Skip to main content
SpringerLink
Log in

Continuous Dependence and Optimal Control for a Class of Variational–Hemivariational Inequalities

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

The paper investigates control problems for a class of nonlinear elliptic variational–hemivariational inequalities with constraint sets. Based on the well posedness of a variational–hemivariational inequality, we prove some results on continuous dependence and existence of optimal pairs to optimal control problems. We obtain some continuous dependence results in which the strong dependence and weak dependence are considered, respectively. A frictional contact problem is given to illustrate our main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barabasz, B., Migórski, S., Schaefer, R., Paszynski, M.: Multi deme, twin adaptive strategy \(hp\)-HGS. Inverse Probl. Sci. Eng. 19, 3–16 (2011)

    Article  MathSciNet  Google Scholar 

  2. Barabasz, B., Gajda-Zagorska, E., Migórski, S., Paszynski, M., Schaefer, R., Smolka, M.: A hybrid algorithm for solving inverse problems in elasticity. Int. J. Appl. Math. Comput. Sci. 24, 865–886 (2014)

    Article  MathSciNet  Google Scholar 

  3. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  4. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer, Boston (2003)

    Book  Google Scholar 

  5. Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. I. An output least-squares approach. Math. Mech. Solids 12, 259–276 (2007)

    Article  MathSciNet  Google Scholar 

  6. Han, W., Migórski, S., Sofonea, M.: A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)

    Article  MathSciNet  Google Scholar 

  7. Hintermüller, M.: Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization. M2AN Math. Model. Numer. Anal. 35, 129–152 (2001)

    Article  MathSciNet  Google Scholar 

  8. Jadamba, B., Khan, A.A., Sama, M.: Inverse problems of parameter identification in partial differential equations. In: Jadamba, B., Khan, A.A. (eds.) Mathematics in Science and Technology, pp. 228–258. World Scientific Publishing, Hackensack (2011)

    Chapter  Google Scholar 

  9. Liu, Z.H., Zeng, B.: Optimal control of generalized quasi-variational hemivariational inequalities and its applications. Appl. Math. Optim. 72, 305–323 (2015)

    Article  MathSciNet  Google Scholar 

  10. Migórski, S.: Sensitivity analysis of inverse problems with applications to nonlinear systems. Dyn. Syst. Appl. 8, 73–89 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Migórski, S.: Identification coefficient problems for elliptic hemivariational inequalities and applications. In: Tanaka, M., Dulikravich, G.S. (eds.) Inverse Problems in Engineering Mechanics II, pp. 513–519. Elsevier, Amsterdam (2000)

    Chapter  Google Scholar 

  12. Migórski, S., Ochal, A.: Inverse coefficient problem for elliptic hemivariational inequality, Chap. 11. In: Gao, D.Y. (ed.) Nonsmooth/Nonconvex Mechanics. Modeling, Analysis and Numerical Methods, pp. 247–261. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  13. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)

    MATH  Google Scholar 

  14. Migórski, S., Ochal, A., Sofonea, M.: A class of variational-hemivariational inequalities in reflexive Banach spaces. J. Elast. 127, 151–178 (2017)

    Article  MathSciNet  Google Scholar 

  15. Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)

    Article  MathSciNet  Google Scholar 

  16. Motreanu, D., Sofonea, M.: Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv. Math. Sci. Appl. 10, 103–118 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc., New York (1995)

    MATH  Google Scholar 

  18. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  Google Scholar 

  19. Sofonea, M.: Optimal control of a class of variational-hemivariational inequalities in reflexive Banach spaces. Appl. Math. Optim. (2017). https://doi.org/10.1007/s00245-017-9450-0

  20. Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  21. Sofonea, M., Han, W., Migórski, S.: Numerical analysis of history-dependent variational inequalities with applications to contact problems. Eur. J. Appl. Math. 26, 427–452 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Sciences, Guangxi University for Nationalities, Nanning, 530006, Guangxi, People’s Republic of China

    Caijing Jiang

  2. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, Guangxi, People’s Republic of China

    Biao Zeng

Authors
  1. Caijing Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Biao Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Biao Zeng.

Additional information

This work was supported by the School-level Research Project in Guangxi University for Nationalities (No. 2018MDQN005), Guangxi Natural Science Foundation (No. 2018GXNSFAA138181), the National Natural Science Foundation of China (No. 11361009) and the High level innovation teams and distinguished scholars in Guangxi Universities.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, C., Zeng, B. Continuous Dependence and Optimal Control for a Class of Variational–Hemivariational Inequalities. Appl Math Optim 82, 637–656 (2020). https://doi.org/10.1007/s00245-018-9543-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-018-9543-4

Keywords

Mathematics Subject Classification

Search

Navigation