# Lapshin A. L.

### Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 776–783

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

### The correlation matrix of random solutions of a dynamical system with Markov coefficients

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 338–348

For dynamical systems which are described by systems of differential or difference equations dependent on a finite-valued Markov process, we suggest a new form of equations for moments of their random solution. We derive equations for a correlation matrix of random solutions.

### Filtration and prediction of random solutions of a system of linear differential equations with coefficients depending on a finite-valued Markov process

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 997–1000

We obtain an equation of optimal filtration for processes of Markov random evolution, which is a solution of systems of linear differential equations with Markov switchings.

### Filtration of random solutions of a system of linear difference equations with coefficients depending on a Markov chain

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 590–592

We solve the problem of the estimation of a random state for a system with discrete time that is described by a system of linear difference equations with coefficients depending on a finite-valued Markov chain.

### Optimization of solutions of a linear control system

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1429–1431

We suggest a new method for optimizing solutions of a linear control system, which is based on the solution of the Lyapunov matrix equation.