Set-Valued and Variational Analysis

, Volume 23, Issue 1, pp 69–86 | Cite as

Discrete Approximations of a Controlled Sweeping Process

  • G. Colombo
  • R. Henrion
  • N. D. Hoang
  • B. S. Mordukhovich
Article

Abstract

The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhedral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of equality and inequality types. It makes challenging and difficult their analysis and optimization. In this paper we establish some existence results for the sweeping process under consideration and develop the method of discrete approximations that allows us to strongly approximate, in the W1,2 topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts.

Keywords

Optimal control Sweeping process Moving controlled polyhedra Dissipative differential inclusions Discrete approximations Variational analysis. 

Mathematics Subject Classifications (2010)

49J52 49J53 49K24 49M25 90C30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adam, L., Outrata, J.V.: On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete Contin. Dyn. Syst. Ser. B 19, 2709–2738 (2014)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program., to appear.Google Scholar
  3. 3.
    Attouch, H., Buttazzo, G., Michelle, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. SIAM, Philadelphia (2005)Google Scholar
  4. 4.
    Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discret. Contin. Dyn. Syst. Ser. B 18, 331–348 (2013)CrossRefMATHGoogle Scholar
  5. 5.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)MATHGoogle Scholar
  6. 6.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process. Dyn. Contin. Discret. Impuls. Syst. Ser. B 19, 117–159 (2012)MATHMathSciNetGoogle Scholar
  7. 7.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. preprint (2014)Google Scholar
  8. 8.
    Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D. Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis. International Press, Boston (2010)Google Scholar
  9. 9.
    Donchev, T., Farkhi, F., Mordukhovich, B.S.: Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces. J. Diff. Eq. 243, 301–328 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)MATHGoogle Scholar
  11. 11.
    Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Impacts in Mechanical Systems, Lecture Notes in Phys., Vol. 551, pp 1–60. Springer, Berlin (2000)Google Scholar
  12. 12.
    Krejčí, P.: Vector hysteresis models. Eur. J. Appl. Math. 2, 281–292 (1991)CrossRefMATHGoogle Scholar
  13. 13.
    Krejčí, P., Vladimirov, A.: Polyhedral sweeping processes with oblique reflection in the space of regulated functions. Set-Valued Anal. 11, 91–110 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Birkhäuser, Boston (1993)CrossRefMATHGoogle Scholar
  15. 15.
    Mordukhovich, B.S.: Discrete approximations and refined Euler-Lagrange conditions for differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  17. 17.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)Google Scholar
  18. 18.
    Moreau, J.J.: Rafle par un convexe variable I. Sém. Anal. Convexe Montpellier Exposé, 15 (1971)Google Scholar
  19. 19.
    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Diff. Eqs. 26, 347–374 (1977)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Moreau, J.J.: An introduction to unilateral dynamics. In: Frémond, M., Maceri, F. (eds.) New Variational Techniques in Civil Engineering. Springer, Berlin (2002)Google Scholar
  21. 21.
    Rindler, F.: Optimal control for nonconvex rate-independent evolution processes. SIAM J. Control. Optim. 47, 2773–2794 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Smirnov, G.V.: Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI (2002)Google Scholar
  23. 23.
    Thibault, L.: Sweeping process with regular and nonregular sets. J. Diff. Eqns. 193, 1–26 (2003)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Tolstonogov, A.A.: Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system. Nonlinear Anal. 75, 4711–4727 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Vinter, R.B.: Optimal Control. Birkhaüser, Boston (2000)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • G. Colombo
    • 1
  • R. Henrion
    • 2
  • N. D. Hoang
    • 3
  • B. S. Mordukhovich
    • 4
  1. 1.Department of MathematicsUniversity of PadovaPaduaItaly
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Departamento de MatemáticaUniversidad Téchnica Federico Santa MaríaValparaísoChile
  4. 4.Department of MathematicsWayne State UniversityDetroitUSA

Personalised recommendations