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Maximum Principle for Delayed Stochastic Switching System with Constraints

  • Charkaz AghayevaEmail author
Chapter
  • 715 Downloads
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 343)

Abstract

This paper is devoted to the stochastic optimal control problem of switching systems with constraints. Dynamic of the system is described by the collection of delayed stochastic differential equations which initial conditions depend on its previous state. The restriction on the system is defined by the functional constraints on the end of each interval. Maximum principle for stochastic control problems of delayed switching system is established. Afterwards, using Ekeland’s Variational Principle the necessary condition of optimality for optimal control problem with constraints is obtained.

Keywords

Stochastic control system Differential equation with delay Switching system Switching law Optimal control problem Maximum principle 

References

  1. 1.
    Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)zbMATHCrossRefGoogle Scholar
  2. 2.
    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publication House, Chichester (1997)zbMATHGoogle Scholar
  3. 3.
    Chojnowska-Michalik, A.: Representation theorem for general stochastic delay equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 26(7), 635–642 (1978)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Kolmanovsky, V.B., Myshkis, A.D.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1992)CrossRefGoogle Scholar
  5. 5.
    Agayeva, C.A., Allahverdiyeva, J.J.: Kiev 13(29), 3–11 (2007)MathSciNetGoogle Scholar
  6. 6.
    El-Bakry, H.M., Mastorakis, N.: Fast packet detection by using high speed time delay neural networks. In: Chen, S., Guan, Q. (eds.) Proceedings of the 10th WSEAS International Conference on Multimedia Systems & Signal Processing, pp. 222–227 (2010)Google Scholar
  7. 7.
    Chernousko, F.L., Ananievski, I.M., Reshmin, S.A.: Control of Nonlinear Dynamical Systems: Methods and Applications (Communication and Control Engineering). Springer, Berlin (2008)CrossRefGoogle Scholar
  8. 8.
    Elsanosi, I., Øksendal, B., Sulem, A.: Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71(1–2), 69–89 (2000)zbMATHCrossRefGoogle Scholar
  9. 9.
    Federico, S., Golds, B., Gozzi, F.: HJB equations for the optimal control of differential equations with delays and state constraints, II: optimal feedbacks and approximations. SIAM J. Control Optim. 49, 2378–2414 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975)zbMATHCrossRefGoogle Scholar
  11. 11.
    Larssen, B.: Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Rep. 74(3–4), 651–673 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Vinter, R.B., Kwong, R.H.: The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J. Control Optim. 19(1), 139–153 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Boukas, E.-K.: Stochastic Switching Systems: Analysis and Design. Birkhauer, Boston (2006)Google Scholar
  14. 14.
    Avezedo, N., Pinherio, D., Weber, G.W.: Dynamic programing for a Markov-switching jump diffusion. J. Comput. Appl. Math. 267, 1–19 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shen, H., Xu, S., Song, X., Luo, J.: Delay-dependent robust stabilization for uncertain stochastic switching systems with distributed delays. Asian J. Control 5(11), 527–535 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Aghayeva, C.A., Abushov, Q.: The maximum principle for the nonlinear stochastic optimal control problem of switching systems. J. Glob. Optim. 56(2), 341–352 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Aghayeva, Ch.A., Abushov, Q.: Stochastic optimal control problem for switching system with controlled diffusion coefficients. In: Ao, S.I., Gelman, L., Hukins, D.W.L. (eds.) Book Series: Lecture Notes in Engineering and Computer Science, vol. 1, pp. 202–207 (2013)Google Scholar
  18. 18.
    Hall, E., Hanagud, S.: Control of nonlinear structural dynamic systems—chaotic vibrations. J. Guid. Control Dyn. 16(3), 470–476 (1993)CrossRefGoogle Scholar
  19. 19.
    Aghayeva, C.A.: Stochastic optimal control problem of switching systems with lag. Trans. ANAS Math. Mech. Ser. Phys.-Tech. Math. Sci. 31(3), 68–73 (2011)Google Scholar
  20. 20.
    Aghayeva, Ch.A.: Necessary condition of optimality for stochastic switching systems with delay. In: Senichenkov, Y., Korablev, V., et al. (eds.) Proceedings of International Conference MMAS’14, pp. 54–58(2014).Google Scholar
  21. 21.
    Kharatatishvili, G., Tadumadze, T.: The problem of optimal control for nonlinear systems with variable structure, delays and piecewise continuous prehistory. Mem. Diff. Equat. Math. Phys. 11, 67–88 (1997)Google Scholar
  22. 22.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Capuzzo, D.I., Evans, L.C.: Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22(1), 143–161 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Bengea, S.C., Raymond, A.C.: Optimal control of switching systems. Automatica 41, 11–27 (2005)zbMATHGoogle Scholar
  25. 25.
    Seidmann, T.I.: Optimal control for switching systems. In: Proceedings of the 21st Annual Conference on Informations Science and Systems, pp. 485–489 (1987)Google Scholar
  26. 26.
    Agayeva, Ch., Abushov, Q.: Necessary condition of optimality for stochastic control systems with variable structure. In: Sakalauskas, L., Weber, G., Zavadskas, E. (eds.) Proceedings of EurOPT 2008, pp. 77–81 (2008)Google Scholar
  27. 27.
    Abushov, Q., Aghayeva, C.: Stochastic maximum principle for the nonlinear optimal control problem of switching systems. J. Comput. Appl. Math. 259, 371–376 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Aghayeva, Ch., Abushov, Q.: Stochastic maximum principle for switching systems. In: AidaZade, K. (eds.) 4th International Conference PCI, vol. 3, pp. 198–201 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Industrial Engineering DepartmentAnadolu UniversityEskisehirTurkey
  2. 2.Institute of Control Systems of ANASBakuAzerbaijan

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