Advertisement

An Optimal Control Problem with a Risk Zone

  • Sergey M. Aseev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10665)

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.

Keywords

Risk zone State constraints Optimal control Differential inclusion Hamiltonian inclusion Stationarity condition 

Notes

Acknowledgements

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

References

  1. 1.
    Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math. 54(6), 1342–1400 (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arutyunov, A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aseev, S.M.: Methods of regularization in nonsmooth problems of dynamic optimization. J. Math. Sci. 94(3), 1366–1393 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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)zbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  10. 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 CrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). (in Russian)zbMATHGoogle Scholar
  14. 14.
    Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar, New York (1961)Google Scholar
  15. 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. 16.
    Pshenichnyi, B.N., Ochilov, S.: On the problem of optimal passage through a given domain. Kibern. Vychisl. Tekh. 99, 3–8 (1993)MathSciNetGoogle Scholar
  17. 17.
    Pshenichnyi, B.N., Ochilov, S.: A special problem of time-optimal control. Kibern. Vychisl. Tekhn. 101, 11–15 (1994)zbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations