Asymptotic stability condition for stochastic Markovian systems of differential equations

Efraim Shmerling

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 4, page 443-448
  • ISSN: 0862-7959

Abstract

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Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by d X ( t ) = A ( ξ ( t ) ) X ( t ) d t + H ( ξ ( t ) ) X ( t ) d w ( t ) , where ξ ( t ) is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.

How to cite

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Shmerling, Efraim. "Asymptotic stability condition for stochastic Markovian systems of differential equations." Mathematica Bohemica 135.4 (2010): 443-448. <http://eudml.org/doc/196332>.

@article{Shmerling2010,
abstract = {Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by $\{\rm d\} X(t) = A(\xi (t))X(t) \{\rm d\} t + H(\xi (t))X(t) \{\rm d\} w(t)$, where $\xi (t)$ is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.},
author = {Shmerling, Efraim},
journal = {Mathematica Bohemica},
keywords = {jump parameter system; Markov process; asymptotic stability; jump parameter system; Markov process; asymptotic stability},
language = {eng},
number = {4},
pages = {443-448},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic stability condition for stochastic Markovian systems of differential equations},
url = {http://eudml.org/doc/196332},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Shmerling, Efraim
TI - Asymptotic stability condition for stochastic Markovian systems of differential equations
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 4
SP - 443
EP - 448
AB - Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by ${\rm d} X(t) = A(\xi (t))X(t) {\rm d} t + H(\xi (t))X(t) {\rm d} w(t)$, where $\xi (t)$ is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.
LA - eng
KW - jump parameter system; Markov process; asymptotic stability; jump parameter system; Markov process; asymptotic stability
UR - http://eudml.org/doc/196332
ER -

References

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  1. Bitmead, R. R., Anderson, B. D. O., 10.1109/TAC.1980.1102427, IEEE Trans. Autom. Control 25 (1980), 782-787. (1980) Zbl0451.93065MR0583456DOI10.1109/TAC.1980.1102427
  2. Blair, W. P. Jr., Sworder, D. D., 10.1109/TSMC.1975.5408409, IEEE Trans. Systems Man Cyber. 5 (1975), 341-346. (1975) Zbl0302.90013MR0469423DOI10.1109/TSMC.1975.5408409
  3. Blankenship, G., 10.1109/TAC.1977.1101612, IEEE Trans. Autom. Control 22 (1977), 834-838. (1977) Zbl0362.93033MR0470386DOI10.1109/TAC.1977.1101612
  4. Blom, H. A. P., Bar-Shalom, Y., 10.1109/9.1299, IEEE Trans. Autom. Control 33 (1988), 780-783. (1988) Zbl0649.93065DOI10.1109/9.1299
  5. Costa, O. L. V., Fragoso, M. D., 10.1006/jmaa.1993.1341, J. Math. Anal. Appl. 179 (1993), 154-178. (1993) Zbl0790.93108MR1244955DOI10.1006/jmaa.1993.1341
  6. Souza, C. E. de, Fragoso, M. D., H ∞ control for linear systems with Markovian jumping parameters, Control Theory Adv. Tech. 9 (1993), 457-466. (1993) MR1236777
  7. Fang, Y., 10.1109/9.557580, IEEE Trans. Autom. Control 42 (1997), 378-382. (1997) Zbl0870.93048MR1435826DOI10.1109/9.557580
  8. Feng, X., Loparo, K. A., Ji, Y., Chizeck, H. J., 10.1109/9.109637, IEEE Trans. Autom. Control 37 (1992), 1884-1892. (1992) Zbl0747.93079MR1139614DOI10.1109/9.109637
  9. Fragoso, M. D., Costa, O. L. V., 10.1137/S0363012903434753, SIAM J. Control Optim. 44 (2005), 1165-1191. (2005) MR2177308DOI10.1137/S0363012903434753
  10. Ji, Y., Chizeck, H. J., 10.1109/9.57016, IEEE Trans. Autom. Control 35 (1990), 777-788. (1990) MR1058362DOI10.1109/9.57016
  11. Ji, Y., Chizeck, H. J., Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control Theory Adv. Tech. 6 (1990), 289-319. (1990) MR1080011
  12. Leizarowitz, A., 10.1080/17442508808833523, Stochastics 24 (1988), 335-356. (1988) Zbl0669.34060MR0972970DOI10.1080/17442508808833523
  13. Loparo, K. A., 10.1080/07362998408809033, Stochastic Anal. Appl. 2 (1984), 193-228. (1984) MR0746436DOI10.1080/07362998408809033
  14. Mao, X., Yuan, C., Stochastic Differential Equations with Markovian Switching, Imperial College Press, London (2006). (2006) Zbl1126.60002MR2256095
  15. Mariton, M., 10.1016/0167-6911(88)90098-9, Systems-Control Letts. 11 (1988), 393-397. (1988) Zbl0672.93073MR0971460DOI10.1016/0167-6911(88)90098-9
  16. Morozan, T., 10.1080/07362998308809016, Stochastic Anal. Appl. 1 (1983), 219-225. (1983) Zbl0525.93070MR0709081DOI10.1080/07362998308809016
  17. Shmerling, E., Hochberg, K. J., 10.1080/17442500802006436, Stochastics 80 (2008), 513-518. (2008) Zbl1153.60389MR2460246DOI10.1080/17442500802006436

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