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We study the asymptotic behavior at infinity of the solutions of stochastic functional-differential equations by the method of asymptotic equivalence according to which a system of ordinary differential equations is constructed on the basis of the original stochastic system so that each solution of the stochastic system can be associated with a solution of the constructed deterministic system such that the difference between these solutions tends to zero as t → ∞ both in the mean square and with probability 1. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem for systems with infinite memory whose models are described by functional-differential equations. We prove the theorem on existence, uniqueness, and continuity of solutions for a system of functional-differential equations in which the delay interval is infinitely large. Sufficient conditions for the existence of optimal controls in the optimal control problems for systems with infinite memory are obtained in terms of the right-hand sides of the equations of motion and the function of performance criterion. [ABSTRACT FROM AUTHOR]

We consider the problem of optimal control formed by a parabolic nonlinear equation with rapidly oscillating coefficients, an additive control function, and a coercive cost functional. It is proved that the optimal value of the perturbed problem is close to the optimal value for the corresponding problem with averaged coefficients. [ABSTRACT FROM AUTHOR]

In this paper we study functional-differential equations on the semi-axis, which are non-linear with respect to the phase variables and linear with respect to the control. Sufficient conditions for existence of optimal control in terms of the right-hand side and the quality criterion are obtained. Relation between the solutions of the problems on infinite and finite intervals is studied and results that about these connections are proven. [ABSTRACT FROM AUTHOR]

For systems of ordinary differential equations, we obtain some sufficient conditions of existence of an optimal control in terms of their right-hand sides and the cost functional with the use of compactness methods. In the theorems proven, either the time interval is finite and a solution belongs to some domain or the time interval coincides with the semiaxis. [ABSTRACT FROM AUTHOR]

In this paper we study the connection between the existence of bounded (on real axis) solutions of differential equations and the corresponding difference equations. We obtain the conditions, under which the existence of bounded solutions of differential equations implies the existence of bounded solution of difference equation andvice versa. [ABSTRACT FROM AUTHOR]

In the paper, we consider problem with impulse actions for system of nonlinear ordinary differential equations (ODEs) in the interval. For solving this problem we use a modification of parameterization method by Dulat Dzhumabaev. In the modification of the method we introduce the parameters as the values of the unknown function at the middle of the subintervals of the partition of the considered interval. The problem with impulse actions transfers to an equivalent problem system of nonlinear ODEs with parameters. Conditions for an existence of solution to the equivalent problem are obtained. Existence theorem for solutions this problem is established by one generalizes a theorem of Hadamard. We also constructed an algorithm for finding of solution to this problem. Finally, we found conditions for solvability to the problem with impulse actions for system of nonlinear ODEs in terms of special matrix composed by the initial data. This method can be applied to various types of nonlinear problems with impulse actions for ODEs. [ABSTRACT FROM AUTHOR]