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Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems. (English) Zbl 0893.60029
In their previous papers V. B. Kolmanovskij and L. E. Shajkhet [see, for example, Differ. Equations 29, No. 11, 1657-1666 (1993); translation from Differ. Uravn. 29, No. 11, 1909-1920 (1993; Zbl 0815.34068) and in: Dynamical systems and applications. World Sci. Ser. Appl. Anal. 4, 397-439 (1995; Zbl 0846.93083)] have proposed a procedure which allows in some sense to formalize the algorithm of construction of the Lyapunov function to investigate the stability problem for stochastic differential equations as well as for stochastic difference equations. Stability conditions are formulated in terms of the coefficients of the equations under consideration. The present paper contains a procedure of construction of the Lyapunov function to investigate the stability problem for stochastic difference equations of the Volterra type.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
34F05 Ordinary differential equations and systems with randomness
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