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Infimal Convolution and Optimal Time Control Problem II: Limiting Subdifferential

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Abstract

This paper is the continuation of our study of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. We study several properties of the limiting subdifferential for the infimum time function and we provide explicit expressions of the limiting subdifferential. Necessary and sufficient conditions for the infimum time function to be lower regular are presented. The sharpness of assumptions is shown through diverse examples.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357, 1275–1301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335–361 (2004)

    MathSciNet  MATH  Google Scholar 

  7. De Blasi, F.S., Myjak, J.: On a generalized best approximation problem. J. Approx. Theory 94, 96–108 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Math. Soc. 224, 432–467 (1979)

    MathSciNet  MATH  Google Scholar 

  9. He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896–910 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ivanov, G.E., Thibault, L.: Infimal convolution and optimal time control problem I: Fréchet and proximal subdifferentials. Set-Valued Var. Anal. (2017). doi:10.1007/s11228-016-0398-z

  11. Ivanov, G.E., Thibault, L.: Infimal convolution and optimal time control problem III: Minimal time projection set, submitted

  12. Li, C., Ni, R.: Derivatives of generalized functions and existence of generalized nearest points. J. Approx. Theory 115, 44–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mordukhovich, B.S.: Variational analysis and generalized differentiation i and II, Springer, New York, Comprehensive Studies in Mathematics, Vol. 330 and 331 (2005)

  14. Mordukhovich, B.S., Nam, N.M.: Subgradients of distance functions with applications to Lipschitzian stability. Math. Program. 104, 635–668 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mordukhovich, B.S., Nam, N.M.: Subgradients of distance functions at out-of-set points. Taiwanese J. Math. 10, 299–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mordukhovich, B.S., Nam, N.M.: Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. 46, 615–633 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moreau, J.J.: Fonctionnelles Convexes, Collège De France, Paris (1967); Second edition. Consiglio Nazionale delle Ricerche and Facoltá di ingegneria Universita di Roma ”Tor Vergata” (2003)

  18. Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)

    MathSciNet  MATH  Google Scholar 

  19. Nam, N.M., Cuong, D.V.: Generalized differentiation and characterizations for differentiability of infimal convolutions. Set-Valued Var. Anal. 23, 333–353 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pereira, F.F., Goncharov, V.V.: Regularity of a kind of marginal functions in Hilbert spaces. In: Butenko, S., Floudas, C.H., Rassias, T.H. (eds.) On global optimization in science and engineering, pp 423–464 (2014)

    Chapter  Google Scholar 

  21. Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wolenski, P.R., Zhuang, Y.: Proximal analysis and the minimal time function. SIAM J. Control Optim. 36, 1048–1072 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow region, 141700, Russia

    Grigorii E. Ivanov

  2. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CC 051, Place Eugène Bataillon, 34095, Montpellier cedex 05, France

    Lionel Thibault

  3. Centro de Modelamiento Matematico, Universidad de Chile, Santiago, Chile

    Lionel Thibault

Authors
  1. Grigorii E. Ivanov
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  2. Lionel Thibault
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Correspondence to Grigorii E. Ivanov.

Additional information

The research of Grigorii E. Ivanov was supported by the Russian Foundation for Basic Research, grant 16-01-00259

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Ivanov, G.E., Thibault, L. Infimal Convolution and Optimal Time Control Problem II: Limiting Subdifferential. Set-Valued Var. Anal 25, 517–542 (2017). https://doi.org/10.1007/s11228-017-0402-2

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