Coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (n ² + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 516–522, April, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42199).     

Relationship between spectral and coefficient criteria of mean-square stability for systems of linear stochastic differential and difference equations

We establish the relationship (equivalence) between the spectral and algebraic (coefficient) criteria (the latter is represented in terms of the Sylvester matrix algebraic equation) of mean-square asymptotic stability for three classes of systems of linear equations with varying random perturbations of coefficients, namely, the ltô differential stochastic equations, difference stochastic equations with discrete time, and difference stochastic equations with continuous time.