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Introduction to Averaging

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Stochastic Averaging and Stochastic Extremum Seeking

Part of the book series: Communications and Control Engineering ((CCE))

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Abstract

Existing results in averaging theory for deterministic and stochastic systems are reviewed. In the stochastic case, the system structures considered include an ODE, which is either forced by a stochastic perturbation process or by the state of a stochastic differential equation. The history of averaging methods is briefly reviewed and the restrictions imposed in the existing stochastic averaging theory are discussed.

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References

  1. Anosov DV (1960) Osrednenie v sistemakh obyknovennykh differentsial’nykh uravenenii s bystro koleblyushchimisya resheniyami. Izv Akad Nauk SSSR, Ser Mat 24(5):721–742

    MathSciNet  MATH  Google Scholar 

  2. Benveniste A, Métivier M, Priouret P (1990) Adaptive algorithms and stochastic approximations. Springer, Berlin

    Book  MATH  Google Scholar 

  3. Blankenship G, Papanicolaou GC (1978) Stability and control of stochastic systems with wide-band noise disturbances. I. SIAM J Appl Math 34:437–476

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogolyubov NN, Mitropolsky YA (1961) Asymptotic methods in the theory of nonlinear oscillation. Gordon & Breach, New York

    Google Scholar 

  5. Bogolyubov NN, Zubarev DN (1955) Metod asimptoticheskogo priblizheniya dlya sistem s vrashchayuschcheisya fazoi i ego primenenie k dvizheniyu zaryazhennykh chastits v magnitnom pole. Ukr Mat Zh 7:5–17

    MathSciNet  MATH  Google Scholar 

  6. Freidlin MI, Wentzell AD (1984) Random perturbations of dynamical systems. Springer, Berlin

    Book  MATH  Google Scholar 

  7. Hale JK, Verduyn Lunel SM (1990) Averaging in infinite dimensions. J Integral Equ Appl 2:463–494

    Article  MathSciNet  MATH  Google Scholar 

  8. Hashemi SN, Heunis AJ (1998) Averaging principle for diffusion processes. Stoch Stoch Rep 62:201–216

    MathSciNet  MATH  Google Scholar 

  9. Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  10. Khas’minskiı̌ RZ (1966) A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab Appl 11:390–406

    Article  Google Scholar 

  11. Khas’minskiı̌ RZ, Yin G (2004) On averaging principles: an asymptotic expansion approach. SIAM J Math Anal 35:1534–1560

    Article  MathSciNet  Google Scholar 

  12. Kifer Y (1992) Averaging in dynamical systems and large deviations. Invent Math 110:337–370

    Article  MathSciNet  MATH  Google Scholar 

  13. Korolyuk VS (1995) Average and stability of dynamical system with rapid stochastic switching. In: Skorokhod AV, Borovskikh YuV (eds) Exploring stochastic laws. Brill, Leiden, pp 219–232

    Google Scholar 

  14. Korolyuk VS (1998) Stability of stochastic systems in the scheme of diffusion approximation. Ukr Mat Zh 50:36–47

    Article  MathSciNet  MATH  Google Scholar 

  15. Korolyuk VS, Chabanyuk YM (2002) Stability of dynamical system with semi-Markov switches under conditions of stability of the averaged system. Ukr Mat Zh 54:239–252

    Article  MathSciNet  Google Scholar 

  16. Krylov NM, Bogolyubov NN (1937) Introduction to nonlinear mechanics. Izd AN UkSSR, Kiev (in Russian)

    Google Scholar 

  17. Kushner HJ, Ramachandran KM (1988) Nearly optimal singular controls for wideband noise driven systems. SIAM J Control Optim 26:569–591

    Article  MathSciNet  MATH  Google Scholar 

  18. Kushner HJ, Yin G (2003) Stochastic approximation and recursive algorithms and applications, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  19. Liptser RS, Shiryaev AN (1989) Theory of martingales. Kluwer Academic, Norwell

    Book  MATH  Google Scholar 

  20. Liptser RS, Stoyanov J (1990) Stochastic version of the averaging principle for diffusion type processes. Stoch Stoch Rep 32:145–163

    MathSciNet  MATH  Google Scholar 

  21. Ljung L (1977) Analysis of recursive stochastic algorithms. IEEE Trans Autom Control 22:551–575

    Article  MathSciNet  MATH  Google Scholar 

  22. Neishtadt AI (1975) Averaging in multifrequency systems, I. Sov Phys Dokl 20(7):492–494

    MATH  Google Scholar 

  23. Neishtadt AI (1976) Averaging in multifrequency systems, II. Sov Phys Dokl 21(2):80–82

    Google Scholar 

  24. Roberts JB, Spanos PD (1986) Stochastic averaging: an approximate method of solving random vibration problems. Int J Non-Linear Mech 21:111–134

    Article  MathSciNet  MATH  Google Scholar 

  25. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  26. Sastry S, Bodson M (1989) Adaptive control: stability, convergence, and robustness. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  27. Skorokhod AV (1989) Asymptotic methods in the theory of stochastic differential equations. Translations of mathematical monographs. Amer Math Soc, Providence

    MATH  Google Scholar 

  28. Skorokhod AV, Hoppensteadt FC, Salehi H (2002) Random perturbation methods with applications in science and engineering. Springer, New York

    MATH  Google Scholar 

  29. Solo V, Kong X (1994) Adaptive signal processing algorithms: stability and performance. Prentice Hall, New York

    Google Scholar 

  30. Spall JC (2003) Introduction to stochastic search and optimization: estimation, simulation, and control. Wiley-Interscience, New York

    Book  MATH  Google Scholar 

  31. Volosov VM (1962) Averaging in systems of ordinary differential equations. Russ Math Surv 17(6):1–126

    Article  MathSciNet  MATH  Google Scholar 

  32. Yin G, Zhang Q (1994) Near optimality of stochastic control in systems with unknown parameter processes. Appl Math Optim 29:263–284

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhu WQ (1988) Stochastic averaging methods in random vibration. Appl Mech Rev 41(5):189–199

    Article  Google Scholar 

  34. Zhu WQ, Yang YQ (1997) Stochastic averaging of quasi-non integrable-Hamiltonian systems. J Appl Mech 64:157–164

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, People’s Republic of China

    Shu-Jun Liu

  2. Department Mechanical & Aerospace Engineering, University of California, San Diego, La Jolla, California, USA

    Miroslav Krstic

Authors
  1. Shu-Jun Liu
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  2. Miroslav Krstic
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Correspondence to Miroslav Krstic .

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© 2012 Springer-Verlag London

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Liu, SJ., Krstic, M. (2012). Introduction to Averaging. In: Stochastic Averaging and Stochastic Extremum Seeking. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-4087-0_1

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  • DOI: https://doi.org/10.1007/978-1-4471-4087-0_1

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-4086-3

  • Online ISBN: 978-1-4471-4087-0

  • eBook Packages: EngineeringEngineering (R0)

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