Vol 23, No 123 (2018)
Articles
ON STABILIZATION OF DIFFERENTIAL SYSTEMS WITH HYBRID FEEDBACK CONTROL
Abstract
In this paper two-dimensional systems of differential equations are considered together with their stabilization by a hybrid feedback control. A stabilizing hybrid control for an arbitrary controlled system that belongs to a certain category within two-dimensional systems is constructed as a result of this study and some stabilization proprieties of the system with the obtained hybrid control are presented.
Russian Universities Reports. Mathematics. 2018;23(123):331-352
331-352
353-360
INTEGRAL MODEL OF DEVELOPING SYSTEM WITHOUT PREHISTORY
Abstract
The paper addresses integral model of the developing system, in which the moment of its origin coincides with the beginning of the modeling, so there is no prehistory and for t = 0 all the age groups of the elements are empty.
Russian Universities Reports. Mathematics. 2018;23(123):361-367
361-367
ON STABILITY CONTROL OF A PARABOLIC SYSTEMS WITH DISTRIBUTED PARAMETERS ON THE GRAPH
Abstract
The work is an attempt to demonstrate the concept of sustainability in the undisturbed state Lyapunov differential system for equations with partial derivatives, and show the ability to use a famous classical results in the studied case.
Russian Universities Reports. Mathematics. 2018;23(123):368-376
368-376
ON GENERALIZATIONS AND APPLICATIONS OF VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS
Abstract
There are considered some classes of functions to which variational principles of nonlinear are applicable. In particular, it is shown that the Bishop-Phelps variational principle is applicable to some unbounded below functions. The properties of locally Lipschitzian mappings are investigated. Conditions for a mapping that is pseudo-Lipschitzian at every point of its graph to be Lipschitzian are derived.
Russian Universities Reports. Mathematics. 2018;23(123):377-385
377-385
ON STABILITY OF DIFFERENCE EQUATIONS IN PARTIALLY ORDERED SPACES
Abstract
We consider implicit difference equations in partially ordered spaces. We define the notion of a stable equilibrium point. The conditions of the stability is obtained. The study is based on the theory of partially ordered mappings.
Russian Universities Reports. Mathematics. 2018;23(123):386-394
386-394
IDENTIFICATION OF HAMMERSTEIN SYSTEMS OF FRACTIONAL ORDER WITH A POLYNOMIAL NONLINEARITY IN THE PRESENCE OF A FRACTIONAL WHITE NOISE
Abstract
An identification algorithm is proposed nonlinear dynamical systems of fractional order of the Hammerstein class. Designed algorithm allows you to get strongly consistent parameter estimates in the presence of observation noise in the form of fractional white noise. The results of numerical experiments showed high efficiency of the proposed identification algorithm in comparison with the least squares method (LS).
Russian Universities Reports. Mathematics. 2018;23(123):395-401
395-401
EEFFECTIVE CRITERIA OF EXPONENTIAL STABILITY OF AUTONOMOUS DIFFERENCE EQUATIONS
Abstract
We obtain stability criteria for several classes of linear autonomous difference equations. The criteria are expressed in explisit analytic form, as well as in the form of belonging values of a vector function of the equation parameters to a domain in three-dimensional space.
Russian Universities Reports. Mathematics. 2018;23(123):402-414
402-414
DECISION-MAKING IN A HYBRID TWO-STEP PROBLEM OF DYNAMIC CONTROL
Abstract
The equations of motion of the controlled system in the two-step problem under consideration at a fixed time interval contain the controls of either one player or two players. In the first step (stage) of the controlled process (from the initial moment to a certain predetermined moment), only the first player controls the system, which solves the problem of optimal control with a given terminal functional. In the second step (stage) of the process, the first player decides whether the second player will participate in the control process for the remainder of the time, or not. It is assumed that for participation the second player must pay the first side payment in a fixed amount. If «yes», then a non-antagonistic positional differential game is played out, in which the Nash equilibrium is taken as the solution. In addition, players can use «abnormal» behaviors, which can allow players to increase their winnings. If « no », then until the end of the process continues to solve the problem optimal control.
Russian Universities Reports. Mathematics. 2018;23(123):415-423
415-423
424-430
HURWITZ MATRIX, LYAPUNOV AND DIRICHLET ON THE SUSTAINABILITY OF LYAPUNOV’S
Abstract
The concepts of Hurwitz, Lyapunov and Dirichlet matrices are introduced for the convenience of the stability of linear systems with constant coefficients. They allow us to describe all the cases of interest in the stability theory of linear systems with constant coefficients. A similar classification is proposed for systems of linear differential equations with periodic coefficients. Monodromy matrices of such systems can be either Hurwitz matrices or Lyapunov matrices or Dirichlet matrices (in the discrete sense) in a stable case. The new material relates to systems with variable coefficients.
Russian Universities Reports. Mathematics. 2018;23(123):431-436
431-436
437-440
ATTAINABLE VALUES OF ON-TARGET FUNCTIONALS FOR A FUNCTIONAL DIFFERENTIAL SYSTEM WITH IMPULSES
Abstract
For a linear functional differential system with aftereffect and impulses, a description of the attainability set is given. The attainability is considered in the term of a given system of on-target functionals in the case of polyhedral constraints with respect to control and impulses.
Russian Universities Reports. Mathematics. 2018;23(123):441-447
441-447
SOLUTION OF THE SYSTEM OF NAVIER-STOKES EQUATIONS LINEARIZED WITH RESPECT TO THE VELOCITY WITH REGARD OF A POWER-LOW DEPENDENCE OF VISCOSITY, THERMAL CONDUCTIVITY AND THE GASEOUS MEDIUM DENSITY ON THE TEMPERATURE
Abstract
We received the solution of the system of Navier-Stokes equations linearized with respect to the velocity in the spheroidal coordinate system with regard of a power-law dependence of viscosity, thermal conductivity and the gaseous medium density on the temperature by means of generalized power series.
