Geodesic
The method of parametric Lyapunov functions for Markov dynamical systems
Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 3, pp. 40-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

The notion of parametric Lyapunov function is introduced for Markov dynamic systems. The existence of a function of this kind is shown to be a necessary and sufficient condition for the strong stochastic stability of an equilibrium. In terms of parametric Lyapunov functions, a sufficient criterion is proved for asymptotic strong stochastic stability in the case of Feller Markov chains. Some examples are given showing the efficiency of the method proposed. 
@article{SJIM_2005_8_3_a5,
     author = {S. M. Dobrovolskii and T. M. Strugova},
     title = {The method of parametric {Lyapunov} functions for {Markov} dynamical systems},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {40--47},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a5/}
}
TY  - JOUR
AU  - S. M. Dobrovolskii
AU  - T. M. Strugova
TI  - The method of parametric Lyapunov functions for Markov dynamical systems
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2005
SP  - 40
EP  - 47
VL  - 8
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a5/
LA  - ru
ID  - SJIM_2005_8_3_a5
ER  - 
%0 Journal Article
%A S. M. Dobrovolskii
%A T. M. Strugova
%T The method of parametric Lyapunov functions for Markov dynamical systems
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2005
%P 40-47
%V 8
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a5/
%G ru
%F SJIM_2005_8_3_a5
S. M. Dobrovolskii; T. M. Strugova. The method of parametric Lyapunov functions for Markov dynamical systems. Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 3, pp. 40-47. http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a5/

[1] Khasminskii R. Z., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969 | MR

[2] Khalanai A., Veksler D., Kachestvennaya teoriya impulsnykh sistem, Mir, M., 1971 | MR

[3] Borovkov A. A., Ergodichnost i ustoichivost sluchainykh protsessov, Editorial URSS, M., 1999 | MR

[4] Meyn S. P., Tweedie R. L., Markov Chains and Stochastic Stability, Springer-Verl., London, 1993 | MR | Zbl

[5] Meyn S. P., Tweedie R. L., “Criteria for stability of Markovian processes III: Foster–Lyapunov criteria for continuous time processes, with examples”, Adv. Appl. Probab., 25 (1993), 518–548 | DOI | MR | Zbl

[6] Kolmanovskii V. V., Nosov V. R., Ustoichivost i periodicheskie rezhimy reguliruemykh sistem s posledeistviem, Nauka, M., 1981 | MR

[7] Kolmanovskii V. W., Shaikhet L. E., “General method of Lyapunov functional construction for stability investigations of stochastic difference equations”, Dynam. Systems Appl., 4 (1995), 397–439 | MR

[8] Shaikhet L. E., “Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations”, Appl. Math. Lett., 10:3 (1997), 111–115 | DOI | MR | Zbl

[9] Florchinger P., “Adaptive stabilization of nonlinear stochastic system”, Appl. Math. Optim., 38 (1998), 109–128 | DOI | MR

[10] Battilotti S., De Santis A., “Stabilization in probability of nonlinear stochastic systems with guaranteed region of attraction and target set”, IEEE Trans. Automat. Control, 48 (2003), 1585–1599 | DOI | MR

[11] Arnold L., Random Dynamical Systems, Springer Verl., Berlin, 1998 | MR

[12] Prokhorov Yu. V., Rozanov Yu. A., Teoriya veroyatnostei. Osnovnye ponyatiya. Predelnye teoremy. Sluchainye protsessy, Nauka, M., 1987 | MR | Zbl

[13] Korenevskii D. G., “Kriterii ustoichivosti reshenii sistem lineinykh determinirovannykh i stokhasticheskikh raznostnykh uravnenii s nepreryvnym vremenem i zapazdyvaniem”, Mat. zametki, 70:2 (2001), 213–229 | MR | Zbl