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Hierarchical Controllability for a Nonlinear Parabolic Equation in One Dimension

Author

Listed:
  • Miguel R. Nuñez-Chávez

    (UFMT)

  • Juan B. Límaco Ferrel

    (IME, UFF)

Abstract

This paper deals with the hierarchical control of a nonlinear parabolic equation in one dimension. The novelty in this work is the appearance of the spatial derivative of the solution instead of considering only the solution in the quasilinear term (nonlinearity), here lies the difficulty of approaching said equation. We use Stackelberg–Nash strategies. As usual, we consider one control called leader and two controls called followers. To each leader, we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves null and trajectory controllability problems. First, we study the linear problem and then, we use the results obtained in the linear case to conclude the nonlinear problem by applying the Right Inverse Function Theorem. Some comments will be put at the end.

Suggested Citation

  • Miguel R. Nuñez-Chávez & Juan B. Límaco Ferrel, 2023. "Hierarchical Controllability for a Nonlinear Parabolic Equation in One Dimension," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 1-48, July.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02217-0
    DOI: 10.1007/s10957-023-02217-0
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    References listed on IDEAS

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    1. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 457-498, March.
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