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Advances in Mathematical Modeling, Optimization and Optimal Control pp 147–201Cite as

Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls: Theory and Applications

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 109))

Abstract

We survey the results on no-gap second-order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs Milyutin and Osmolovskii (Calculus of Variations and Optimal Control. Translations of Mathematical Monographs. American Mathematical Society, Providence, 1998) and Osmolovskii and Maurer (Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM Series Design and Control, vol. DC 24. SIAM Publications, Philadelphia, 2012), and discuss their further development. First, we formulate such conditions for broken extremals in the simplest problem of the Calculus of Variations and then, we consider them for discontinuous controls in optimal control problems with endpoint and mixed state-control constraints, considered on a variable time interval. Further, we discuss such conditions for bang-bang controls in optimal control problems, where the control appears linearly in the Pontryagin-Hamilton function with control constraints given in the form of a convex polyhedron. Bang-bang controls induce an optimization problem with respect to the switching times of the control, the so-called Induced Optimization Problem. We show that second-order sufficient condition for the Induced Optimization Problem together with the so-called strict bang-bang property ensures second-order sufficient conditions for the bang-bang control problem. Finally, we discuss optimal control problems with mixed control-state constraints and control appearing linearly. Taking the mixed constraint as a new control variable we convert such problems to bang-bang control problems. The numerical verification of second-order conditions is illustrated on three examples.

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Author information

Authors and Affiliations

  1. University of Technology and Humanities, ul. Malczewskiego 20a, 26-600, Radom, Poland

    Nikolai P. Osmolovskii

  2. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Nikolai P. Osmolovskii

  3. Moscow State University of Civil Engineering, Jaroslavskoe shosse 26, 129337, Moscow, Russia

    Nikolai P. Osmolovskii

  4. Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms–Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    Helmut Maurer

Authors
  1. Nikolai P. Osmolovskii
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  2. Helmut Maurer
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Correspondence to Nikolai P. Osmolovskii .

Editor information

Editors and Affiliations

  1. Institut de Mathématiques, Université Paul Sabatier, Toulouse, France

    Jean-Baptiste Hiriart-Urruty

  2. Department of Automatics and Bio. Engin., AGH University of Science and Technology, Kraków, Poland

    Adam Korytowski

  3. University of Münster, Institute of Computational and Appl. Mat, Münster, Nordrhein-Westfalen, Germany

    Helmut Maurer

  4. Department of Automatics and Bio.Eng., AGH University of Science and Technology, Kraków, Poland

    Maciej Szymkat

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Osmolovskii, N.P., Maurer, H. (2016). Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls: Theory and Applications. In: Hiriart-Urruty, JB., Korytowski, A., Maurer, H., Szymkat, M. (eds) Advances in Mathematical Modeling, Optimization and Optimal Control. Springer Optimization and Its Applications, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-319-30785-5_6

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