# Stanzhitskii A. N.

### Application of the method of averaging to the problems of optimal control over functional-differential equations

Koval’chuk T. V., Kravets V. I., Mohyl'ova V. V., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 206-215

We study the application of the method of averaging to the problems of optimal control over functional-differential equations. The procedure of averaging allows us to replace the original problem with the problem of optimal control by a system of ordinary differential equations. It is proved that the optimal control over the averaged problem is almost optimal for the exact problem. The optimal control problems are investigated on finite and infinite horizons.

### Viscous solutions for the Hamilton – Jacobi – Bellman equation on time scales

Danilov V. Ya., Lavrova O. E., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 933-950

We introduce the concept of viscous solution for the Bellman equation on time scales and establish сonditions for the existence and uniqueness of this solution.

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Oscillation of solutions of the second-order linear functional-difference equations

Karpenko O. V., Kravets V. I., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 226-235

We establish conditions for the oscillation of solutions of functional difference linear equations and discrete difference linear equations of the second order in the case where the corresponding solutions of their differential analogs are oscillating on a segment.

### On asymptotic equivalence of solutions of stochastic and ordinary equations

Novak I. H., Samoilenko A. M., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1103-1127

For a weakly nonlinear stochastic system, we construct a system of ordinary differential equations the behavior of solutions of which at infinity is similar to the behavior of solutions of the original stochastic system.

### Global attractor for the autonomous wave equation in *R*_{n } with continuous nonlinearity

_{n }

Horban’ N. V., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 260–267

We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow.

### Asymptotic equivalence of impulsive systems

Misyats O. O., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 514–521

We present sufficient conditions for the asymptotic equivalence of a nonlinear impulsive system and a nonlinear system without pulses. We also consider the case of the asymptotic equivalence of a weakly nonlinear impulsive system and a linear system with pulses.

### Dissipativity of differential equations and the corresponding difference equations

Stanzhitskii A. N., Tkachuk A. M.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1249–1256

We establish conditions under which the existence of a bounded solution of a difference equation yields the existence of a bounded solution of the corresponding differential equation. We investigate the relationship between the dissipativities of differential and difference equations in terms of Lyapunov functions.

### Investigation of exponential dichotomy of linear Itô stochastic systems with random initial data by using quadratic forms

Krenevych A. P., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 543–553

We study conditions for the mean-square exponential dichotomy of linear Itô stochastic systems. We prove that a sufficient condition for exponential dichotomy is the existence of a quadratic form whose derivative along the solutions of a system is negative definite. The converse theorem is also proved.

### On the Relationship between Properties of Solutions of Difference Equations and the Corresponding Differential Equations

Stanzhitskii A. N., Tkachuk A. M.

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 989–996

We establish conditions under which the existence of a periodic solution of a differential equation is preserved if a solution of the corresponding difference equation possesses the same property. We prove the convergence of periodic solutions of a system of difference equations to a periodic solution of a system of differential equations. Analogous problems are considered for bounded solutions.

### On Invariant Tori of Itô Stochastic Systems

Samoilenko A. M., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 501-513

By using the Green–Samoilenko function, we establish conditions for the existence of invariant sets of Itô stochastic systems that are extensions of dynamical systems on a torus.

### Investigation of Exponential Dichotomy of Itô Stochastic Systems by Using Quadratic Forms

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1545-1555

For linear stochastic systems, we obtain sufficient conditions for mean-square exponential dichotomy in terms of Lyapunov functions that are quadratic forms.

### On Reduction Principle in Stability Theory for Systems with Random Perturbations

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1232-1240

For stochastic systems, we obtain an analog of the reduction principle that enables one to reduce the analysis of the stability of a system with random perturbations to the analysis of the stability of a deterministic system.

### Investigation of Invariant Sets of Itô Stochastic Systems with the Use of Lyapunov Functions

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 282-285

By using Lyapunov functions, we obtain conditions for the invariance and stochastic stability of invariant sets of Itô-type systems.

### Investigation of invariant sets with random perturbations with the use of the Lyapunov function

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 309–312

We consider invariant sets of the form V(*t, x*) *=* 0, where V(*t, x*) is the Lyapunov function of the corresponding deterministic system.

### Periodic solutions of systems of differential equations with random right-hand sides

Danilov V. Ya., Martynyuk D. I., Stanzhitskii A. N.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 223–227

We prove a theorem on the existence of periodic solutions of a system of differential equations with random right-hand sides and small parameter of the form *dx/dt=εX(t, x, ξ(t))* in a neighborhood of the equilibrium state of the averaged deterministic system *dx/dt*=ε*X* _{0}(*t*).

### On invariant tori of stochastic systems on a plane

Kopas' I. N., Stanzhitskii A. N.

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1125-1129

We study invariant tori of stochastic systems of the ltd type on a plane and present conditions for stability of such sets in probability.

### On the second Bogolyubov theorem for systems with random perturbations

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1104–1109

For systems of differential equations with random right-hand sides, we establish conditions for the existence of periodic solutions in the neighborhoods of equilibrium points of the averaged system.