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Coupled Variational Inequalities: Existence, Stability and Optimal Control

Author

Listed:
  • Jinjie Liu

    (Chongqing Normal University)

  • Xinmin Yang

    (Chongqing Normal University)

  • Shengda Zeng

    (Yulin Normal University
    Nanjing University
    Faculty of Mathematics and Computer Science)

  • Yong Zhao

    (Chongqing Jiaotong University
    Chongqing University)

Abstract

In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity arguments, and a stability result for the solutions of CVI is proposed, through the perturbations of duality mappings. Furthermore, an optimal control problem governed by CVI is introduced, and a solvability result for the optimal control problem is established. Finally, to illustrate the applicability of the theoretical results, we study a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively.

Suggested Citation

  • Jinjie Liu & Xinmin Yang & Shengda Zeng & Yong Zhao, 2022. "Coupled Variational Inequalities: Existence, Stability and Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 877-909, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01995-9
    DOI: 10.1007/s10957-021-01995-9
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    References listed on IDEAS

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    1. Xing Wang & Nan-jing Huang, 2014. "A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 633-648, August.
    2. Stanisław Migórski & Shengda Zeng, 2018. "A class of differential hemivariational inequalities in Banach spaces," Journal of Global Optimization, Springer, vol. 72(4), pages 761-779, December.
    3. Y. C. Liou & X. Q. Yang & J. C. Yao, 2005. "Mathematical Programs with Vector Optimization Constraints," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 345-355, August.
    4. Gaoxi Li & Xinmin Yang, 2021. "Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 724-743, March.
    5. Guo-ji Tang & Nan-jing Huang, 2013. "Existence theorems of the variational-hemivariational inequalities," Journal of Global Optimization, Springer, vol. 56(2), pages 605-622, June.
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