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Applied Mathematics & Optimization

, Volume 68, Issue 3, pp 445–473 | Cite as

Optimal Control of Evolution Mixed Variational Inclusions

  • Gonzalo AlduncinEmail author
Article

Abstract

Optimal control problems of primal and dual evolution mixed variational inclusions, in reflexive Banach spaces, are studied. The solvability analysis of the mixed state systems is established via duality principles. The optimality analysis is performed in terms of perturbation conjugate duality methods, and proximation penalty-duality algorithms to mixed optimality conditions are further presented. Applications to nonlinear diffusion constrained problems as well as quasistatic elastoviscoplastic bilateral contact problems exemplify the theory.

Keywords

Constrained optimal control Evolution mixed variational inclusions Duality principles Composition duality methods Perturbation conjugate duality methods Proximation penalty-duality algorithms Set-valued variational analysis 

References

  1. 1.
    Alduncin, G.: Primal and dual evolution macro-hybrid mixed variational inclusions. Int. J. Math. Anal. 5, 1631–1664 (2011) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alduncin, G.: Composition duality principles for evolution mixed variational inclusions. Appl. Math. Lett. 20, 734–740 (2007) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alduncin, G.: Composition duality methods for evolution mixed variational inclusions. Nonlinear Anal. Hybrid Syst. 1, 336–363 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Akagi, G., Ôtani, M.: Evolution inclusions governed by subdifferentials in reflexive Banach spaces. J. Evol. Equ. 4, 519–541 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Migórski, S.: Evolution hemivariational inequalities in infinite dimension and their control. Nonlinear Anal. 47, 101–112 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Azé, D., Bolintinéanu, S.: Optimality conditions for constrained convex parabolic control problems via duality. J. Convex Anal. 7, 1–17 (2000) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod/Gauthier-Villars, Paris (1969) zbMATHGoogle Scholar
  8. 8.
    Alduncin, G.: Evolution mixed variational inclusions with optimal control. Math. Res. 4, 64–79 (2012) MathSciNetGoogle Scholar
  9. 9.
    Ekeland, I., Temam, R.: Analyse Convexe et Problèmes Variationnels. Dunod/Gauthier-Villars, Paris (1974) zbMATHGoogle Scholar
  10. 10.
    Ernst, E., Théra, M.: On the necessity of the Moreau-Rockafellar-Robinson qualification condition in Banach spaces. Math. Program., Ser. B 117, 149–161 (2009) CrossRefzbMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T.: Duality and stability in extremum problems involving convex functions. Pac. J. Math. 21, 167–187 (1967) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974) CrossRefGoogle Scholar
  13. 13.
    Heins, W., Mitter, S.K.: Conjugate convex functions, duality and optimal control problems I: systems governed by ordinary differential equations. Inf. Sci. 2, 211–243 (1970) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Alduncin, G.: Duality and variational principles of potential boundary value problems. Comput. Methods Appl. Mech. Eng. 64, 469–485 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jeyakumar, V.: Duality and infinite dimensional optimization. Nonlinear Anal. 15, 1111–1122 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010) zbMATHGoogle Scholar
  17. 17.
    Simons, S.: The occasional distributivity of • over \(\stackrel {\textstyle+}{e}\) and the change of variables for conjugate functions. Nonlinear Anal. 14, 1111–1120 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fortin, M., Glowinski, R. (eds.): Méthodes de Lagrangien Augmenté: Applications à la Résolution Numérique de Problèmes aux Limites. Dunod/Bordas, Paris (1982) Google Scholar
  19. 19.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989) CrossRefzbMATHGoogle Scholar
  20. 20.
    Alduncin, G.: On Gabay’s algorithms for mixed variational inequalities. Appl. Math. Optim. 35, 21–44 (1997) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Alduncin, G.: Composition duality methods for mixed variational inclusions. Appl. Math. Optim. 52, 311–348 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gabay, D.: Application de la méthode des multiplicateurs aux inéquations variationnelles. In: Fortin, M., Glowinski, R. (eds.) Méthodes de Lagrangien Augmenté, pp. 279–307. Dunod/Bordas, Paris (1982) Google Scholar
  23. 23.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) CrossRefzbMATHGoogle Scholar
  24. 24.
    Alduncin, G.: Numerical resolvent methods for constrained problems in mechanics. J. Approx. Theory Appl. 12(4), 1–25 (1996) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Alduncin, G.: Parallel proximal-point algorithms for constrained problems in mechanics. In: Yang, L.T., Paprzycki, M. (eds.) Practical Applications of Parallel Computing, pp. 69–88. Nova Science, New York (2003) Google Scholar
  26. 26.
    Duvaut, G., Lions, J.-L.: Les Inéquations en Méchanique et en Physique. Dunod, Paris (1972) Google Scholar
  27. 27.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) Google Scholar
  28. 28.
    Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007) CrossRefzbMATHGoogle Scholar
  29. 29.
    Le Tallec, P.: Numerical Analysis of Viscoelastic Problems. Masson, Paris (1990) zbMATHGoogle Scholar
  30. 30.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986) CrossRefzbMATHGoogle Scholar
  31. 31.
    Alduncin, G.: Variational formulations of nonlinear constrained boundary value problems. Nonlinear Anal. 72, 2639–2644 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Alduncin, G.: Composition duality principles for mixed variational inequalities. Math. Comput. Model. 41, 639–654 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Temam, R.: A generalized Norton-Hoff model and the Prandt-Reuss law of plasticity. Arch. Ration. Mech. Anal. 95, 137–183 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Perzyna, P.: Fundamental problems in viscoplasticity. Rec. Adv. Appl. Mech. 9, 243–377 (1966) CrossRefGoogle Scholar
  35. 35.
    Sofonea, M., Renon, N., Shillor, M.: Stress formulation for frictionless contact of an elastic-perfectly-plastic body. Appl. Anal. 83, 1157–1170 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Alduncin, G.: Composition duality methods for quasistatic evolution elastoviscoplastic variational problems. Nonlinear Anal. Hybrid Syst. 5, 113–122 (2011) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de Recursos Naturales, Instituto de GeofísicaUniversidad Nacional Autónoma de MéxicoMéxicoMéxico

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