## No 1 (2023)

**Year:**2023**Articles:**8**URL:**https://ered.pstu.ru/index.php/amcs/issue/view/355**DOI:**https://doi.org/10.15593/2499-9873/2023.1

Investigation of the stability of one difference equation with complex coefficients

#### Abstract

We study the stability of a linear autonomous difference equation with two (generally speaking, complex) coefficients. The starting point of the study is the Schur-Kohn theorem on the location of the roots of the characteristic equation with respect to the unit disk in the complex plane. To construct the domain of exponential stability in the parameter space, we use the D decompodition method, which consists in constructing curves (or surfaces) such that the number of roots of the characteristic equation outside the unit disk changes when passing through the curves; then the area is determined, which corresponds to the zero number of such roots; this is the area of stability. We implement this scheme for the above difference equation: geometric stability criteria are found and the domains of exponential stability in a four-dimensional space of coefficients are described, as well as their three-dimensional, two-dimensional and one-dimensional sections. The Lyapunov stability is studied separately, which is corresponded by the domain of exponential stability supplemented by a part of its boundary. To describe Lyapunov stability exactly we use a "multiplicity curve", which is a line such that all its points correspond to multiple roots of the characteristic equation. In addition, we find and construct a domain of absolute stability with respect to one of the parameters of the initial equation. For this domain, we formulate criteria of exponential stability and Lyapunov stability.The results obtained can be applied to the study of processes in physics, technology, economics, biology, which are modeled using discrete models in the form of difference equations.

**Applied Mathematics and Control Sciences**. 2023;(1):6-25

The turbulence models review for aerodynamic, thermal and structural problems in jet engine turbines.

#### Abstract

Analysis of the aerodynamic, thermal and aeroelasticity (including vibration) processes in gas turbine stage is principal for a design and creating turbines and jet engine integrally. In this article was the review showed for the basic problems, which are solved by engineers for the gas turbines design and take a numerical analysis of flow. The special aspects was showed which appearance in the vanes and blades channels for turbojets turbine. The fundamental structure of the boundary layer and vortices on the low and high pressure faces of blade was demonstrated also. A target of this article is a review of current classification of the turbulence models and a review of best practices which include this turbulence models usage for the scientific and engineering problems. Authors describe the basic classification and the small specification with the application domain for each turbulence model (for aircraft). The general equation was demonstrated for URANS with a description of the Spalart – Allmaras, k-ε, k-ω, Menter’s SST, SST Gamma-ReTheta turbulence models. For LES, DES and DNS was showed description with the features only.The article contains the publications analysis with the turbulence models for three basic problems: aerodynamics, cooling problem (heat exchange) and structural problems (aeroelasticity, vibration resistance, stress limits, fatigue resistance). The turbulence models with high frequency of usage was presented in the current review moreover review contains the reasons of the models usage, the models hypothesizes, the changes of models (if it exists) and the solution accuracy. The solution accuracy was analyzed between different models of turbulence and between softwares for CFD. All arguments and conclusions was based on the different publications summary which was used for the review creating, and on data (numerical and graphical results) which was included in the researched CFD-articles. A little bit of the comparative analysis between CFD-softwares was created with the software’s user guides, the public reports and articles which includes the numerical researches in this problem. All results of the current review was presented in the general conclusions in end of article.

**Applied Mathematics and Control Sciences**. 2023;(1):26-47

Application of neuro-fuzzy systems technology for control the density of asphalt concrete mixtures in the paving process

