L. Cesari, Semicontinuità e convessità nel calcolo delle variazioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964), 389–423.MathSciNetzbMATHGoogle Scholar
  14. [14]
    L. Cesari, Optimization-Theory and Applications, Springer-Verlag, New York 1983.CrossRefzbMATHGoogle Scholar
  15. [15]
    F.H. Clarke, Admissible relaxation in variational and control problems, J. Math. Anal. Appl. 51 (1975), 557–576.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York 1983.zbMATHGoogle Scholar
  17. [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. [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. [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.MathSciNetGoogle Scholar
  20. [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.MathSciNetCrossRefGoogle Scholar
  21. [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.MathSciNetzbMATHGoogle Scholar
  22. [22]
    I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam 1976.zbMATHGoogle Scholar
  23. [23]
    C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A.D. Ioffe, On lower semicontinuity of integral functionals I, SIAM J. Control Optim. 15 (1977), 521–538.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A.D. Ioffe, On lower semicontinuity of integral functionals II, SIAM J. Control Optim. 15 (1977), 991–1000.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems, North-Holland, Amsterdam (1979).zbMATHGoogle Scholar
  27. [27]
    E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley and Sons, London 1968.Google Scholar
  28. [28]
    P. Marcellini and C. Sbordone, Dualità e perturbazione di funzionali integrali, Ricerche Mat. 26 (1977), 383–421.MathSciNetzbMATHGoogle Scholar
  29. [29]
    P. Marcellini and C. Sbordone, Semicontinuity problems in the calculus of variations, Nonlinear Anal. 4 (1980), 241–257.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    E.J. McShane, Relaxed controls and variational problems, SIAM J. Control Optim. 5 (1967), 438–485.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin 1966.zbMATHGoogle Scholar
  32. [32]
    F. Murat, Contre-exemples pour divers problèmes où le controle intervient dans les coefficients, Ann. Mat. Pura Appl. 112 (1977), 49–68.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    C. Olech, Weak lower semicontinuity of integral functionals, J. Optim. Theory Appl. 19 (1976), 3–16.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Y. Reshetniak, General theorems on semicontinuity and on convergence with a functional, Siberian Math. J. 8 (1967), 801–816.CrossRefGoogle Scholar
  35. [35]
    R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1972.zbMATHGoogle Scholar
  36. [36]
    J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139–167.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Tonelli, Fondamenti di Calcolo delle Variazioni, Vols. 1,2, Zanichelli, Bologna 1921–23.Google Scholar
  38. [38]
    J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 111–128.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J. Warga, Necessary conditions for minimum in relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 129–145.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    L.C. Young, Lectures on the Calculus of Variations, W.B. Saunders, Philadelphia 1969.zbMATHGoogle Scholar
  41. [41]
    T. Zolezzi, Book in preparation.Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Giuseppe Buttazzo
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly

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