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Stability of the linear stochastic system

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Stability of Stochastic Dynamical Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 294))

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References

  1. A. Rosenbloom, Analysis of linear systems with randomly time-varying parameters, Proc. Symp. Inf. Nets., Vol. III, Poly. Inst. Brooklyn, p. 145 (1954).

    Google Scholar 

  2. R. L. Stratonovich, et al., "Non-Linear Transformations of Stochastic Processes," Pergamon Press, Oxford (1965).

    MATH  Google Scholar 

  3. J.E. Bertram and P.E. Sarachik, "Stability of circuits with randomly time-varying parameters," Trans. IRE, PGIT-5, Special Supplement, p. 260 (1959).

    Google Scholar 

  4. I.I. Kats and N.N. Krasovskii, "On the stability of systems with random parameters," Prkil. Met. Mek. 24, 809 (1960).

    Google Scholar 

  5. J.C. Samuels, "On the mean square stability of random linear systems," Trans. IRE, PGIT-5, Special Supplement, p. 248 (1959).

    Google Scholar 

  6. T.K. Caughey and A.H. Cray, Jr., "On the almost sure stability of linear dynamic systems with stochastic coefficients," ASME J. Appl. Mech. 32, 365 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  7. T.K. Caughey, Comments on "On the stability random systems," J. Acoust. Soc. Am. 32, 1356 (1960).

    Article  Google Scholar 

  8. H. Kushner, "Stochastic Stability and Control," Academic Press, New York (1967).

    MATH  Google Scholar 

  9. R.Z. Khas'minskii, "On the stability of the trajectories of Markov processes (in English), Prikl. Mat. Mek. 26, 1554 (1962).

    MathSciNet  Google Scholar 

  10. N.B. Nevel'son and R.Z. Khas'minskii, "Stability of a linear system with random disturbances of its parameters (in English). Prikl. Mat. Mek. 30, 487 (1966).

    MathSciNet  MATH  Google Scholar 

  11. E.F. Infante, "On the stability of some linear non-autonamous random systems," ASME paper 67-WA/APM-25 (1967).

    Google Scholar 

  12. F. Kozin, "On almost sure stability of linear systems with random coefficients," M.I.T. J. Math. Phys. 43, 59 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Kozin, "On almost sure asymptotic sample properties of diffusion processes defined by stochastic differential equations," J. Math. Kyoto Univ., 4, 575 (1965).

    MathSciNet  MATH  Google Scholar 

  14. F. Kozin, "A survey of stability of stochastic systems," Automatika, J. IFAC, 5 (1969), pp. 95–112.

    Article  MathSciNet  MATH  Google Scholar 

  15. B.H. Bharucha, "On the stability of randomly varying systems," Ph.D. Thesis Dept. of Elec. Eng., Univ. Calif., Berkeley, July (1961).

    Google Scholar 

  16. F. Kozin, "On relations between moment properties and almost sure Lyapunov stability for linear stochastic systems," J. Math. Anal. Appl. 10, 342 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Kozin, "Stability of stochastic systems," Paper 3A, 3rd IFAC Congr., London (1966).

    Google Scholar 

  18. L. Cesari, "Asymptotic Behavior and Stability Problems in Ordinary Differential Equations," Springer, Berlin (1959).

    Book  MATH  Google Scholar 

  19. F. Kozin, C.M. Wu, "On the stability of linear stochastic differential equations," (to appear ASME Journ. of App. Mech. (1972)).

    Google Scholar 

  20. R.Z. Khaz'minskii, "Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems," Th. Prob. Appls. 1, 144 (1967).

    Article  Google Scholar 

  21. K. Ito, "On stochastic differential equations," Mem. Am. Math. Soc. No. 4 (1951).

    Google Scholar 

  22. K. Ito, "On a formula concerning stochastic differentials," Nagoya Math. Jr., Japan 3, 55 (1951).

    Google Scholar 

  23. R.Z. Khaz'minskii, "Stability of Systems of differential equations with random parametric excitation," Book in Russian (1969).

    Google Scholar 

  24. K. Ito and H.P. McKean, Jr., "Diffusion Processes and their sample paths," Springer-Verlag/Academic Press, New York 1965.

    Book  MATH  Google Scholar 

  25. F. Kozin, S. Prodromou, "Necessary and sufficient conditions for almost sure sample stability of linear Ito equations," SIAM Journ. Appl. Math. Vol. 21, No. 3, pp. 413–424 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Wong and M. Zakai, "On the convergence of orginary integrals to stochastic integrals," Ann. Math. Statist., 36 (1965), pp. 1560–1564.

    Article  MathSciNet  MATH  Google Scholar 

  27. R.L. Stratonovich, "A new representation for stochastic integrals," SIAM J. Control, 4 (1966), pp. 362–371.

    Article  MathSciNet  Google Scholar 

  28. G.F. McDonough, "Stability Problems in the Control of Saturn Launch Vehicles," Dynamic Stability of Structures, Proc. Int. Conf. Pergamon, New York, p. 113 (1967).

    Google Scholar 

  29. S. Prodromou, "Necessary and sufficient conditions for stability of stochastic systems," Doctoral thesis, System Sciences, Polytechnic Institute of Brooklyn, New York, 1970.

    Google Scholar 

  30. R.R. Mitchell, "Necessary and sufficient conditions for sample stability of second order stochastic differential equations," Doctoral thesis, School of Aero Astro and Engineering Sciences, Purdue University, (1970).

    Google Scholar 

  31. J.C. Samuels, A.C. Eringen, "On stochastic linear systems," MIT J. Math. Phys. 38, p. 83 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  32. S.T. Ariaratnam, "Dynamic stability of a column under random loading," Dynamic Stability of Structures, Proc. Int. Conf., Pergamon, New York, p. 267, (1967).

    Google Scholar 

  33. J.A. Lepore, and H.C. Shah, "Dynamic stability of axially loaded columns subjected to stochastic excitation," AIAA Journal, vol. 6, No. 8, August 1968, pp. 1515–1521.

    Article  MATH  Google Scholar 

  34. P.K.C. Wang, "On the almost sure stability of linear stochastic distributed parameter dynamical systems," A.S.M.E. Transactions, Journal of Applied Mechanics, March, 1966, pp. 182–186.

    Google Scholar 

  35. E.F. Infante, and R.H. Plaut, "Stability of a column subjected to a time-dependent axial load," AIAA Journal, vol. 7, No. 4, April 1969, pp. 766–768.

    Article  MATH  Google Scholar 

  36. R.H. Plaut, and E.F. Infante, "On the stability of some continuous systems subjected to random excitation," A.S.M.E. Transactions, Journal of Applied Mechanics, September 1970, pp. 623–627.

    Google Scholar 

  37. A.H. Gray, Jr., "Frequency dependent almost sure stability conditions for a parametrically excited random vibrational system," ASME paper 67-APM-9 (1967).

    Google Scholar 

  38. C.S. Hsu, T.H. Lee, "A stability study of continuous systems under parametric excitation via Lyapunov's direct method," IUTAM Symposium on Instability of Continuous Systems, July 1969.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Polytechnic Institute of Brooklyn, U.S.A.

    F. Kozin

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  1. F. Kozin
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© 1972 Springer-Verlag

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Kozin, F. (1972). Stability of the linear stochastic system. In: Stability of Stochastic Dynamical Systems. Lecture Notes in Mathematics, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064943

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  • DOI: https://doi.org/10.1007/BFb0064943

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06050-5

  • Online ISBN: 978-3-540-38000-9

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