#### Abstract

The increase in the number of cars and trucks, and the intensity of their use, directly affects the asphalt-concrete road surfaces of roads. This is manifested in the reduction of overhaul terms of operation and large financial costs. Improving the quality and increasing the service life of asphalt concrete pavements of roads is a national economic problem. This problem is solved by improving the regulatory framework, improving the properties of road materials, optimizing and automating technological processes in road construction. Road roller density monitoring and control systems are based on Intelligent Compaction and Continuous Compaction Control technologies. In the Russian Federation, the quality of road surface construction largely depends on the results of the work of pavers, who ensure the acceptance, laying and compaction of mixtures. Many defects in the operation of road surfaces are eliminated by high-quality compaction. Automatic control and control of the sealing process for asphalt pavers have not been developed.The aim of the study is to build a density management system for asphalt pavers based on an intelligent automatic control system with feedback. Real-time prediction of the volume density of the layer, effective manual (operator) and automatic density control is organized to achieve the required quality indicators.The results of the development of a new system for intelligent control of the density asphalt concrete mixtures by pavers is considered. The automatic control system with feedback includes a continuous density control system designed to calculate the seal quality index based on the implementation of the neural network structure algorithm in real time, as well as a neuro-fuzzy PID regulator. The study considers a system with models of control objects of high order – the fourth and sixth orders. The structure of a neuro-fuzzy network of the ANFIS type is proposed. The generation of a fuzzy output system is performed on the basis of the lattice partition method. ANFIS training is performed by a hybrid method on an array of variables obtained as a result of modeling an automatic control system with an analog PID controller. Possible modes of use an intelligent compaction control system: continuous automatic control with manual control of the operating modes of compaction; automatic density control. Automatic measure of density and control compaction modes is aimed at improving the quality of asphalt concrete pavements of roads and improving the quality of asphalt pavements efficiency of technological processes road construction.

**Applied Mathematics and Control Sciences**. 2023;(1):48-60

Retrospective analysis of the impact of the COVID-19 pandemic on the socio-economic development of the region (on the example of the Sverdlovsk region).

#### Abstract

The COVID-19 pandemic has affected many aspects of people's lives and territories. It has a destabilizing effect on the economy due to both direct and indirect factors. In order to develop an effective policy to combat such emergencies, the problem of studying its consequences seems urgent.To form effective responses to the consequences of the pandemic, adequate scientific tools are needed to assess them. In many of the studies conducted, the impact of the pandemic was assessed without considering the previously existing socio-economic trends in the regions.The purpose of the article is to conduct a retrospective analysis of the consequences of economic instability caused by the COVID-19 pandemic on the socio-economic development of the region on the example of the Sverdlovsk region. The monthly values of twelve main socio-economic indicators from January 2016 to December 2021 were analyzed. The study is based on the use of regression models with dummy variables and seasonal autoregression for forecasting, which allow considering the seasonal dependence of socio-economic indicators.To determine the seasonality of individual indicators, a graphical method, autocorrelation correlograms and spectral analysis were used. When determining the uniformity between the predicted and actual values during the pandemic, the parametric Student's criterion and the nonparametric Wilcoxon criterion were applied, which allowed us to obtain a statistically significant result regardless of the normality of the compared aggregates.The retrospective analysis made it possible to establish a significant heterogeneous impact of the COVID-19 pandemic on most of the analyzed spheres of life in the Sverdlovsk region.

**Applied Mathematics and Control Sciences**. 2023;(1):61-71

The model of rational stimulation of project team members based on game theory tools

#### Abstract

This work is devoted to the development of recommendations for managing a project team at the conflict stage of its formation using game theory tools. In the course of the study, the main approaches to modeling the activities of the project team were described, as well as the features of the process of its formation were considered in detail. To model business activity at the conflict stage of team formation, a game-theoretic model of a hierarchical type was selected and adapted. The constructed model of team stimulation, as a kind of hierarchical game, allows us to consider the interaction of the management center both with the team as a whole and with individual participants. The adapted model is used in the task of finding a common team solution. Unlike the basic model, the adapted model does not use the type of each of the team members, which significantly reduces the amount of information needed for calculations, and also simplifies the calculations themselves. At the same time, information about the complexity of the task, the relationship between team members, as well as the priority of the task assigned to the participants was added to the basic model. In comparison with the empirical decision-making on incentives, the developed model allows the management center not only to achieve the tasks assigned to the project team with guarantee and on time, but also to use the company's monetary resources rationally.

**Applied Mathematics and Control Sciences**. 2023;(1):72-89

The management of the proceeds of the enterprise, taking into account the effectiveness of its activities based on the SFA model

