Advertisement

Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 581–606 | Cite as

Infimal Convolution and Optimal Time Control Problem I: Fréchet and Proximal Subdifferentials

  • Grigorii E. Ivanov
  • Lionel Thibault
Article

Abstract

We consider a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. If the target function is the indicator function of a closed set, this problem is a minimal time problem for a target set, studied previously in particular by Colombo, Goncharov and Mordukhovich. We investigate several properties of the Fréchet and proximal subdifferentials for the infimum time function. Also explicit expressions of the above mentioned subdifferentials as well as various directional derivatives are obtained. We provide some examples to show the essentiality of assumptions of our theorems.

Keywords

Fréchet subdifferential Proximal subdifferential Dini directional derivative Generalized directional derivative Minimal time function Minimal time projection Infimal convolution 

Mathematics Subject Classification (2010)

49J52 46N10 58C20 28B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez, O., Koike, S., Nakayama, I.: Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem. SIAM J. Control Optim. 38, 470–481 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357, 1275–1301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Borwein, J.M., Giles, J.R.: The proximal normal formula in Banach space. Trans. Amer. Math. Soc. 302, 371–381 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burke, J.V., Ferris, M.C., Qian, M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199–213 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Clarke, F.H.: Optimization and nonsmooth analysis, Wiley, New York, 1983 (Republished in 1990: Vol 5, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa.)Google Scholar
  7. 7.
    Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the lower- C 2 property. J. Convex Anal. 2, 117–144 (1995)MathSciNetMATHGoogle Scholar
  8. 8.
    Colombo, G., Goncharov, V.V., Mordukhovich, B.S.: Well-posedness of minimal time problem with constant dynamics in Banach space. Set-Valued Var. Anal. 18(3-4), 349–372 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Global Optim. 28, 269–282 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335–361 (2004)MathSciNetMATHGoogle Scholar
  11. 11.
    Correa, R., Jofre, A., Thibault, L.: Characterization of lower semicontinuous convex functions. Proc. Amer. Math. Soc. 116, 67–72 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    De Blasi, F.S., Myjak, J.: On a generalized best approximation problem. J. Approx. Theory 94, 96–108 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Goncharov, V.V., Pereira, F.F.: Neighbourhood retractions of nonconvex sets in a Hilbertspace via sublinear functionals. J. Convex Anal. 18, 1–36 (2011)MathSciNetMATHGoogle Scholar
  14. 14.
    Goncharov, V.V., Pereira, F.F.: Geometric conditions for regularity in a time-minimum problem with constant dynamics. J. Convex Anal. 19, 631–669 (2012)MathSciNetMATHGoogle Scholar
  15. 15.
    Goncharov, V.V., Pereira, F.F.: Geometric conditions for regularity of viscosity solution to the simplest Hamilton-Jacobi equation. In: Hómberg, D., Tróltzsch, F. (eds.) Proc. of the 25th IFIP TC7 conference system modeling and optimization, CSMO 2011, Berlin, Springer, pp 245–254 (2012)Google Scholar
  16. 16.
    He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896–910 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41, 175–192 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ivanov, G.E., Thibault, L.: Infimal convolution and optimal time control problem II: Limiting subdifferential, submittedGoogle Scholar
  19. 19.
    Ivanov, G.E., Thibault, L.: Infimal convolution and optimal time control problem III: Minimal time projection set, submittedGoogle Scholar
  20. 20.
    Jiang, Y., He, Y.: Subdifferentials of a minimum time function in normed spaces. J. Math. Anal. Appl. 358, 410–418 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kruger, A.Y.: Epsilon-semidifferentials and epsilon-normal elements, pp 1331–81. Depon. VINITI, Moscow (1981)Google Scholar
  22. 22.
    Li, C., Ni, R.: Derivatives of generalized functions and existence of generalized nearest points. J. Approx. Theory 115, 44–55 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I and II, Springer, New York, Comprehensive Studies in Mathematics, Vol. 330 and 331 (2005)Google Scholar
  24. 24.
    Mordukhovich, B.S., Nam, N.M.: Subgradients of distance functions with applications to Lipschitzian stability. Math. Program. 104, 635–668 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mordukhovich, B.S., Nam, N.M.: Subgradients of distance functions at out-of-set points. Taiwanese J. Math. 10, 299–326 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mordukhovich, B.S., Nam, N.M.: Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. 46, 615–633 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mordukhovich, B.S., Nam, N.M.: Subgradients of minimal time functions under minimal assumptions. J. Convex Anal. 18, 915–947 (2011)MathSciNetMATHGoogle Scholar
  28. 28.
    Moreau, J.J.: Fonctionnelles convexes, Collège de France, Paris (1967), 2nd edition. Consiglio Nazionale delle Ricerche and Facoltá di ingegneria Universita di Roma ”Tor Vergata” (2003)Google Scholar
  29. 29.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)MathSciNetMATHGoogle Scholar
  30. 30.
    Nam, N.M.: Subdifferential formulas for a class of nonconvex infimal convolutions. Optimization 64, 2213–2222 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Nam, N.M., Cuong, D.V.: Generalized differentiation and characterizations for differentiability of infimal convolutions. Set-Valued Var. Anal. 23, 333–353 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Pereira, F.F., Goncharov, V.V.: Regularity of a kind of marginal functions in Hilbert spaces. In: Butenko, S., Floudas, C., Rassias, T. (eds.) On global optimization in science and engineering, pp 423–464 (2014)Google Scholar
  33. 33.
    Rockafellar, R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Canad. J. Math. 32, 257–280 (1980)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wolenski, P.R., Zhuang, Y.: Proximal analysis and the minimal time function. SIAM J. Control Optim. 36, 1048–1072 (1998)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zhang, Y., He, Y., Jiang, Y.: Subdifferentials of a perturbed minimal time function in normed spaces. Optim. Lett. 8, 1921–1930 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Institut Montpelliérain Alexander GrothendieckUniversité de MontpellierMontpellier Cedex 05France
  3. 3.Centro de Modelamiento MatematicoUniversidad de ChileSantiagoChile

Personalised recommendations