Stanzhitskii A. N.
Application of the method of averaging to the problems of optimal control over functional-differential equations
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.
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.
Boichuk A. 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
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.
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.
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.
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.
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
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
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.
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.
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.
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.
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.
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.
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=εX 0(t).
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.
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.