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Discrete Approximations of a Controlled Sweeping Process

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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 W 1,2 topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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.

  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. 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)

    Article  MATH  Google Scholar 

  5. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  7. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. preprint (2014)

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)

    MATH  Google Scholar 

  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)

  12. Krejčí, P.: Vector hysteresis models. Eur. J. Appl. Math. 2, 281–292 (1991)

    Article  MATH  Google Scholar 

  13. Krejčí, P., Vladimirov, A.: Polyhedral sweeping processes with oblique reflection in the space of regulated functions. Set-Valued Anal. 11, 91–110 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Birkhäuser, Boston (1993)

    Book  MATH  Google Scholar 

  15. Mordukhovich, B.S.: Discrete approximations and refined Euler-Lagrange conditions for differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  17. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  18. Moreau, J.J.: Rafle par un convexe variable I. Sém. Anal. Convexe Montpellier Exposé, 15 (1971)

  19. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Diff. Eqs. 26, 347–374 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  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. Rindler, F.: Optimal control for nonconvex rate-independent evolution processes. SIAM J. Control. Optim. 47, 2773–2794 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Smirnov, G.V.: Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI (2002)

  23. Thibault, L.: Sweeping process with regular and nonregular sets. J. Diff. Eqns. 193, 1–26 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  25. Vinter, R.B.: Optimal Control. Birkhaüser, Boston (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Padova, Padua, Italy

    G. Colombo

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    R. Henrion

  3. Departamento de Matemática, Universidad Téchnica Federico Santa María, Valparaíso, Chile

    N. D. Hoang

  4. Department of Mathematics, Wayne State University, Detroit, Michigan, USA

    B. S. Mordukhovich

Authors
  1. G. Colombo
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  2. R. Henrion
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  3. N. D. Hoang
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  4. B. S. Mordukhovich
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Corresponding author

Correspondence to B. S. Mordukhovich.

Additional information

Research of this author was partially supported by the CARIPARO project “Nonlinear Partial Differential Equations: Models, Analysis, and Control-Theoretical Problems” and by the University of Padova research project “Some Analytic and Differential Geometric Aspects in Nonlinear Control Theory with Applications to Mechanics.

Research of this author was partially supported by the DFG Research Center MATHEON.

Research of this author was partially supported by FONDECYT Nos. 3140060 and Basal Project, CMM, Universidad de Chile.

Research of this author was partially supported by the USA National Science Foundation under grant DMS-1007132.

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Colombo, G., Henrion, R., Hoang, N.D. et al. Discrete Approximations of a Controlled Sweeping Process. Set-Valued Var. Anal 23, 69–86 (2015). https://doi.org/10.1007/s11228-014-0299-y

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  • DOI: https://doi.org/10.1007/s11228-014-0299-y

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