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Necessary Second-Order Conditions for a Strong Local Minimum in a Problem with Endpoint and Control Constraints

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  • Nikolai Pavlovich Osmolovskii

    (Polish Academy of Sciences)

Abstract

The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a first-order necessary condition for a strong local minimum) using the local maximum principle (a first-order necessary condition for a weak local minimum). In the present work, we use this method to derive second-order necessary conditions for a strong local minimum on the base of such conditions for a weak local minimum. For simplicity, we confine ourselves to the consideration of the Mayer problem with endpoint equality and inequality constraints and control inequality constraints given by a finite number of twice smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide a rather short proof of second-order necessary conditions for a strong local minimum.

Suggested Citation

  • Nikolai Pavlovich Osmolovskii, 2020. "Necessary Second-Order Conditions for a Strong Local Minimum in a Problem with Endpoint and Control Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 1-16, April.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:1:d:10.1007_s10957-020-01647-4
    DOI: 10.1007/s10957-020-01647-4
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    References listed on IDEAS

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    1. Nikolai P. Osmolovskii, 2015. "On Second-Order Necessary Conditions for Broken Extremals," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 379-406, February.
    2. Alexander D. Ioffe, 2019. "On Generalized Bolza Problem and Its Application to Dynamic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 285-309, July.
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