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Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems

Author

Listed:
  • Jinxia Cen

    (Yulin Normal University)

  • Tahar Haddad

    (Université Mohammed Seddik Benyahia)

  • Van Thien Nguyen

    (FPT University)

  • Shengda Zeng

    (Yulin Normal University
    Nanjing University
    Jagiellonian University in Krakow)

Abstract

The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.

Suggested Citation

  • Jinxia Cen & Tahar Haddad & Van Thien Nguyen & Shengda Zeng, 2022. "Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems," Journal of Global Optimization, Springer, vol. 84(3), pages 783-805, November.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01155-x
    DOI: 10.1007/s10898-022-01155-x
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    References listed on IDEAS

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    1. Yarui Duan & Song Wang & Yuying Zhou, 2021. "A power penalty approach to a mixed quasilinear elliptic complementarity problem," Journal of Global Optimization, Springer, vol. 81(4), pages 901-918, December.
    2. Ramadurai, Gitakrishnan & Ukkusuri, Satish V. & Zhao, Jinye & Pang, Jong-Shi, 2010. "Linear complementarity formulation for single bottleneck model with heterogeneous commuters," Transportation Research Part B: Methodological, Elsevier, vol. 44(2), pages 193-214, February.
    3. E. B. Mermri & W. Han, 2012. "Numerical Approximation of a Unilateral Obstacle Problem," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 177-194, April.
    4. Huang, N.J. & Yang, X.Q. & Chan, W.K., 2007. "Vector complementarity problems with a variable ordering relation," European Journal of Operational Research, Elsevier, vol. 176(1), pages 15-26, January.
    5. Mahdi Boukrouche & Domingo Tarzia, 2012. "Convergence of distributed optimal control problems governed by elliptic variational inequalities," Computational Optimization and Applications, Springer, vol. 53(2), pages 375-393, October.
    6. Y. Zhou & S. Wang & X. Yang, 2014. "A penalty approximation method for a semilinear parabolic double obstacle problem," Journal of Global Optimization, Springer, vol. 60(3), pages 531-550, November.
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