On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces

  • Patrick MehlitzEmail author


Sequential normal compactness is one of the most important properties in terms of modern variational analysis. It is necessary for the derivation of calculus rules for the computation of generalized normals to set intersections or preimages of sets under transformations. While sequential normal compactness is inherent in finite-dimensional Banach spaces, its presence has to be checked in the infinite-dimensional situation. In this paper, we show that broad classes of sets in Lebesgue and Sobolev spaces which are reasonable in the context of optimal control suffer from an intrinsic lack of sequential normal compactness.


Decomposable set Optimal control Sequential normal compactness 

Mathematics Subject Classification (2010)

28B05 49J53 


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This work is supported by the DFG grant Analysis and Solution Methods for Bilevel Optimal Control Problems within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev spaces. Elsevier Science, Oxford (2003)zbMATHGoogle Scholar
  2. 2.
    Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aubin, J.-P., Frankowska, H.: Set-valued analysis. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2009). ISBN 978-0-8176-4847-3. Reprint of the 1990 editionCrossRefGoogle Scholar
  5. 5.
    Bogachev, V.I.: Measure Theory. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Borwein, J.M., Strojwas, H.M.: Tangential approximations. Nonlinear Anal. Theory Methods Appl. 9(12), 1347–1366 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borwein, J.M., Lucet, Y., Mordukhovich, B.S.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7(2), 375–393 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fabian, M., Mordukhovich, B.S.: Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. Theory Methods Appl. 54 (6), 1057–1067 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo, L., Ye, J.J.: Necessary optimality conditions for optimal control problems with equilibrium constraints. SIAM J. Control. Optim. 54(5), 2710–2733 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Harder, F., Wachsmuth, G.: The limiting normal cone of a complementarity set in Sobolev spaces. Preprint TU Chemnitz, pp. 1–31. (2017)
  11. 11.
    Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7(1), 149–182 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hintermüller, M., Mordukhovich, B.S., Surowiec, T.M.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146(1), 555–582 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ioffe, A.D.: Coderivative compactness, metric regularity and subdifferential calculus. In: Théra, M. (ed.) Experimental, Constructive and Nonlinear Analysis, volume 27 of Canadian Mathematical Society of Conference Proceedings, pp. 123–164. AMS, Providence, RI (1990)Google Scholar
  14. 14.
    Jarušek, J., Outrata, J.V.: On sharp necessary optimality conditions in control of contact problems with strings. Nonlinear Anal. Theory Methods Appl. 67(4), 1117–1128 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Loewen, P.D.: Limits of Fréchet normals in nonsmooth analysis. In: Ioffe, A.D., Marcus, L., Reich, S. (eds.) Optimization and Nonlinear Analysis, volume 244 of Pitman Research Notes Math. Ser., pp. 178–188. Longman, Harlow, Essex, UK (1992)Google Scholar
  16. 16.
    Megginson, R.E.: An introduction to Banach space theory. Graduate texts in mathematics. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Mehlitz, P., Wachsmuth, G.: On the limiting normal cone to pointwise defined sets in Lebesgue spaces. Set-valued and variational analysis (2016).
  18. 18.
    Mehlitz, P., Wachsmuth, G.: The weak sequential closure of decomposable sets in Lebesgue spaces and its application to variational geometry. Set-valued and variational analysis (2017).
  19. 19.
    Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183(1), 250–288 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. Springer-Verlag, Berlin (2006)Google Scholar
  21. 21.
    Mordukhovich, B.S., Shao, Y.: Nonconvex differential calculus for infinite-dimensional multifunctions. Set-Valued Anal. 4(3), 205–236 (1996a).
  22. 22.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996b).
  23. 23.
    Mordukhovich, B.S., Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications. SIAM J. Control. Optim. 35(1), 285–314 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mordukhovich, B.S., Wang, B.: Calculus of sequential normal compactness in variational analysis. J. Math. Anal. Appl. 282 (1), 63–84 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Outrata, J., Jarušek, J., Stará, J.: On optimality conditions in control of elliptic variational inequalities. Set-Valued and Variational Anal. 19(1), 23–42 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Outrata, J.V., Römisch, W.: On optimality conditions for some nonsmooth optimization problems over L p spaces. J. Optim. Theory Appl. 126 (2), 411–438 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.: Handbook of applied analysis, volume 19 of advances in mechanics and mathematics. Springer, New York. ISBN 978-0-387-78906-4 (2009)Google Scholar
  28. 28.
    Rockafellar, R.T.: Integrals which are convex functionals. Pacific J. Math. 24(3), 525–539 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, volume 317 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1998)Google Scholar
  30. 30.
    Simonnet, M.: Measures and probabilities. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  31. 31.
    Wachsmuth, G.: Towards M-stationarity for optimal control of the obstacle problem with control constraints. SIAM J. Control. Optim. 54(2), 964–986 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yost, D.: Asplund spaces for beginners. Acta Universitatis Carolinae Mathematica et Physica 34(2), 159–177 (1993)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceTechnische Universität Bergakademie FreibergFreibergGermany

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