Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Investigation of a system of linear differential equations with random coefficients

We investigate a system of linear differential equations with random coefficients that depend on a periodic Markov process. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1137–1143, August, 1998.     

On one method of factorization of polynomials

We propose and justify a numerical method of factorization of polynomials with complex coefficients. We construct and algorithm of factorization of polynomials with real coefficients into real factors in the case of multiple roots. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1281–1286, September, 1999.     

Optimization of a system of linear differential equations with random coefficients

We consider a system of differential equations with controls that are linearly contained in the right-hand sides. We establish a necessary condition for the optimal control that minimizes a quadratic functional. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 556–561, April, 1999.     

Derivation of Moment Equations for Solutions of a System of Differential Equations Dependent on a Semi-Markov Process

We present a new method for the derivation of moment equations for solutions of a system of nonlinear differential equations dependent on a finite-valued semi-Markov process. For systems of linear equations, we compare the results obtained with known ones.     

Investigation of stabilization of a mathematical model of a dynamical system with random influence in the resonance case

We construct and investigate a mathematical model of a dynamical system with random influence stabilized by increasing the frequency of random influence.     

Derivation of Moment Equations for Solutions of a System of Nonlinear Difference Equations Dependent on a Semi-Markov Process

We propose a method for the derivation of moment equations for solutions of a system of nonlinear difference equations that depends on a finite-valued semi-Markov process. For systems of linear equations, we compare the results obtained with known ones.     

On one generalization of the averaging method

We consider one case where it is possible to establish sufficient conditions for the convergence and analyticity of matrix series used for the construction of a system of moment equations. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 906–911, July, 1997.     

Optimization of Nonlinear Systems of Stochastic Difference Equations

We present new results concerning the synthesis of optimal control for systems of difference equations that depend on a semi-Markov or Markov stochastic process. We obtain necessary conditions for the optimality of solutions that generalize known conditions for the optimality of deterministic systems of control. These necessary optimality conditions are obtained in the form convenient for the synthesis of optimal control. On the basis of Lyapunov stochastic functions, we obtain matrix difference equations of the Riccati type, the integration of which enables one to synthesize an optimal control. The results obtained generalize results obtained earlier for deterministic systems of difference equations.     

Second-order moment equations for a system of differential equations with random right-hand side

We present a method for the derivation of second-order moment equations for solutions of a system of nonlinear equations that depends on a finite-valued semi-Markov or Markov process. For systems of linear differential equations with random coefficients, the case where the inhomogeneous part contains white noise is considered.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 687–690, May, 2004.