Vol 23, No 124 (2018)

In this paper, we consider the system of equations that describes air motion in a chimney. By introducing a method based on the «second approximation», we prove a theorem on the existence and uniqueness of the global solution.

Asymptotic behavior of the value function is studied in an infinite horizon optimal control problem with an unlimited integrand index discounted in the objective functional. Optimal control problems of such type are related to analysis of trends of trajectories in models of economic growth. Stability properties of the value function are expressed in the infinitesimal form. Such representation implies that the value function coincides with the generalized minimax solution of the Hamilton-Jacobi equation. It is shown that that the boundary condition for the value function is substituted by the property of the sublinear asymptotic behavior. An example is given to illustrate construction of the value function as the generalized minimax solution in economic growth models.

In this paper we review the results of the authors related to the study of the stability property of solutions for nonlinear systems of differential equations, on the right-hand side of which there are terms containing products of discontinuous functions and distributions. The solutions of such systems are formalized by the closure of the set of smooth solutions in the space of functions of bounded variation. For such systems, sufficient conditions are obtained for the asymptotic stability of unperturbed solutions.

We study locally injective covering mappings between metric spaces. We show that under natural assumptions these mappings have continuation property.

In this paper we consider some problems in combinatorial analysis related to placements without neighbours on graphs, namely, we find numbers and probabilities of such placements for simplest graphs (segment, two segments, cycle), and also (which is more difficult) we solve the same problems for a cycle up to rotations.

On the basis of previous works of authors new signs of regularity and stability of vector-matrix differential equations with a variable main part are specified.

We investigate the models of dynamics of the harvested population, given by the control systems with impulse influences depending on random parameters. We assume that in the absence of harvesting population development is described by system of the differential equations x =f x and in time moments kd , d >0 from population are taken some random share of a resource ω k = ω 1 k ,…, ω n k ∈ Ω , k =1, 2, …, that leads to sharp (impulse) reduction of its quantity. Considered resource x ∈ R+ n is non-uniform, that is or it consists of separate kinds x 1 ,…, x n , or it is divided on n age groups. In particular, it is possible to assume that we make harvesting of n various kinds of fishes between which there are competition relations for food or dwelling places.We describe the probability model of a competition of two kinds for which we receive the estimations of average time benefit from the resource extraction, fulfilled with probability one.

The paper presents a recurrent algorithm for estimating the parameters of multidimensional discrete linear dynamical systems of different orders with input errors, described by white noise. It is proved that the obtained estimates using stochastic gradient algorithm for minimization of quadratic forms are highly consistent

We consider an abstract hybrid system of two equations with two unknowns: a vector function x that is absolutely continuous on each finite interval 0, T , T>0, and a sequence of numerical vectors y. The study uses the W -method N.V. Azbelev. As a model, a system containing a functional differential equation with respect to x is used, and a difference equation with respect to y. Solutions spaces are studied. For a hybrid system, the Bohl-Perron theorem on asymptotic stability and the converse theorem are obtained.

A survey of the results obtained in the theory of optimization of distributed systems by the method of Volterra functional-operator equations is given. Topics are considered: the conditions for preserving the global solvability of controllable initial-boundary value problems, optimality conditions, singular controlled systems in the sense of J.L. Lions, singular optimal controls, numerical optimization methods substantiation and others.

The risk model of multidimensional stochastic systems is described. It is based on the hypothesis that the risk is characterized by probabilistic properties of components of multidimensional stochastic system which are used as risk factors. The case of the Gaussian stochastic systems is investigated. The model of risk monitoring allows to estimate the current risk of system and the contribution of all its components. Models of risk management are optimizing tasks. As the target functions the conditional minimum of risk and achievement of the given level by it can be used at minimum changes of probabilistic characteristics of the system.

A combined (jointing analytical methods and computational procedures) approach to the construction of solutions in a class of boundary-value problems for a Hamiltonian-type equation is proposed. In the class of problems under consideration, the minimax (generalized) solution coincides with the Euclidean distance to the boundary set. The properties of this function are studied depending on the geometry of the boundary set and the differential properties of its boundary. Methods are developed for detecting pseudo-vertices of a boundary set and for constructing singular solution sets with their help. The methods are based on the properties of local diffeomorphisms and use partial one-sided limits. The effectiveness of the research approaches developed is illustrated by the example of solving a planar timecontrol problem for the case of a nonconvex target set with boundary of variable smoothness.

The Banach algebra of complex operators that are used in the study of linear differential equations with constant bounded operator coefficients in a Banach space is consiuder.

In this work we give explicit expressions of covariant symbols in polynomial quantization on para-Hermitian symmetric spaces.

We give strict justification for derivative formulas of functionals in problems approximating free time optimal control problems in the frame of sliding nodes method and control parametrization technique. As example we present results of numerical solution for landing on the Moon problem.

We propose procedure to solve the optimal stabilization problem for linear periodic systems of differential equations. Stabilizing controls, formed as a feedback, are defined by the system states at the fixed instants of time. Equivalent discrete-time linear periodic problem of optimal stabilization is considered. We propose a special procedure for the solution of discrete periodic Riccati equation. We investigate the relation between continuous-time and discrete-time periodic optimal stabilization problems. The proposed method is used for stabilization of mechanical systems.