## No 3 (2021)

ANALOGUE OF THE DISCRETE MAXIMUM PRINCIPLE AND THE NECESSARY OPTIMALITY CONDITION OF SINGULAR CONTROLS IN ONE TWO-PARAMETRIC DISCRETE OPTIMAL CONTROL PROBLEM

#### Abstract

A two-stage (stepwise) optimal control problem for linear two-parameter systems with distributed control functions is considered. The aim of the work is to establish the necessary optimality condition under the assumption that the convexity of the set of admissible controls is satisfied and the connection condition is nonlinear. Using increments of the quality functional in the form of two-dimensional linear inhomogeneous systems of difference equations, a formula is obtained that allows one to obtain both a discrete analogue of the Pontryagin maximum principle and to study the case of its degeneration. A theorem is formulated that is an analogue of the discrete Pontryagin maximum principle for the problem under consideration. In the case of special controls, the discrete maximum principle degenerates and, therefore, becomes ineffective, including in the verification sense. Therefore, it is necessary to have new necessary conditions for optimality. A special, in the sense of the Pontryagin maximum principle, case of a discrete maximum condition, under which admissible controls are considered special, is studied. A necessary condition for optimality of singular controls is established.

**Applied Mathematics and Control Sciences**. 2021;(3):7-34

ON THE CONTROLLABILITY OF LINEAR EVOLUTIONARY FUNCTIONAL-DIFFERENTIAL SYSTEMS

#### Abstract

New theorem on the controllability of linear functional-differential system of evolutionary type and algorithm for practical verification of controllability are obtained, which can be applied even in the case when the coefficients of system are not continuously differentiable over the time interval under consideration. Special cases of this system are nonstationary systems with distributed and concentrated delay, integro-differential systems with a Voltaire integral and ordinary differential systems. The main results obtained are formulated in the form of 12 theorems and 3 corollaries. On their basis an algorithm for practical verification of the controllability of the system under consideration using a computer is built. Examples illustrating the operability of the obtained theorems and the algorithm are given. The algorithm is implemented in the Maple 17 package for examples of second-order differential systems with delay.

**Applied Mathematics and Control Sciences**. 2021;(3):35-59

URBAN HEATING SYSTEM VIRTUAL STAND UNDER DATA INCOMPLETE CONDITIONS

#### Abstract

The problem of heat load optimal regulation in a boiler house considered to provide heat carrier to end users, taking into account the permissible temperature deviations according to the temperature schedule, is relevant for a heat supplying organization. Regulation in a boiler house that feeds a heating network with unknown thermophysical characteristics is a problem that belongs to the class of poorly formalized problems. This is due, not least, to the specific features of heating system exploitation. The problem relatively easily solved at different ambient temperatures and at different times in the data completeness case on the above specific features. Unfortunately, collecting such information is almost impossible. We have to pose and solve the problem under conditions of uncertainty. Therefore, decided to develop an urban heating system virtual stand under data incomplete conditions, which will allow creating a heating system models with various conditions of heat loss, with heat consumers of different categories and different heat energy consumption. It will allow to simulate a changes made without making factual changes in an urban heating system structure. The object of the study is a closed urban heating system with a complex topology, and the subject is calculation models of a heating system, which allow determining the temperature loss of the coolant in the sections of the heating system based on data processing from general house metering devices (GHMD). The aim of the work is to improve the efficiency of an urban heating system management based on data obtained using a virtual stand. A conceptual model developed and the main functionality of the virtual stand implemented. Based on a number of numerical experiments carried out, an algorithm formulated and implemented within the framework of a virtual stand, which makes it possible to determine total thermal resistances values at each section of the heating system from GHMD data. The stand tested on the urban heating system of the Lysva city. The deviation of the temperature values calculated within the framework of the model implemented in the virtual stand from the temperatures obtained from the GHMD did not exceed 5 %. Consequently, this virtual stand can used for more efficient urban heating system management, taking into account the calculation error.

