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Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers

Author

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  • Giovanni Fusco

    (Department of Mathematics “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste, 63, 35121 Padova, Italy
    These authors contributed equally to this work.)

  • Monica Motta

    (Department of Mathematics “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste, 63, 35121 Padova, Italy
    These authors contributed equally to this work.)

Abstract

We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. Therefore, it is relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in nonsmooth analysis, we investigate the relation between the occurrence of an infimum gap and the abnormality of necessary conditions. For the notion of a local minimizer based on control distance and an extension, including the impulsive one, we prove that (i) a local extended minimizer that is not a local minimizer of the original problem, and (ii) a local strict-sense minimizer that is not a local minimizer of the extended problem both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now, it has only been shown that a strict-sense minimizer that is not an extended minimizer is abnormal for an ‘averaged version’ of the maximum principle.

Suggested Citation

  • Giovanni Fusco & Monica Motta, 2024. "Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers," Mathematics, MDPI, vol. 12(7), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:943-:d:1362210
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    References listed on IDEAS

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    1. Fernando A. C. C. Fontes & Hélène Frankowska, 2015. "Normality and Nondegeneracy for Optimal Control Problems with State Constraints," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 115-136, July.
    2. A. V. Arutyunov & D. Yu. Karamzin & F. L. Pereira, 2015. "State Constraints in Impulsive Control Problems: Gamkrelidze-Like Conditions of Optimality," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 440-459, August.
    3. A. Arutyunov & V. Dykhta & F. Lobo Pereira, 2005. "Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 55-77, January.
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