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Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 379–401 | Cite as

Boundary Optimal Control for a Frictional Contact Problem with Normal Compliance

  • Andaluzia Matei
  • Sorin Micu
Article
First Online:

Abstract

We consider the contact between an elastic body and a deformable foundation. Firstly, we introduce a mathematical model for this phenomenon by means of a normal compliance contact condition associated with a friction law. Then, we propose a variational formulation of the model in a form of a quasi-variational inequality governed by a non-differentiable functional and we briefly discuss its well-possedness. Nextly, we address an optimal control problem related to this model in order to led the displacement field as close as possible to a given target by acting with a localized boundary control. By using some mollifiers of the normal compliance functions, we introduce a regularized model which allows us to establish an optimality condition. Finally, by means of asymptotic analysis tools, we show that the solutions of the regularized optimal control problems converge to a solution of the initial optimal control problem.

Keywords

Frictional contact Elastic body Deformable foundation Normal compliance Quasi-variational inequality Weak solution Boundary optimal control Regularization Optimality condition 

Mathematics Subject Classification

74M10 74M15 49J20 49K20 
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Notes

Acknowledgements

This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project number PN-II-ID-PCE-2011-3-0257 and by a LEA MATH-MODE Project CNRS-IMAR.

References

  1. 1.
    Amassad, A., Chenais, D., Fabre, C.: Optimal control of an elastic contact problem involving Tresca friction law. Nonlinear Anal. 48, 1107–1135 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andersson, L.-E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. TMA 16, 347–370 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barbu, V.: Optimal Control of Variational Inequalities. Pitman Advanced Publishing, Boston (1984)zbMATHGoogle Scholar
  4. 4.
    Bartosz, K., Kalita, P.: Optimal control for a class of dynamic viscoelastic contact problems with adhesion. Dyn. Syst. Appl. 21, 269–292 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bonnans, J.F., Tiba, D.: Pontryagin’s principle in the control of semiliniar elliptic variational inequalities. Appl. Math. Optim. 23, 299–312 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Capatina, A., Timofte, C.: Boundary optimal control for quasistatic bilateral frictional contact problems. Nonlinear Anal. 94, 84–99 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Denkowski, Z., Migorski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems, a special issue in honour of Professor S. Rolewicz, invited paper. Control Cybern. 36, 611–632 (2007)zbMATHGoogle Scholar
  8. 8.
    Denkowski, Z., Migorski, S., Ochal, A.: A class of optimal control problems for piezoelectric frictional contact models. Nonlinear Anal. Real World Appl. 12, 1883–1895 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Friedman, A.: Optimal Control for Variational Inequalities. SIAM J. Control Optim. 24, 439–451 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefGoogle Scholar
  11. 11.
    Kimmerle, S.J., Moritz, R.: Optimal control of an elastic tyre-damper system with road contact. ZAMM, 14, 875–876 (2014). In: Steinmann, P., Leugering, G. (eds.), Special Issue: 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Erlangen (2014)Google Scholar
  12. 12.
    Klarbring, A., Mikelic, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klarbring, A., Mikelič, A., Shillor, M.: On friction problems with normal compliance. Nonlinear Anal. 13, 935–955 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Laursen, T.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)zbMATHGoogle Scholar
  15. 15.
    Lions, J.-L.: Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968)zbMATHGoogle Scholar
  16. 16.
    Martins, J.A.C., Oden, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. TMA 11, 407–428 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Matei, A., Micu, S.: Boundary optimal control for nonlinear antiplane problems. Nonlinear Anal. 74(5), 1641–1652 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mignot, R.: Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22, 130–185 (1976)CrossRefGoogle Scholar
  19. 19.
    Mignot, R., Puel, J.-P.: Optimal control in some variational inequalities. SIAM J. Control Optim. 22, 466–476 (1984)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Neittaanmaki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems: Theory and Applications. Springer Monographs in Mathematics, Springer, New York (2006)zbMATHGoogle Scholar
  21. 21.
    Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Popov, V.L.: Contact Mechanics and Friction. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shillor, M., Sofonea, M., Telega, J.J.: Models and Variational Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)CrossRefGoogle Scholar
  25. 25.
    Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Advances in Mechanics and Mathematics, vol. 18. Springer, New York (2009)zbMATHGoogle Scholar
  26. 26.
    Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics, London Mathematical Society, Lecture Note Series 398. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  27. 27.
    Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1991)zbMATHGoogle Scholar
  28. 28.
    Strömberg, N.: Thermomechanical Modelling of Tribological Systems, Ph.D. Thesis 497, Linköping University, Linköping (1997)Google Scholar
  29. 29.
    Strömberg, N., Johansson, L., Klarbring, A.: Generalized standard model for contact friction and wear. In: Raous, M., Jean, M., Moreau, J.J. (eds.) Contact Mechanics. Plenum Press, New York (1995)zbMATHGoogle Scholar
  30. 30.
    Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact friction and wear. Int. J. Solids Struct. 33, 1817–1836 (1996)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Touzaline, A.: Optimal control of a frictional contact problem. Acta Math. Appl. Sin. Engl. Ser. 31(4), 991–1000 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Willner, K.: Kontinuums- und Kontaktmechanik. Springer, Berlin (2003)CrossRefGoogle Scholar
  33. 33.
    Wriggers, P., Laursen, T.: Computational Contact Mechanics: CISM Courses and Lectures, vol. 298. Springer, New York (2007)CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

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