Vol 23, No 124 (2018)
Articles
STUDY OF THE EXISTENCE OF THE GLOBAL SOLUTION TO THE SYSTEM OF EQUATIONS OF THE VERTICAL MOTION OF THE AIR IN A CHIMNEY
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):583-594
583-594
NEW SUFFICIENT CONDITIONS IN THE GENERALIZED SPECTRUM APPROACH TO DEAL WITH SPECTRAL POLLUTION
Abstract
In this work, we propose new sufficient conditions to solve the spectralpollution problem by using the generalized spectrum method. We give the theoretical foundation of the generalized spectral approach, as well as illustrate its effectivenessby numerical results.
Russian Universities Reports. Mathematics. 2018;23(124):595-604
595-604
ASYMPTOTICS OF VALUE FUNCTION IN MODELS OF ECONOMIC GROWTH
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):605-616
605-616
THE STRONG-NORM CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD OF SOLUTION OF A PARABOLIC EQUATION WITH THE PERIODIC CONDITION ON THE SOLUTION
Abstract
A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.
Russian Universities Reports. Mathematics. 2018;23(124):617-623
617-623
ON THE STABILITY OF SOLUTIONS OF NONLINEAR SYSTEMS WITH IMPULSE STRUCTURE
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):624-636
624-636
ON EQUATIONS GENERATED BY NONLINEAR NILPOTENT MAPPINGS
Abstract
A generalization of a nilpotent linear operator concept is proposed for nonlinear mapping acting from R2 to R2 . The properties of nonlinear nilpotent mappings are investigated. Criterions of nilpotence for differentiable and polynomial mappings are obtained.
Russian Universities Reports. Mathematics. 2018;23(124):637-642
637-642
643-647
DECLUSTERZATION OF NEIGHBORHOOD STRUCTURES
Abstract
Neighborhood structures (digraphs of a special kind) can have vertex or relational sets of equipping variables. Vertex variables correspond to the vertices of the structure, while the relational ones correspond to the arcs. The article describes an algorithm for the canonical transformation (declusterization) of the relational structures into the vertex ones. This transformation establishes a connection between two types of control metasystems on neighborhood structures.
Russian Universities Reports. Mathematics. 2018;23(124):648-654
648-654
PLACEMENTS WITHOUT NEIGHBOURS
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):655-665
655-665
СLOSEDNESS OF THE TECHNOLOGY SET IN DYNAMICAL PRODUCTION MODELS
Abstract
The paper is a study of some properties of the technology set in dynamical production models. The models under consideration are treated as a linear dynamical control systems, where the input is the non-productive consumption function, which takes values from a convex closed finitely generated cone.
Russian Universities Reports. Mathematics. 2018;23(124):666-673
666-673
674-678
679-684
ABOUT ONE STOCHASTIC HARVESTING MODEL OF A RENEWED RESOURSE
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):685-695
685-695
ON OSCILLATION OF SOLUTIONS FOR SOME NONLINEAR EQUATIONS OF POPULATION DYNAMICS
Abstract
Several nonlinear equations being models of population dynamics and hematopoiesis are considered in this paper. For these equations conditions of oscillation for solutions about nontrivial equilibrium position are obtained
Russian Universities Reports. Mathematics. 2018;23(124):696-706
696-706
RECURSIVE ALGORITHM FOR ESTIMATING THE PARAMETERS OF MULTIDIMENSIONAL DISCRETE LINEAR DYNAMIC SYSTEMS OF DIFFERENT ORDERS WITH ERRORS ON THE INPUT
Abstract
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
Russian Universities Reports. Mathematics. 2018;23(124):707-716
707-716
ON THE EXISTENCE OF A NON-ANTICIPATING SELECTION OF NON-ANTICIPATING MULTIVALUED MAPPING
Abstract
The existence of a non-anticipating selection of a non-anticipating multifunction is considered. For the case ordinary used in applications, it is shown that every non-anticipating multifunction with non-empty compact values has a nonanticipating selection.
Russian Universities Reports. Mathematics. 2018;23(124):717-725
717-725
THE THEOREM OF BOHL-PERRON ON THE ASIMPTOTIC STABILITY OF HYBRID SYSTEMS AND INVERSE THEOREM
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):726-737
726-737
APPROXIMATION OF HYPERBOLIC DIFFERENTIAL INCLUSIONS OF FRACTIONAL ORDER WITH IMPULSES
Abstract
In this paper there are considered hyperbolic differential inclusions of fractional order with impulses. Here we represent the concept of approximate solution ( δ -solution) for a hyperbolic differential inclusion of fractional order with impulses. The asymptotic properties of solutions sets to approximating differential inclusions of fractional order with external disturbance are derived.
