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Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

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Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

We present optimality conditions for bilevel optimal control problems where the upper level is a scalar optimal control problem to be solved by a leader and the lower level is a multiobjective convex optimal control problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing amongst efficient optimal controls. We deal with the so-called optimistic case, when the followers are assumed to choose the best choice for the leader amongst their best responses, as well with the so-called pessimistic case, when the best response chosen by the followers can be the worst choice for the leader. This paper continues the research initiated in Bonnel (SIAM J. Control Optim. 50(6), 3224–3241, 2012) where existence results for these problems have been obtained.

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Notes

  1. 1.

    If A and B f do not depend on t, it is well known that this system is controllable if, and only if, \(\mathrm{rank}\,(\mathbf{B_{f}},A\mathbf{B_{f}},{A}^{2}\mathbf{B_{f}},\ldots,{A}^{n-1}\mathbf{B_{f}}) = n\).

  2. 2.

    Note that the embedding \(H_{1}^{n}([t_{0},T]) \subset L_{2}^{n}([t_{0},T])\) is continuous.

  3. 3.

    In the sense that there exists a function \(\tilde{u}_{l}\) continuous at t 1 and \(\bar{u}_{l}(t) =\tilde{ u}_{l}(t)\) a.e. on [t 0 ,T]. Note that by Lusin’s theorem, we can find measurable sets of arbitrarily small positive measure and such functions \(\tilde{u}_{l}\) which are continuous on the complement of those sets.

  4. 4.

    We identify the Hilbert space \(L_{2}^{m_{l}}([t_{0},T])\) with its dual according to Riesz-Fréchet theorem; hence \(\nabla _{u_{l}}\hat{J}_{l}(\theta,t_{1},u_{l}) \in L_{2}^{m_{l}}([t_{0},T])\) (see, e.g. [7, p. 38]).

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Acknowledgements

Henri Bonnel thanks the University of Naples Federico II and CSEF for its support and the Department of Mathematics and Statistics for its hospitality.

Author information

Authors and Affiliations

  1. Université de la Nouvelle-Calédonie, ERIM, BP. R4, F98851, Nouméa Cédex, New Caledonia, France

    Henri Bonnel

  2. Department of Economy and Statistics & CSEF, University of Naples Federico II, Complesso Universitario di Monte S. Angelo, Via Cintia, 80126, Napoli, Italy

    Jacqueline Morgan

Authors
  1. Henri Bonnel
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  2. Jacqueline Morgan
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Corresponding author

Correspondence to Henri Bonnel .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Communicated By Michel Théra.

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Bonnel, H., Morgan, J. (2013). Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_4

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