#### Abstract

Actual, which requires an effective solution, is considered the task of managing the revenue of the enterprise. In the domestic literature, this topic is not given enough attention. In practice, most enterprises introduce the revenue management methodology based on foreign experience.Revenue is the main source of cash receipts specifically from the main activity of the enterprise, as well as one of the main factors affecting the functioning of the enterprise. As a result, the amount of revenue is extremely important for the company - it should be sufficient in order to ensure the repayment of all expenses of the company and the formation of the necessary volume of profit. However, the value is not the only important characteristic of revenue, the stability of revenue in time and the regularity of its receipt are no less important.The aim of this work is to develop a dynamic model of the management of the enterprise, which differs from the model known in the literature, taking into account the parameter of the efficiency of the enterprise.As a method to assess the effectiveness of the enterprise, the parametric method of Stochastic Frontier Analysis is used. As input and output, financial indicators are used.The model was checked for 9 russian enterprises (6 operating enterprises and3 bankrupt enterprises) for the period from 2013 to 2020, related to one all - Russian classifier of types of economic activity. Data collection is made using the SPARK system, which allows you to select enterprises for research on the status of an enterprise (bankrupt / current), by the size of the enterprise (large / medium / small / micro), etc.As an example, two enterprises were considered, of which one current, another bankrupt. The calculations on the built modified model showed the possibility of using the management of the proceedings of the enterprise with the desired pace of changes and with the parameter of the effectiveness of the activity.

**Applied Mathematics and Control Sciences**. 2023;(1):89-101

A method for constructing non-elementary Cobb – Douglas production functions

#### Abstract

The actual scientific problem in machine learning is the development of new interpretable mathematical models, as well as methods and programs for their construction. Many well-known regression models have good interpretative properties, for example, linear and power models (Cobb – Douglas production functions). Previously, the author developed non-elementary linear regressions, the problem of constructing which was reduced to the mixed integer 0–1 linear programming problem.Based on non-elementary linear regressions, in this paper, for the first time, non-elementary Cobb-Douglas production functions are proposed, which include not only explanatory variables in degrees, but also all their possible pair combinations, transformed using binary operations min and max. The proposed models are linearized, which makes it possible to apply for their construction the mixed integer 0–1 linear programming problem formulated in the same way as for non-elementary linear regressions. As a result of its solution, the optimal structure model is automatically determined. The advantage of such a formulation is that the solution of the problem can be obtained faster than using enumeration procedures, and also that the signs of the estimates of the constructed model are guaranteed to be consistent with the meaningful meaning of the factors. At the same time, it is possible to control the requirements for the structure of the model using linear constraints on binary variables. In particular, the problem can be used to select the optimal structures for traditional elementary Cobb – Douglas production functions.The problem of modeling the gross regional product of the Tomsk region is solved. The following variables were chosen as explanatory variables: average per capita cash income of the population, investments in fixed capital, costs of innovative activities of organizations, average annual number of employees, cost of fixed assets, internal costs for research and development. The LPSolve package was chosen as the solver of the mixed integer 0–1 linear programming problem. As a result of solving this problem, the optimal structure of the non-elementary Cobb – Douglas production function was chosen, which contains all six explanatory variables in three regressors. The coefficient of determination for the constructed model turned out to be 0.997. All regression coefficients turned out to be significant according to Student's t-test, and their signs satisfy the meaningful meaning of the factors. An interpretation for the constructed model is given.

**Applied Mathematics and Control Sciences**. 2023;(1):102-115

Valuation of used machinery based on the new model of its degradation

#### Abstract

We propose a new model of equipment degradation. In it, the machine is subjected to random latent failures, the danger of which depends on the equipment condition and after each failure the intensity of the equipment's benefits decreases by a random amount. Equipment that brings negative benefits is subject to decommissioning. The model parameters are found based on known information about the average value and the coefficient of variation of the equipment lifetime. Market value of equipment is determined by discounting the flow of benefits from its future use. This allows us to find the dependence of the equipment’s market value on the benefits it brings. Assessing the market value of new equipment is usually not difficult, but it is much more difficult to do this for used equipment items. Appraisers are usually unable to estimate the value of the work performed by equipment, and when valuing a used equipment, they have to rely on its age. To do this, the market value of a similar new equipment is usually reduced by a depreciation factor or multiplied by Percent Good Factor (PGF, relative value), depending on the age of equipment being valued. However, equipment of the same age may be in different conditions and, therefore, have a different market value. Therefore, such PGFs, in fact, relate to the average equipment that has survived to the appropriate age. Appraisers determine them by formulas or tables that are usually not supported by proper justifications. The proposed model makes it possible to build the dependence of the average PGF on age and calculate the market value of the work performed by machines, even if such works are not traded on the market. It turns out that it is possible to take into account the influence of the utilization cost of the machine and inflation in the model. The results of experimental calculations performed using the model (with calibration parameters selected appropriately) are in good agreement with the market prices of some types of construction equipment.

**Applied Mathematics and Control Sciences**. 2023;(1):116-132