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A Relaxation Approach to Vector-Valued Allen–Cahn MPEC Problems

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Abstract

In this paper we consider a vector-valued Allen–Cahn MPEC problem. To derive optimality conditions we exploit a regularization–relaxation technique. The optimality system of the regularized–relaxed subproblems are investigated by applying the classical result of Zowe and Kurcyusz. Finally we show that the stationary points of the regularized–relaxed subproblems converge to weak stationary points of the limit problem.

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References

  1. Barbu, V.: Optimal Control of Variational Inequalities. Research Notes in Mathematics 100. Pitman Advanced Publishing Program, Boston (1984)

    MATH  Google Scholar 

  2. Barbu, V.: Analysis and Control of Non Linear Infinite Dimensional Systems. Mathematics in Science and Engineering 190. Academic Press, Boston (1993)

    Google Scholar 

  3. Bergounioux, M.: Optimal control of an obstacle problem. Appl. Math. Optim. 36, 147–172 (1997)

    Article  MathSciNet  Google Scholar 

  4. Bergounioux, M., Lenhart, S.: Optimal control of bilateral obstacle problems. SIAM J. Control Optim. 43, 240–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergounioux, M., Zidani, H.: Pontryagin maximum principle for optimal control of variational inequalities. SIAM J. Control Optim. 37, 1273–1290 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bermudez, A., Saguez, C.: Optimal Control of Variational Inequalities: Optimality Conditions and Numerical Methods, Free Boundary Problems: Applications and Theory, Vol IV, Bossavit, A., Damlamian, A. and Fremond, M. (eds.), pp. 478–487. Pitman London (1985)

  7. Bonnans, J.F., Tiba, D.: Pontryagin’s principle in the control of semilinear elliptic variational inequalities. Appl. Math. Optim. 23, 299–312 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blank, L., Garcke, H., Sarbu, L., Styles, V.: Primal-dual active set methods for Allen–Cahn variational inequalities with non-local constraints. Numer. Methods PDEs 29(3), 999–1030 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Blank, L., Garcke, H., Sarbu, L., Styles, V.: Non-local Allen-Cahn systems: Analysis and a primal dual active set method. IMA J. Numer. Anal. (2013). doi:10.1093/imanum/drs039

  10. Bossavit, A., Damlamian, A. and Fremond, M. (eds.): Free Boundary Problems: Applications and Theory Pitman Research Notes in Mathematics 121 Pitman Boston London Melbourne Pitman, London (1985)

  11. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121. Springer, New York (1996)

    Book  MATH  Google Scholar 

  12. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  13. Farshbaf-Shaker, M.H.: A penalty approach to optimal control of Allen–Cahn variational inequalities: MPEC-view. Numer. Func. Anal. Opt. 33, 1321–1349 (2012)

  14. Fasano, A. and Primicerio, M. (eds.): Free Boundary Problems, Theory Applications Pitman Research Notes in Mathematics 79 Pitman Boston London Melbourne Pitman, London (1983)

  15. Friedman, A.: Optimal control for variational inequalities. SIAM J. Control Optim. 24, 439–451 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Friedman, A.: Optimal control for parabolic variational inequalities. SIAM J. Control Optim. 25, 482–497 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Garcke, H.: On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation Thesis, University Bonn, (2000)

  18. Garcke, H.: On a Cahn–Hilliard model for phase separation with Elastic Misfit. Ann. de l’IHP - Analyse non linéaire 22(2), 165–185 (2005)

  19. He, Z.X.: State constrained control problems governed by variational inequalities. SIAM J. Control Optim. 25, 1119–1144 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Herzog, R., Meyer, C.: Optimal control of static plasticity with linear kinematic hardening. ZAMM 91(10), 777–794 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hintermüller, M., Tber, M.H.: An inverse problem in american options as a mathematical program with equilibrium constraints: C-stationarity and an active-set-newton solver. SIAM J. Control Optim. 48(7), 4419–4452 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ito, K., Kunisch, K.: Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41, 343–364 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ito, K., Kunisch, K.: Optimal control of parabolic variational inequalities. J. Math. Pures Appl. 93, 329–360 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  26. Mignot, F., Puel, J.P.: Optimal control in some variational inequalities. SIAM J. Control Optim. 22(3), 466–476 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mini-Workshop: Control of Free Boundaries. Abstracts from the mini-workshop held February 11–17. 2007. Organized by Charles M. Elliott, M. Hinze and V. Styles. Oberwolfach Reports. Vol. 4, No.1 (2007)

  28. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author wishes to thank Luise Blank and Harald Garcke for many fruitful hints and discussions. This work was supported by the SPP 1253 ”Optimization with Partial Differential Equations” of the German Science Foundation (DFG) through the Grant GA 695/5-2.

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  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    M. Hassan Farshbaf-Shaker

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  1. M. Hassan Farshbaf-Shaker
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Correspondence to M. Hassan Farshbaf-Shaker.

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Farshbaf-Shaker, M.H. A Relaxation Approach to Vector-Valued Allen–Cahn MPEC Problems. Appl Math Optim 72, 325–351 (2015). https://doi.org/10.1007/s00245-014-9282-0

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