Lyapunov functionals construction for stochastic difference second-kind Volterra equations with continuous time

Leonid Shaikhet

doi:10.1155/s1687183904308022

Abstract

The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem, this method can be successfully used also for stochastic difference Volterra equations with continuous time usable in mathematical models. The theoretical results are illustrated by numerical calculations.

Highlights

The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered
Construction of Lyapunov functionals is usually used for investigation of stability of hereditary systems which are described by functional differential equations or Volterra equations and have numerous applications [3, 4, 8, 21]
It is shown that after some modification of the basic Lyapunov-type stability theorem, this method can be used for stochastic difference Volterra equations with continuous time, which are popular enough in researches [1, 14, 15, 16, 20]

Summary

Stability theorem

Construction of Lyapunov functionals is usually used for investigation of stability of hereditary systems which are described by functional differential equations or Volterra equations and have numerous applications [3, 4, 8, 21]. (ii) asymptotically mean square stable if it is mean square stable and for each initial function φ, condition (1.6) holds;. (iii) asymptotically mean square quasistable if it is mean square stable and for each initial function φ and each t ∈ [t0, t0 + h0), condition (1.7) holds. 2. From here and (1.8), it follows that the solution of (1.2), (1.3) is uniformly mean square summable. The solution of (1.2) for each initial function (1.3) is mean square integrable. The trivial solution of (1.10) is asymptotically mean square quasistable. It means that the trivial solution of is asymptotically mean square quasistable. From Theorem 1.5 and Remark 1.6, it follows that an investigation of the solution of (1.2) can be reduced to the construction of appropriate Lyapunov functionals.

Formal procedure of Lyapunov functionals construction

Linear Volterra equations with constant coefficients

84 Difference Volterra equations with continuous time η 1

86 Difference Volterra equations with continuous time x 2 1