**Applied Mathematics and Control Sciences**. 2021;(3):63-78

POWER CABLE INSULATION PRODUCTION LINE CONTROL

#### Abstract

The paper considers the problem of control the cable insulation continuous vulcanization line. The axisymmetric mathematical model of the power cable rubber insulation vulcanization technological process has been developed. The model describes the heat transfer process inside the vulcanization pipe and the vulcanization process kinetics inside the insulation. The mathematical model is based on the energy conservation equation and it is presented in the form of a differential equations system closed by boundary conditions The mathematical model makes it possible to estimate the influence of various technological and design parameters on the temperature distribution along the radius and length of the cable. Based on the temperature distribution curves, the cable insulation vulcanization degree is calculated The study of the raw rubber compositions vulcanization properties was carried out by rotary rheometer, where the samples of the material were subjected to an oscillating load at different temperatures. As a result of the experiment, the curves family of the torque dependence on the curing time at different temperatures was obtained. To describe the vulcanization process kinetics, a parametric expression is used based on the experimentally obtained torque dependences. The obtained temperature dependence of the parametric expression coefficients makes it possible to describe the vulcanization process taking into account the uneven temperature distribution along the cable radius and length. The developed model makes it possible to calculate the speed at which the vulcanization degree reaches a given value. Algorithms for calculating and correcting line speed and extruder flow rate are proposed on the basis of a mathematical model. It allows us to determine the speed and flow rate of the extruder for various: cable designs, insulation materials, abnormal operating conditions of the production line. The results can be used at enterprises engaged in the production of cable and wire products with cross linked insulation. In cases where it is required to quickly calculate a new technological mode, make changes to the cable design, change the insulation material, and also take into account possible pressure drops inside the vulcanization pipe and deviations in the cable properties during the production process.

**Applied Mathematics and Control Sciences**. 2021;(3):81-94

COMBINED NEURAL NETWORK METHOD FOR DETERMINING THE BRILLOUIN SPECTRUM MAXIMUM IN DISTRIBUTED FIBER-OPTIC SENSORS

#### Abstract

Distributed fiber-optic sensors are becoming an increasingly common solution for monitoring and diagnosing extended information transmission lines, industrial devices, buildings and devices. One of the directions - Brillouin reflectometry, which allows diagnosing a fiber line for changes in ambient temperature or mechanical deformation, is becoming increasingly popular for engineers and researchers. However, modern standards impose increasingly strict requirements on diagnostic systems in terms of the accuracy of determined parameters. For Brillouin reflectometry, where the value of the environmental parameters is determined by the position of the maximum of the Brillouin gain spectrum, the task of more accurately determining this maximum becomes the main one. The paper considers modern computer-computational methods for detecting the maximum of the Brillouin gain spectrum in an optical fiber. The authors note that imperfections in the shape of the optical spectrum, such as the signal-to-noise ratio, as well as possible digital defects that occur during digitization, can significantly impair the accuracy of the system. The authors consider three approaches to detecting the maximum of the spectrum: the classical method of Lorentz curve fitting, the method of cross-correlation with the ideal Lorentzian function, as well as the method of inverse correlation developed by the authors earlier. To combine the results of the work of the three methods, a neural network model was developed that accepts the input data of each method, together with the parameters of noise and distortion of the spectrum. The model is presented in the form of a four-layer perceptron with two hidden layers. As a result, the authors achieved an increase in the accuracy of determining the position of the maximum of the spectrum by 10% on the model data.

**Applied Mathematics and Control Sciences**. 2021;(3):95-106

BUILDING A MATHEMATICAL MODEL BASED ON ESTIMATES OF THE COMPETITIVENESS OF THE REGIONS OF THE RUSSIAN FEDERATION FOR AUTOMATION SUPPORT FOR DECISION-MAKING IN SOCIAL STRUCTURES

#### Abstract

The problem of constructing a mathematical model for assessing the competitiveness of the regions of the Russian Federation has been formulated and solved. For this, a structuring method based on hierarchical trees is proposed. Their leaves are statistical indicators of socio-economic activity according to official data. These indicators are combined using integral characteristics. An example of the analysis of networks of socio-economic indicators based on the construction of minimal sections of the Kolmogorov - Chapman equations for the “Innovation” indicator is given. To describe the leaf vertices of the indicator trees, it is proposed to use status functions that represent complex-valued functions. The proposed mathematical model represents a system of integrodifferential equations, including the status function for the integral indicator of the competitiveness of the region, functions for each of the integral indicators, polynomials that are obtained as a result of interpolation of statistical data, and management influences. The analysis of the obtained graphs of the normalized values of a number of static indicators, the assessment of trends is given. The possibility of using numerical methods of nonlinear dynamics based on status functions to take into account the cross-section and mutual influence of the parameters of the risks of competitiveness of the regions of the Russian Federation for use in automating decision support in social structures is shown.

