Abstract

Continuing earlier research [``Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals'', J. Convex Analysis 18 (2011) 1–36] on local well-posedness of a time-minimum problem associated to a closed target set C⊂HC\subset H (HH is a Hilbert space) and a convex constant dynamics F⊂HF\subset H we study the Lipschitz (or, in general, Hölder) regularity of the (unique) point πCF(x)\pi_{C}^{F}\left( x\right) in CC achieved from xx for a minimal time. As a consequence, smoothness of the value function is proved, and an explicit formula for its derivative is given.

Author information

Contact details are reproduced from the original publication and may be historical.

Vladimir V. Goncharov

CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal

goncha@uevora.pt

Fátima F. Pereira

CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal

fmfp@uevora.pt

Keywords

  • Time-minimum problem
  • Hölder continuity
  • proximal
  • Fréchet subdifferential
  • Clarke subdifferential
  • duality mapping
  • curvature
  • proximal smoothness

Mathematics Subject Classification

49J52, 49N15

References

  1. M. V. Balashov, G. E. Ivanov: Properties of the metric projection on weakly Vial-convex sets and parametrization of set-valued mappings with weakly convex images, Math. Notes 80 (2006) 461–467.https://doi.org/10.1007/s11006-006-0163-yGoogle Scholar
  2. M. Bounkhel, L. Thibault: On various notions of regularity of sets in nonsmooth analysis, Nonlinear Anal. 48 (2002) 223–246.https://doi.org/10.1016/S0362-546X(00)00183-8Google Scholar
  3. J. V. Burke, M. C. Ferris, M. Qian: On the Clarke subdifferential of the distance function of a closed set, J. Math. Anal. Appl. 166 (1992) 199–213.https://doi.org/10.1016/0022-247X(92)90336-CGoogle Scholar
  4. A. Canino: On p-convex sets and geodesics, J. Differ. Equations 75 (1988) 118–157.https://doi.org/10.1016/0022-0396(88)90132-5Google Scholar
  5. A. Canino: Local properties of geodesics on p-convex sets, Ann. Mat. Pura Appl., IV. Ser. 159 (1991) 17–44.https://doi.org/10.1007/BF01766291Google Scholar
  6. C. K. Chui, P. W. Smith: Unique best nonlinear approximation in Hilbert spaces, Proc. Amer. Math. Soc. 49 (1975) 66–70.https://doi.org/10.1090/S0002-9939-1975-0361575-XGoogle Scholar
  7. C. K. Chui, E. R. Rozema, P. W. Smith, J. D. Ward: Metric curvature, folding and unique best approximation, SIAM J. Math. Anal. 7 (1976) 436–449.https://doi.org/10.1137/0507035Google Scholar
  8. F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley, New York (1983).Google Scholar
  9. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and the lower-C 2 property, J. Convex Analysis 2 (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  10. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Springer, New York (1998).Google Scholar
  11. G. Colombo, V. V. Goncharov: Variational inequalities and regularity properties of closed sets in Hilbert spaces, J. Convex Analysis 8 (2001) 197–221.https://doi.org/10.68381/jca08009Google Scholar
  12. G. Colombo, V. V Goncharov, B. S. Mordukhovich: Well-posedness of minimal time problems with constant dynamics in Banach spaces, Set-Valued Var. Anal. 18 (2010) 349–372.https://doi.org/10.1007/s11228-010-0151-yGoogle Scholar
  13. G. Colombo, P. R. Wolenski: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space, J. Glob. Optim. 3–4 (2004) 269–282.https://doi.org/10.1023/B:JOGO.0000026460.10505.ddGoogle Scholar
  14. G. Colombo, P. R. Wolenski: Variational analysis for a class of minimal time functions in Hilbert spaces, J. Convex Analysis 11 (2004) 335–361.https://doi.org/10.68381/jca11021Google Scholar
  15. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  16. V. V. Goncharov, F. F. Pereira: Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals, J. Convex Analysis 18 (2011) 1–36.https://doi.org/10.68381/jca18001Google Scholar
  17. R. E. Megginson: An Introduction to Banach Space Theory, Graduate Texts in Math. 183, Springer, New York (1998).https://doi.org/10.1007/978-1-4612-0603-3Google Scholar
  18. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Science) 330, Springer, Berlin (2006).Google Scholar
  19. B. S. Mordukhovich, N. M. Nam: Limiting subgradients of minimal time functions in Banach spaces, J. Glob. Optim. 46 (2010) 615–633.https://doi.org/10.1007/s10898-009-9446-7Google Scholar
  20. H. V. Ngai, J.-P. Penot: Approximately convex sets, J. Nonlinear Convex Anal. 8 (2007) 337–355.Google Scholar
  21. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  22. R. T. Rockafellar: Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear Anal. 3 (1979) 145–154.https://doi.org/10.1016/0362-546X(79)90044-0Google Scholar
  23. E. R. Rozema, P. W. Smith: Nonlinear approximation in uniformly smooth Banach spaces, Trans. Amer. Math. Soc. 188 (1974) 199–211.https://doi.org/10.1090/S0002-9947-1974-0330875-5Google Scholar
  24. A. Shapiro: Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4 (1994) 130–141.https://doi.org/10.1137/0804006Google Scholar
  25. J.-P. Vial: Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983) 231–259.https://doi.org/10.1287/moor.8.2.231Google Scholar
  26. P. R. Wolenski, Yu Zhuang: Proximal analysis and the minimal time function, SIAM J. Control Optim. 36 (1998) 1048–1998.https://doi.org/10.1137/S0363012996299338Google Scholar
  27. Yiran He, Kung Fu Ng: Subdifferentials of a minimum time function in Banach spaces, J. Math. Anal. Appl. 321 (2006) 896–910.https://doi.org/10.1016/j.jmaa.2005.09.009Google Scholar

Suggested citation

V. V. Goncharov, F. F. Pereira. “Geometric Conditions for Regularity in a Time-Minimum Problem with Constant Dynamics.” Journal of Convex Analysis 19 (2012), No. 3, 631–669. https://doi.org/10.68381/jca19035

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.