Set-Valued and Variational Analysis

, Volume 27, Issue 2, pp 549–568 | Cite as

Sensitivity Properties of Parametric Nonconvex Evolution Inclusions with Application to Optimal Control Problems

  • Samir AdlyEmail author
  • Taron Zakaryan


The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first order involving a general class of nonconvex functions. Using recent results on the stability of the subdifferentials, with respect to the Gamma convergence, of the associated sequence of subsmooth or semiconvex functions, we give some continuity properties of the solution set associated to these problems. The particular case of the parametric sweeping process involving uniformly subsmooth or uniformly prox-regular sets is studied in details. As an application, we study the sensitivity analysis of the generalized Bolza/Mayer problem governed by a nonsmooth dynamic of a sweeping process type.


Sensitivity analysis Evolution inclusions Primal lower nice functions Semi-convex functions Sweeping process Prox-regular sets Bolza/Mayer problem Optimal control 

Mathematics Subject Classification (2010)

34A60 49J15 49J52 49J53 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We are very grateful to the anonymous referee for his/her careful reading and relevant suggestions which considerably improved the paper.


  1. 1.
    Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman, London (1984)Google Scholar
  2. 2.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)Google Scholar
  3. 3.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357, 1275–1301 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balder, E.: Necessary and sufficient conditions for l 1-strong-weak lower semicontinuity of integral functionals. Nonlinear Anal. 11, 1399–1404 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Breides, A: A handbook of Γ-convergence. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3, Elsevier (2006)Google Scholar
  7. 7.
    Brezis, H.: Functional Analysis, Sobolev Spaces, Equations, Partial Differential, Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  8. 8.
    Bounkhel, M., Thibault, L.: Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6, 359–374 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Castaing, C., Marcellin, S.: Evolution inclusions with pln functions and application to viscosity and control. J. Nonlinear Convex Anal. 8(2), 227–255 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Clarke, F.H: Optimization and Nonsmooth Analysis. Wiley-Interscience (1983)Google Scholar
  11. 11.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, Berlin (1998)zbMATHGoogle Scholar
  12. 12.
    Colombo, G., Goncharov, V.V.: The sweeping process without convexity. Set-Valued Anal. 7, 357–374 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-Valued Var. Anal. 23(1), 69–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19(1-2), 117–159 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ. 260(4), 3397–3447 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Colombo, G., Thibault, L.: Prox-Regular Sets and Applications, Handbook of Nonconvex Analysis and Applications, p 99182. Int. Press, Somerville (2010)Google Scholar
  17. 17.
    Dal Maso, G.: An Introduction to Γ-Convergence. Birkhäuser, Boston (1993)CrossRefzbMATHGoogle Scholar
  18. 18.
    Edmond, J.F., Thibault, L.: Relaxation of an optimal control problem involving a perturbed sweeping process. Math. Program., Ser. B 104, 347–373 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gwinner, J.: On Differential Variational Inequalities and Projected Dynamical Systems-Equivalence and Stability Result. Discrete and Continuous Dynamical Systems Supplement (2007)Google Scholar
  20. 20.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. II: Applications. Kluwer, Dordrecht (2000)Google Scholar
  21. 21.
    Jourani, A., Vilches, E.: Positively α-Far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23(3), 775–821 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mordukhovich, B.M.: Variational Analysis and Generalized, Differentiation. I. Basic Theory. Grundlehren Der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)CrossRefGoogle Scholar
  23. 23.
    Levy, A., Poliquin, R.A., Thibault, L.: A partial extension of Attouch’s theorem and its applications to second-order differentiation. Trans. Amer. Math. Soc. 347, 1269–1294 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Marcellin, S., Thibault, L.: Evolution problems associated with primal lower nice functions. J. Convex Anal. 13, 385–421 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, Grundlehren Der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2005)Google Scholar
  26. 26.
    Papageorgiou, N.S.: Convergence theorems for Banach space-valued integrable multifunctions. Internat J. Math, and Math. Sci. 10, 433–442 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Papageorgiou, N.S.: Nonconvex evolution inclusions driven by time dependent subdifferntials. Math. Japon. 37, 1087–1099 (1992)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Papageorgiou, N.S.: On parametric evolution inclusions of the subdifferential type with applications to optimal control problems. Trans. Amer. Math. Soc. 347, 203–231 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Poliquin, R.A.: Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17, 385–398 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)Google Scholar
  32. 32.
    Serea, O.S., Thibault, L.: Primal lower nice property of value functions in optimization and control problems. Set-Valued Var. Anal. 8(3-4), 569–600 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Thibault, L., Zakaryan, T.: Convergence of subdifferentials and normal cones in locally uniformly convex Banach space. Nonlinear Anal. 98, 110–134 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Vial, J.P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zheng, X.Y., Wei, Z.: Convergence of the associated sequence of normal cones of a Mosco convergent sequence of sets. SIAM J. Optim. 22(3), 758–771 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.XLIM UMR-CNRS 7252Université de LimogesLimogesFrance

Personalised recommendations