Volume 27, 2021
|Number of page(s)||30|
|Published online||16 December 2021|
Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization
Avignon Université, Laboratoire de Mathématiques d’Avignon (EA 2151),
2 UMR MISTEA, Univ. Montpellier, INRAE, Institut Agro, 34060 Montpellier, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 23 November 2021
We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than the usual one encountered in the hybrid setting, which requires that any optimal solution should cross the boundary of the constraint set transversely. Doing so, we apply the Pontryagin Maximum Principle to a sequence of regular optimal control problems whose integral cost approximates the time of crisis. Optimality conditions are derived by passing to the limit in the Hamiltonian system (without the use of the hybrid maximum principle). This convergence result essentially relies on the boundedness of the sequence of adjoint vectors in L∞. Our main contribution is to relate this property to the boundedness in L1 of a suitable sequence which allows to avoid the use of the transverse hypothesis on optimal paths. An example with non-transverse trajectories for which necessary conditions are derived highlights the use of this new condition.
Mathematics Subject Classification: 34H05 / 49K15 / 49J45 / 34A38
Key words: Time crisis / Necessary optimality conditions / Hybrid Maximum Principle / Regularization / Transverse crossing time
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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