**Applied Mathematics and Control Sciences**. 2021;(3):109-129

CLASSIFICATION OF THE GAME THEORY MODELS IN PROJECT MANAGEMENT: LITERATURE REVIEW

#### Abstract

The article provides a review of existing project management models and methods using particular game theory tools. The purpose of the review is to determine the current level of the research in this scientific topic, and to determine the most applicable classes of game-theoretic models and methods for the study of particular strategic interactions types that arise at certain stages of the project life cycle. Scientific papers published in international peer-reviewed journals and conferences proceedings over the past 10 years are used as an information base for analytical research. The first part of the article is devoted to the description of the most successful examples of the game-theoretic models’ application for the study of strategic interactions within project management processes. Based on this part, a conclusion is made about the locality of the game theory tools application for solving project management problems, as well as the expediency of developing an integrated approach that allows describing strategic interactions in project management processes in an invariant form. The second part of the article is devoted to the definition of criteria for the existing game-theoretic models’ classification. The classification criteria selected in this part of the article are used later in the third and fourth parts. The third part of the article is devoted to the classification of existing models by the composition of agents participating in the strategic interaction. According to this criterion, the models and methods are divided into the classes "Public-private partnership", "Contractor - contractor", "Contractor - subcontractor", "Subcontractor - subcontractor", "Interactions within the team", "Others". Based on the results of this classification, a conclusion is made about the dominant attention of researchers to the problems of organizing contractual relations and a clear lack of attention to non-contractual relations of performers within the project team. The fourth part of the article is devoted to the classification of existing models by their type. In this classification method, game-theoretic models are divided in two changes: strategic interactions are divided into simultaneous and sequential, as well as cooperative and non-cooperative (within the last class, into zero-sum games and non-zero-sum games). Based on the results of this classification, certain classes of game-theoretic models are determined that are most applicable in the study of strategic interactions in project management processes. The results of the work confirm the prospects of developing both particular game-theoretic models for describing specific strategic interactions that arise during the project implementation, and an invariant approach for describing a project as a complex of the stakeholders’ strategic interactions. The results of the work can be used by researchers to substantiate the relevance of their own research on this topic.

**Applied Mathematics and Control Sciences**. 2021;(3):130-153

MODELING AND FORECASTING THE NUMBER OF UNEMPLOYED IN THE PERM REGION

#### Abstract

This article is devoted to the study of changes in the main indicators of the level of the unemployed population. The dynamics of the officially registered number of unemployed aged 15-72 years is analyzed. The relevance of this work was that the problem of unemployment is a very important problem in our society. Without solving the problem of unemployment, it is impossible to find solutions to improve not only the economic, but also the moral, moral and spiritual situation in Russia. The purpose of this work was to build a forecast model for the development of the number of unemployed in the Perm Region. It is shown that there was a good correlation between the wages of employees, the number of the population, the growth of high-performance jobs, the cost of a fixed set of consumer goods and services, the age composition of the population older than the able-bodied, the migration growth of the population and the growth rate of the number of unemployed in the Perm Region. The analysis made it possible to construct a linear multivariate model and a model in the state space. It is shown that these models cannot be used to predict the number of unemployed, due to poor predictive properties (the sum of the squares of deviations for a linear multivariate model in this case is 0.295, for a model in the state space - 2.354). At the same time, a regression-differential model of the change in the number of unemployed aged 15-72 years in the Perm Region was built. The obtained data showed that for this model, the sum of the squared deviations is 0.091. This suggests that the regression-differential model best describes the number of unemployed people aged 15-72 years in the Perm Region. Based on the obtained model, a forecast of the distribution of the number of unemployed in the region under consideration for 2021 and 2022 was made. The results obtained indicate that there will be a positive trend in reducing the number of unemployed. Over 7 years, it will decrease by 21.8 %.

**Applied Mathematics and Control Sciences**. 2021;(3):154-168