# zbMATH — the first resource for mathematics

Criteria for practical stability in the $$p$$th mean of nonlinear stochastic systems. (English) Zbl 0755.93080
Summary: A new concept, practical stability, in the $$p$$th mean, is proposed for nonlinear stochastic systems. By using the Lyapunov-like functions and the basic comparison principle for stochastic systems, criteria are established for various types of practical stability in the $$p$$th mean of nonlinear stochastic systems. These criteria make it possible to determine the practical stability in the $$p$$th mean of the nonlinear stochastic systems by testing the practical stability of the corresponding auxiliary deterministic systems.

##### MSC:
 93E15 Stochastic stability in control theory 93C10 Nonlinear systems in control theory 93E03 Stochastic systems in control theory (general)
Full Text:
##### References:
 [1] LaSalle, J.P.; Lefschetz, S., Stability by Lyapunov’s direct method with applications, (1961), Academic Press New York [2] Martynyuk, A.A., Practical stability of motion, (1983), Naukava Dumka Kiev, (in Russian) · Zbl 0539.70031 [3] Martynyuk, A.A., Methods and problems of practical stability of motion theory, Nonlinear vibr. problems, 22, 19-46, (1984) · Zbl 0569.34046 [4] Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Practical stability of nonlinear systems, (1990), World Scientific Singapore · Zbl 0753.34037 [5] Ladde, G.S., Systems of differential inequalities and stochastic differential equations II, J. math. physics, 16, 894-900, (1975) · Zbl 0302.34077 [6] Ladde, G.S.; Lakshmikantham, V., Random differential inequalities, (1980), Academic Press New York [7] Gikhman, I.I.; Skorohod, A.V., Stochastic differential equations and their applications, (1982), Naukava Dumka Kiev · Zbl 0557.60041 [8] Has’minskii, R.Z., Stochastic stability of differential equations, (1980), Alphen ann Rijn Holland, Sijthoff and Noordhoff · Zbl 0276.60059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.