Russian Universities Reports. Mathematics. 2018;23(123):448-455
448-455
ON THE STABILITY OF A POPULATION DYNAMICS MODEL WITH DELAY
Abstract
We consider a model of the dynamics of an isolated population whose individuals pass through the three stages of evolution. We use a nonlinear autonomous differential equation with concentrated and distributed delay for description of the model. Effective sufficient conditions for the asymptotic stability of the nontrivial equilibrium point are obtained.
Russian Universities Reports. Mathematics. 2018;23(123):456-465
456-465
ON EXACT SOLUTION OF OPTIMIZATION TASK GENERATED BY THE LAPLACE EQUATION
Abstract
A one-parameter family of finite-dimensional spaces consisting of special two-dimensional splines of Lagrangian type is defined (the parameter N is related to the dimension of the space). The Laplace equation generates in each such space the problem of minimizing the residual functional. The existence and uniqueness of optimal splines are proved. For their coefficients and residuals, exact formulas are obtained. It is shown that with increasing N ; the minimum of the residual functional is ON -5 ; and the special sequence consisting of optimal splines is fundamental.
Russian Universities Reports. Mathematics. 2018;23(123):466-472
466-472
DISCRETE SYSTEMS AND NEIGHBORING STRUCTURES
Abstract
In the article, neighborhood structures (digraphs of a special type) are defined and their relationship with discrete control systems is discussed. The archetypes of the neighborhood structures and the control systems corresponding to these archetypes are listed.
Russian Universities Reports. Mathematics. 2018;23(123):473-478
473-478
NEIGHBORHOOD METASYSTEMS ON DIGRAPHS
Abstract
The article discusses neighborhood systems on oriented graphs. The concepts of the neighborhood metasystem and metastructural identification are introduced. The questions of identification of control systems related to these concepts are considered.
Russian Universities Reports. Mathematics. 2018;23(123):479-487
479-487
STABILITY OF ONE-PARAMETER SYSTEMS OF LINEAR AUTONOMOUS DIFFERENTIAL EQUATIONS WITH BOUNDED DELAY
Abstract
We consider a system of linear autonomous differential equations with bounded delay in the case when its characteristic function depends linearly on one scalar parameter. The application of the D-subdivision method to the problem of constructing the stability region for this equation was developed.
Russian Universities Reports. Mathematics. 2018;23(123):488-502
488-502
KOLMOGOROV MATRIX, AND A CONTINUOUS MARKOV CHAIN WITH A FINITE NUMBER OF STATES
Abstract
In terms of ergodicity of averaged systems with constant coefficients (and Kolmogorov matrix), the signs of ergodicity of continuous Markov chains with a finite number of States with periodic and almost periodic coefficients are indicated.
Russian Universities Reports. Mathematics. 2018;23(123):503-509
503-509
510-516
ON THE CONVERGENCE AND RATE OF THE CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD FOR APPROXIMATE SOLVING A PARABOLIC EQUATION WITH WEIGHT INTEGRAL CONDITION
Abstract
In the Hilbert space the abstract linear parabolic equation with nonlocal weight integral condition for the solution is resolved approximately by projectiondifference method using time-implicit Euler’s method. Approximation of the problem by spatial variables is oriented on the finite element method. Errors estimations of approximate solutions, convergence of approximate solution to exact one and orders of rate of convergence are established.
Russian Universities Reports. Mathematics. 2018;23(123):517-523
517-523
524-530
ON OBTAINING EFFECTIVE CONDITIONS FOR THE SOLVABILITY OF A SYSTEM OF FUNCTIONAL-DIFFERENTIAL EQUATIONS DETERMINATED ON A GEOMETRIC GRAPH
Abstract
This paper is devoted to consideration of a boundary value problem for a system of functional differential equations determined on a geometric graph. The boundary conditions of the problem are determined by the conditions for the connection of the edges of the graph. There is an algorithm that reduces the system of equations on the graph to the system determined on the set Θ of disjoint segments of the real axis. The Azbelev’s W -method is applied to the system determined on the set Θ ; what makes it possible to obtain effective conditions for the unique solvability of the original system. An example is given.
Russian Universities Reports. Mathematics. 2018;23(123):531-538
531-538
539-546
ON SETS OF METRIC REGULARITY OF MAPPINGS IN SPACES WITH VECTOR-VALUED METRIC
Abstract
Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.
Russian Universities Reports. Mathematics. 2018;23(123):547-554
547-554
ON THE NUMERICAL METHOD OF CONSTRUCTION OF UNSTABLE SOLUTIONS OF DYNAMICAL SYSTEMS WITH QUADRATIC NONLINEARITIES
Abstract
In this paper, the author considers the modification of the method of power series for the numerical construction of unstable solutions of systems of ordinary differential equations of chaotic type with quadratic nonlinearities in general form. A region of convergence of series is found and an algorithm for constructing approximate solutions is proposed.
Russian Universities Reports. Mathematics. 2018;23(123):555-565
555-565
ABOUT EXISTENCE AND ESTIMATES OF SOLUTIONS OF THE IMPLICIT DIFFERENTIAL EQUATION WITH AUTOADJUSTABLE DEVIATION ARGUMENT
Abstract
Conditions of a solubility and assessment of solutions of an implicit differential equation with autoadjustable (that is depending on required function) argument deviation are received. Results about the covering displays of partially ordered spaces are used.
Russian Universities Reports. Mathematics. 2018;23(123):566-574
566-574