Construction of lyapunov functionals for stochastic hereditary systems: a survey of some recent results

It is well known that many processes in automatic regulation, physics, mechanics, biology, economy, ecology, etc., can be modelled by hereditary systems. Many stability results in the theory of hereditary systems and their applications were obtained by construction of appropriate Lyapunov functionals (see, for instance, [1–4]). The construction of every such functional was a long time an art of its author. In this paper, formal procedure for construction of Lyapunov functionals for stochastic difference and differential equations and some results on asymptotic mean square stability conditions are considered. More details on these results are presented in [5–52]. The bibliography does not contain works of other researchers since this paper is a short survey of the authorsʹ works.