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An Optimal Control Problem with a Risk Zone

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Large-Scale Scientific Computing (LSSC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10665))

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Abstract

We consider an optimal control problem for an autonomous differential inclusion with free terminal time in the situation when there is a set M (“risk zone”) in the state space \(\mathbb {R}^n\) which is unfavorable due to reasons of safety or instability of the system. Necessary optimality conditions in the form of Clarke’s Hamiltonian inclusion are developed when the risk zone M is an open set. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of problems with state constraints, this allows one to get conditions guaranteeing nondegeneracy of the developed necessary optimality conditions.

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Notes

  1. 1.

    Recall, that \(t\in [0,T)\), \(T>0\), is a point of right approximate continuity of a real function \(\xi (\cdot )\) defined on [0, T] if there is a Lebesgue measurable set \(E\subset [t,T]\) such that t is its density point, and the function \(\xi (\cdot )\) is continuous from the right at t along E (see [14, Chapt. 9, Sect. 5]).

References

  1. Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math. 54(6), 1342–1400 (1991)

    Article  MATH  Google Scholar 

  2. Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  3. Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optim. 35(3), 930–952 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theor. Appl. 149(3), 474–493 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aseev, S.M.: Methods of regularization in nonsmooth problems of dynamic optimization. J. Math. Sci. 94(3), 1366–1393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aseev, S.M.: Optimization of dynamics of a control system in the presence of risk factors. Trudy Inst. Mat. i Mekh UrO RAN 23(1), 27–42 (2017). (in Russian)

    Article  MathSciNet  Google Scholar 

  7. Aseev, S.M., Smirnov, A.I.: The Pontryagin maximum principle for the problem of optimal crossing of a given domain. Dokl. Math. 69(2), 243–245 (2004)

    MATH  Google Scholar 

  8. Aseev, S.M., Smirnov, A.I.: Necessary first-order conditions for optimal crossing of a given region. Comput. Math. Model. 18(4), 397–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  10. Cesari, L.: Optimization - Theory and Applications: Problems with Ordinary Differential Equations. Springer, New York (1983). https://doi.org/10.1007/978-1-4613-8165-5

    Book  MATH  Google Scholar 

  11. Ferreira, M.M.A., Vinter, R.B.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl. 187(2), 438–467 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fontes, F.A.C.C., Frankowska, H.: Normality and nondegeneracy for optimal control problems with state constraints. J. Optim. Theor. Appl. 166(1), 115–136 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). (in Russian)

    MATH  Google Scholar 

  14. Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar, New York (1961)

    Google Scholar 

  15. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes. Pergamon, Oxford (1964)

    Google Scholar 

  16. Pshenichnyi, B.N., Ochilov, S.: On the problem of optimal passage through a given domain. Kibern. Vychisl. Tekh. 99, 3–8 (1993)

    MathSciNet  Google Scholar 

  17. Pshenichnyi, B.N., Ochilov, S.: A special problem of time-optimal control. Kibern. Vychisl. Tekhn. 101, 11–15 (1994)

    MATH  Google Scholar 

  18. Smirnov, A.I.: Necessary optimality conditions for a class of optimal control problems with discontinuous integrand. Proc. Steklov Inst. Math. 262, 213–230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the Russian Science Foundation under grant 14-50-00005.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Sergey M. Aseev

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  1. Sergey M. Aseev
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Correspondence to Sergey M. Aseev .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

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Aseev, S.M. (2018). An Optimal Control Problem with a Risk Zone. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_19

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  • DOI: https://doi.org/10.1007/978-3-319-73441-5_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73440-8

  • Online ISBN: 978-3-319-73441-5

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