Russian Universities Reports. Mathematics. 2018;23(124):738-744
738-744
VOLTERRA FUNCTIONAL-OPERATOR EQUATIONS AND DISTRIBUTED OPTIMIZATION PROBLEMS
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):745-756
745-756
WHY REGULARIZATION OF LAGRANGE PRINCIPLE AND PONTRYAGIN MAXIMUM PRINCIPLE IS NEEDED AND WHAT IT GIVES
Abstract
We consider the regularization of the classical Lagrange principle and the Pontryagin maximum principle in convex problems of mathematical programming and optimal control. On example of the “simplest” problems of constrained infinitedimensional optimization, two main questions are discussed: why is regularization of the classical optimality conditions necessary and what does it give?
Russian Universities Reports. Mathematics. 2018;23(124):757-775
757-775
MODELS OF MONITORING AND MANAGEMENT OF RISK IN GAUSSIAN STOCHASTIC SYSTEMS
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):776-783
776-783
ASYMPTOTIC SOLUTION OF FIRST-ORDER EQUATION WITH SMALL PARAMETER UNDER THE DERIVATIVE WITH PERTURBED OPERATOR
Abstract
The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.
Russian Universities Reports. Mathematics. 2018;23(124):784-796
784-796
EUCLIDEAN DISTANCE TO A CLOSED SET AS A MINIMAX SOLUTION OF THE DIRICHLET PROBLEM FOR THE HAMILTON-JACOBI EQUATION
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):797-804
797-804
SOME PROPERTIES OF THE GENERALIZED SOLUTIONS OF AN INITIAL VALUE PROBLEM FOR FUNCTIONAL-DIFFERENTIAL INCLUSION WITH MULTIPLE-VALUED IMPULSES
Abstract
Deviation estimates in space of piecewise continuous functions of a set of the generalized decisions from beforehand given function are received. The continuous dependence of the generalized decisions on starting conditions is established.
Russian Universities Reports. Mathematics. 2018;23(124):805-812
805-812
813-823
ON SOME PROPERTIES OF QUASI CONVEX FUNCTIONS AND SETS
Abstract
The connection between quasi convexity and proximal smoothness (also known as low C 2 property) of functions is verified. For compact sets, it is proved that the properties of quasi convexity and proximal smoothness are equivalent. The Bouligand cones of tangent directions for the sets that are defined by convex functions are constructed.
Russian Universities Reports. Mathematics. 2018;23(124):824-837
824-837
838-845
MAXIMAL LINKED SYSTEMS AND ULTRAFILTERS OF WIDELY UNDERSTOOD MEASURABLE SPACES
Abstract
Two types of set families (ultrafilters or maximal filters and maximal linked systems) for widely understood measurable space are considered. The resulting sets of ultrafilters and maximal linked systems are equipped with the pair of comparable topologies (within the meaning of «Wallman» and «Stone»). As a result, two bitopological spaces are realized; one of them turns out a subspace of another. More precisely, ultrafilters are maximal linked systems and the totality of the latter forms a cumulative bitopological space. With employment of topological constructions some characteristic properties of ultrafilters and (in smaller power) maximal linked systems are obtained (the question is necessary and sufficient conditions of maximality of filters and linked systems).
Russian Universities Reports. Mathematics. 2018;23(124):846-860
846-860
ON DIFFERENTIATION OF FUNCTIONALS OF APPROXIMATING PROBLEMS IN THE FRAME OF SOLUTION OF FREE TIME OPTIMAL CONTROL PROBLEMS BY THE SLIDING NODES METHOD
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):861-876
861-876
THE VALUE FUNCTION OF A DIFFERENTIAL GAME WITH SIMPLE MOTIONS AND AN INTEGRO-TERMINAL COST
Abstract
An antagonistic positional differential game of two persons is considered. The dynamics of the system is described by a differential equation with simple motions, and the payoff functional is integro-terminal. For the case when the terminal function and the Hamiltonian are piecewise linear, and the dimension of the state space is two, a finite algorithm for the exact construction of the value function is proposed.
Russian Universities Reports. Mathematics. 2018;23(124):877-890
877-890
DISCRETE PROCEDURE OF OPTIMAL STABILIZATION FOR PERIODIC LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
Abstract
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.
Russian Universities Reports. Mathematics. 2018;23(124):891-906
891-906