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State Constraints in Impulsive Control Problems: Gamkrelidze-Like Conditions of Optimality

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Abstract

An impulsive control problem with state constraints is considered. A Pontryagin maximum principle in the framework of R.V. Gamkrelidze is derived, being its proof based on a certain penalization technique and on the application of Ekeland’s variational principle. This approach is distinct from the more usual ones in Impulsive Control theory based on a reduction to a conventional control problem and exhibits the advantage of allowing to address problems with dynamics which are merely measurable in the time variable. Controllability assumptions to ensure the non-degeneracy of the conditions are provided in the impulsive control context. An example demonstrating the significance of the conditions is given.

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Acknowledgments

This research was supported by the Russian Foundation for Basic Research, Grant Numbers 13-01-00494 and 15-01-04601, by the Ministry of Education and Science of the Russian Federation, Grant Number 1.333.2014/K, and by the Foundation for Science and Technology (Portugal), projects PEst-OE/EEI/UI0147/2014, PTDC/EEI-AUT/1450/2012, PIIF-GA-2011-301177.We are thankful to the anonymous referees for useful remarks.

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia, Miklukho-Maklaja st. 6, 117198, Moscow, Russia

    A. V. Arutyunov

  2. Computing Centre of the Russian Academy of Sciences, Vavilova st.,40, 119991, Moscow, Russia

    D. Yu. Karamzin

  3. SYSTEC, Faculty of Engineering, University of Porto, Dr. Roberto Frias st., s/n, 4200-465, Porto, Portugal

    F. L. Pereira

Authors
  1. A. V. Arutyunov
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  2. D. Yu. Karamzin
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  3. F. L. Pereira
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Correspondence to D. Yu. Karamzin.

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Communicated by Günter Leugering.

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Arutyunov, A.V., Karamzin, D.Y. & Pereira, F.L. State Constraints in Impulsive Control Problems: Gamkrelidze-Like Conditions of Optimality. J Optim Theory Appl 166, 440–459 (2015). https://doi.org/10.1007/s10957-014-0690-8

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