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Erschienen in:
Normal Numbers and Pseudorandom Generators Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

4. Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

verfasst von : Henri Bonnel, Jacqueline Morgan

Erschienen in: Computational and Analytical Mathematics

Verlag: Springer New York

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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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Vorheriges Kapitel Champernowne’s Number, Strong Normality, and the X Chromosome
Nächstes Kapitel Monotone Operators Without Enlargements
Fußnoten
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
Note that the embedding \(H_{1}^{n}([t_{0},T]) \subset L_{2}^{n}([t_{0},T])\) is continuous.
 
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
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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Metadaten
Titel
Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems
verfasst von
Henri Bonnel
Jacqueline Morgan
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Computational and Analytical Mathematics

Print ISBN: 978-1-4614-7620-7

Electronic ISBN: 978-1-4614-7621-4

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-7621-4_4