The averaging method is applied to the investigation of the problem of existence of solutions of boundary-value problems for systems of differential and integrodifferential equations. It is shown that if the averaged boundary-value problem has a solution, then the original problem also has a solution. Note that, in this case, the system obtained as a result of averaging of a system of integrodifferential equations has the form of a simpler system of ordinary differential equations.

We prove a theorem on the existence and uniqueness of a mild solution to the Cauchy problem for a stochastic differential equation of neutral type in the weighted Hilbert space.

We prove the existence of an optimal control for systems of stochastic differential equations without solving the Bellman dynamic programming equation. Instead, we use direct methods for solving extremal problems.

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.

We establish results concerning the global existence, uniqueness, and controllability of mild solutions for a neutral functional stochastic differential equations with variable delay in a real separable Hilbert space. The results are obtained by imposing a so-called Caratheodory condition on the nonlinearities, which is weaker than the classical Lipschitz condition. Examples illustrating the applicability of the general theory are also provided.

For impulsive systems, in terms of Lyapunov functions, we obtain conditions for the existence of invariant sets and study the stability of these sets.

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.

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.

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 ).

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

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.

ABSTRACT. We establish results concerning the global existence, uniqueness, and controllability of mild solutions for a neutral functional stochastic differential equations with variable delay in a real separable Hilbert space. The results are obtained by imposing a so-called Caratheódory condition on the nonlinearities, which is weaker than the classical Lipschitz condition. Examples illustrating the applicability of the general theory are also provided.