# Korenevsky D. G.

### Destabilizing effect of random parametric perturbations of the white-noise type in some quasilinear continuous and discrete dynamical systems

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1719–1724

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

### On the Impossibility of Stabilization of Solutions of a System of Linear Deterministic Difference Equations by Perturbations of Its Coefficients by Stochastic Processes of “White-Noise” Type

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 285-288

We consider the problem of mean-square stabilization of solutions of a system of linear deterministic difference equations with discrete time by perturbations of its coefficients by a stochastic “white-noise” process. The answer is negative and is based on the analysis of the corresponding matrix algebraic Sylvester equation introduced earlier by the author in the theory of stability of stochastic systems. At the same time, we answer the same question for a vector matrix system of linear difference equations with continuous time and for a vector matrix system of differential equations.

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

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 228-233

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.

### Criteria of the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1073–1081

We obtain spectral and algebraic coefficient criteria and sufficient conditions for the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay. We consider the case of a rational correlation between delays and a “white-noise”-type stochastic perturbation of coefficients. We use the method of Lyapunov functions. Most results are presented in terms of the Sylvester and Lyapunov matrix algebraic equations.

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

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 516–522

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).

### Ivan Aleksandrovich Lukovskii (on his 60th birthday)

Korenevsky D. G., Koshlyakov V. N., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1136-1137

### Algebraic criteria for absolute (relative to nonlinearity) stability of stochastic automatic control systems with nonlinear feedback

Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 731-736

### Matrix algebraic criteria and sufficient conditions for asymptotic stability and boundedness with probability 1 of the solutions of a system of linear stationary integrodifferential stochastic Ito equations

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 723–728

### Algebraic criteria and sufficient conditions for asymptotic stability and boundedness with probability 1 for the solutions of a system of linear stochastic difference equations

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 447–452

### Algebraic conditions for the absolute stability with probability 1 of the solutions of systems of linear stochastic ito equations with aftereffect. The case of a vector Wiener process and several delays

Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 23–27

### Algebraic coefficient conditions for absolute (not depending on delay) asymptotic stability with probability 1, for solutions of a system of linear stochastic into equations with contagion

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 791–795

### On the principle of averaging for second-order hyperbolic equations with functionally perturbed argument

Ukr. Mat. Zh. - 1971. - 23, № 2. - pp. 147–156

### On the Cauchy problem for a hyperbolic equation with functionally perturbed argument

Feshchenko S. F., Korenevsky D. G.

Ukr. Mat. Zh. - 1969. - 21, № 1. - pp. 108–110

### Systems with distributed parameters and time delays

Feshchenko S. F., Korenevsky D. G.

Ukr. Mat. Zh. - 1967. - 19, № 4. - pp. 57–66