Advertisement

Feasibility in finite time

  • S. D. Flåm
  • J.-B. Hiriart-Urruty
  • A. Jourani
Article

Abstract

It is common to tolerate that a system’s performance be unsustainable during an interim period. To live long however, its state must eventually satisfy various constraints. In this regard we design here differential inclusions that generate, in one generic format, two distinct phases of system dynamics. The first ensures feasibility in finite time; the second maintains that property forever after.

Key words and phrase

Differential inclusions generalized subdifferentials duality mapping distance function prox-regularity finite-time absorption sweeping processes 

2000 Mathematics Subject Classification

28B05 34A60 37C10 37F05 

References

  1. 1.
    E. Asplund, Chebyshev sets in Hilbert space. Trans. Am. Math. Soc. 144 (1969), 235–240.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. P. Aubin, Viability theory. Birkhäuser, Basel (1991).zbMATHGoogle Scholar
  3. 3.
    J. P. Aubin and A. Cellina, Differential inclusions. Springer-Verlag, Berlin (1984).zbMATHGoogle Scholar
  4. 4.
    J. P. Aubin and I. Ekeland, Applied nonlinear analysis. Wiley, New York (1984).zbMATHGoogle Scholar
  5. 5.
    J. P. Aubin and H. Frankowska, Set-valued analysis. Birkhäuser, Basel (1990).zbMATHGoogle Scholar
  6. 6.
    G. Beer, Topologies on closed and closed convex sets. Kluwer Academic, Dordrecht (1993).zbMATHGoogle Scholar
  7. 7.
    H. Benabdellah, Existence of solutions to the nonconvex sweeping process. J. Differ. Equations 164 (2000), 286–295.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. M. Borwein and S. P. Fitzpatrick, Existence of nearest points in Banach space. Can. J. Math. 41 (1989), No. 4, 702–720.zbMATHMathSciNetGoogle Scholar
  9. 9.
    J. M. Borwein, S. P. Fitzpatrick, and J. R. Giles, The differentiability of real functions on normed linear spaces using generalized subgradients. J. Math. Anal. Appl. 128 (1987) 512–534.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Bounkhel and L. Thibault, On various notions of regularity of sets in nonsmooth analysis. Nonlin. Anal. TMA 48 (2002), 223–246.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Brogliato, A. Daniilidis, C. Maréchal, and V. Acary, On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55 (2006) 45–51.zbMATHCrossRefGoogle Scholar
  12. 12.
    C. Castaing, Version aléatoire du problème de raffle par un convexe. Sém. Anal. Conv. Montpellier 1 (1974).Google Scholar
  13. 13.
    C. Castaing, T. X. Duc Ha, and M. Valadier, Evolution equations governed by the sweeping process. Set-Valued Anal. A (1993) 109–139.CrossRefMathSciNetGoogle Scholar
  14. 14.
    C. Castaing and M. D. P. Monteiro Marques, Evolution problems associated with nonconvex closed moving sets. Port. Math. 53 (1996) 73–87.zbMATHMathSciNetGoogle Scholar
  15. 15.
    F. H. Clarke, R. J. Stern, and P. R. Wolenski, Proximal smoothness and the lower C 2 property. J. Convex Anal. 2 (1995) 117–144.zbMATHMathSciNetGoogle Scholar
  16. 16.
    F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory. Springer-Verlag, Berlin (1998).zbMATHGoogle Scholar
  17. 17.
    K. Deimling, Multivalued differential equations. De Gruyter, Berlin (1992).zbMATHGoogle Scholar
  18. 18.
    J.-B. Hiriart-Urruty, Ensembles de Tchebychev vs. ensembles convexes: l’etat de la situation vu par l’analyse convexe non lisse. Ann. Sci. Math Que. 22 (1998) 47–62.zbMATHMathSciNetGoogle Scholar
  19. 19.
    A. Jourani, Weak regularity of functions and sets in Asplund spaces. Nonlin. Anal. TMA 65 (2006) 660–676.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    A. Jourani and L. Thibault, Metric regularity and subdifferential calculus in Banach spaces. Set-Valued Anal. 3 (1995) 87–100.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    B. S. Mordukhovich, Approximation methods in problems of optimization and control [in Russian]. Nauka, Moscow (1988).Google Scholar
  22. 22.
    _____, Variational analysis and generalized differentiation. Springer-Verlag, Berlin (2006).Google Scholar
  23. 23.
    J. J. Moreau, Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 266 (1962) 238–240.MathSciNetGoogle Scholar
  24. 24.
    _____, Rafle par un convexe variable, I. Sém. Anal. Conv. Montpellier 15 (1971).Google Scholar
  25. 25.
    _____, Problèmes d’évolution associé a un convexe mobile d’un espace hilbertien. C. R. Acad. Sci. Paris 276 (1973), 791–794.zbMATHMathSciNetGoogle Scholar
  26. 26.
    _____, Evolution problems associated with a moving convex set in a Hilbert space. J. Differ. Equations 26 (1977) 347–374.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. A. Poliquin, R. T. Rockafellar, and L. Thibault, Local differentiability of distance functions. Trans. Am. Math. Soc. 352 (2000), No. 11, 5231–5249.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    R. T. Rockafellar and R. J.-B. Wets, Variational analysis. Springer-Verlag, Berlin (1998).zbMATHCrossRefGoogle Scholar
  29. 29.
    T. Schwartz, Farthest points and monotonicity methods in Hilbert spaces. In: Proc. of Int. Conf. on Approximation and Optimization, Cluj-Napoca (1997), Vol. I, 351–356.Google Scholar
  30. 30.
    L. Thibault, Sweeping process with regular and nonregular sets. J. Differ. Equations 193 (2003) 1–26.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • S. D. Flåm
    • 1
  • J.-B. Hiriart-Urruty
    • 2
  • A. Jourani
    • 3
  1. 1.Economics DepartmentBergen UniversityBergenNorway
  2. 2.Laboratoire MIPUniversité Paul SabatierToulouseFrance
  3. 3.Institut de Mathématiques de BourgogneUniversité de BourgogneDijonFrance

Personalised recommendations