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Maximum Principle for Delayed Stochastic Switching System with Constraints

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Computational Problems in Science and Engineering

Part of the book series: Lecture Notes in Electrical Engineering ((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.

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References

  1. Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  2. Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publication House, Chichester (1997)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  4. Kolmanovsky, V.B., Myshkis, A.D.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1992)

    Book  Google Scholar 

  5. Agayeva, C.A., Allahverdiyeva, J.J.: Kiev 13(29), 3–11 (2007)

    MathSciNet  Google Scholar 

  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. Chernousko, F.L., Ananievski, I.M., Reshmin, S.A.: Control of Nonlinear Dynamical Systems: Methods and Applications (Communication and Control Engineering). Springer, Berlin (2008)

    Book  Google Scholar 

  8. Elsanosi, I., Øksendal, B., Sulem, A.: Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71(1–2), 69–89 (2000)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975)

    Book  MATH  Google Scholar 

  11. Larssen, B.: Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Rep. 74(3–4), 651–673 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Boukas, E.-K.: Stochastic Switching Systems: Analysis and Design. Birkhauer, Boston (2006)

    Google Scholar 

  14. Avezedo, N., Pinherio, D., Weber, G.W.: Dynamic programing for a Markov-switching jump diffusion. J. Comput. Appl. Math. 267, 1–19 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Hall, E., Hanagud, S.: Control of nonlinear structural dynamic systems—chaotic vibrations. J. Guid. Control Dyn. 16(3), 470–476 (1993)

    Article  Google Scholar 

  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. 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. 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. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  23. Capuzzo, D.I., Evans, L.C.: Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22(1), 143–161 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. Bengea, S.C., Raymond, A.C.: Optimal control of switching systems. Automatica 41, 11–27 (2005)

    MATH  Google Scholar 

  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. 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. Abushov, Q., Aghayeva, C.: Stochastic maximum principle for the nonlinear optimal control problem of switching systems. J. Comput. Appl. Math. 259, 371–376 (2014)

    Article  MathSciNet  Google Scholar 

  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 

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Correspondence to Charkaz Aghayeva .

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Aghayeva, C. (2015). Maximum Principle for Delayed Stochastic Switching System with Constraints. In: Mastorakis, N., Bulucea, A., Tsekouras, G. (eds) Computational Problems in Science and Engineering. Lecture Notes in Electrical Engineering, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-319-15765-8_10

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  • DOI: https://doi.org/10.1007/978-3-319-15765-8_10

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15764-1

  • Online ISBN: 978-3-319-15765-8

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