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Well-Posed Problems in the Calculus of Variations

  • T. Zolezzi
Part of the Mathematics and Its Applications book series (MAIA, volume 331)

Abstract

A scalar minimization problem is called well-posed if there exists a unique solution which either attracts every minimizing sequence (according to a definition firstly isolated by Tikhonov), or depends continuously upon problem’s data (according to the classical notion which goes back to Hadamard), or both.

Keywords

Optimal Control Problem Differentiability Property Lagrange Problem Nonlinear Optimal Control Problem Lavrentiev Phenomenon 
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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References

  1. 1.
    Zolezzi, T.: Wellposedness criteria in optimization with application to the calculus of variations, to appear on Nonlinear Anal. TMA. Google Scholar
  2. 2.
    Dontchev, A. L. and Zolezzi, T.: Well-posed optimization problems, Lecture Notes in Math. 1543, Springer Verlag, Berlin, 1993.zbMATHGoogle Scholar
  3. 3.
    Kuratowski, C.: Topologie, vol.1., Warszawa, 1958.zbMATHGoogle Scholar
  4. 4.
    Attouch, H., and Wets, R. J.-B.: Quantitative stability of variational systems II. A framework for nonlinear conditioning, SIAM J. Optim. 3 (1993), 359–381.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Lemaire, B.: Bonne position, conditionnement, et bon comportement asymptotique, Sém. Anal. Convexe, Montpellier (1992), exp.5.Google Scholar
  6. 6.
    Fitzpatrick, S.: Metric projection and the differentiability of distance functions, Bull. Austral. Math. Soc. 22 (1980), 291–312.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Zolezzi, T.: Wellposedness of optimal control problems, Control Cybernet. 23 (1994), 289–301.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Lions, P. L.: Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69, Pitman, Boston, 1982.zbMATHGoogle Scholar
  9. 9.
    Fleming, W. H. and Soner, M. H.: Controlled Markov processes and viscosity solutions, Springer Verlag, Berlin, 1993.zbMATHGoogle Scholar
  10. 10.
    Clarke, F. H. and Wolenski, P. R.: The sensitivity of optimal control problems to time delav. SIAM J. Control Optim. 29 (1991), 1176–1215.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dal Maso, G.: An introduction to I-convergence, Birkhäuser Verlag, Basel, 1993.CrossRefGoogle Scholar
  12. 12.
    Zolezzi, T.: Well-posedness and the Lavrentiev phenomenon, SIAM J. Control Optirn. 30 (1992), 787–799.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Buttazzo, G.: The gap phenomenon for integral functionals: results and open questions, in Variational methods, nonlinear analysis and differential equations, Proc. Int. Workshop, Genova Nervi 1993. ECIG 1994.Google Scholar
  14. 14.
    Kutznetzov, N. N. and Siskin, A. A.: On a many dimensional problem in the theory of quasilinear equations, Z. Vycisl. Mat. Mat. Fiz. 4 (1964), 192–205.Google Scholar
  15. 15.
    Fleming, W. H.: The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations 5 (1969), 515–530.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dacorogna, B.: Direct methods in the calculus of variations, Springer Verlag, Berlin, 1989.zbMATHCrossRefGoogle Scholar
  17. 17.
    Marcellini, P.: Nonconvex integrals of the calculus of variations, Lecture Notes in Math., Vol. 1446, Springer Verlag, Berlin, 1990, pp. 16–57.Google Scholar
  18. 18.
    Olech, C.: The Lyapounov theorem: its extensions and applications, Lecture Notes in Math., Vol. 1446, Springer Verlag, Berlin, 1990, pp.84–103.Google Scholar
  19. 19.
    Zolezzi, T.: Wellposed problems of the calculus of variations for non convex integrals, Submitted.Google Scholar
  20. 20.
    Malanowski, K.: Stability and sensitivity of solutions to nonlinear optimal control problems, to appear in Appl. Math. Optim. Google Scholar
  21. 21.
    Malanowski, K.: Regularity of solutions in stability analysis of optimization and optimal control problems, Control Cybernet. 23 (1994), 61–86.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • T. Zolezzi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

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