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A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem

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  • Fernando Lobo Pereira

    (Porto University)

  • Nathalie T. Khalil

    (Porto University)

Abstract

In this article, we investigate a time-optimal state-constrained bilevel optimal control problem whose lower-level dynamics feature a sweeping control process involving a truncated normal cone. By bilevel, it is meant that the optimization of the upper level problem is carried out over the solution set of the lower level problem.This problem instance arises in structured crowd motion control problems in a confined space. We establish the corresponding necessary optimality conditions in the Gamkrelidze’s form. The analysis relies on the smooth approximation of the lower level sweeping control system, thereby dealing with the resulting lack of Lipschitzianity with respect to the state variable inherent to the sweeping process, and on the flattening of the bilevel structure via an exact penalization technique. Necessary conditions of optimality in the Gamkrelidze’s form are applied to the resulting standard approximating penalized state-constrained single-level problem, and the main result of this article is obtained by passing to the limit.

Suggested Citation

  • Fernando Lobo Pereira & Nathalie T. Khalil, 2022. "A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 1022-1051, March.
    • Handle: RePEc:spr:joptap:v:192:y:2022:i:3:d:10.1007_s10957-022-02004-3
    DOI: 10.1007/s10957-022-02004-3
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    References listed on IDEAS

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    1. Aram Arutyunov & Dmitry Karamzin, 2020. "A Survey on Regularity Conditions for State-Constrained Optimal Control Problems and the Non-degenerate Maximum Principle," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 697-723, March.
    2. A. V. Arutyunov & D. Y. Karamzin & F. L. Pereira, 2011. "The Maximum Principle for Optimal Control Problems with State Constraints by R.V. Gamkrelidze: Revisited," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 474-493, June.
    3. Nguyen D. Hoang & Boris S. Mordukhovich, 2019. "Extended Euler–Lagrange and Hamiltonian Conditions in Optimal Control of Sweeping Processes with Controlled Moving Sets," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 256-289, January.
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