## No 2 (2020)

REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL

#### Abstract

One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

**Applied Mathematics and Control Sciences**. 2020;(2):7-26

METHODS FOR PRESENTING PETRI NETS MODELING RESULTS OF PRODUCTION SYSTEMS

#### Abstract

The article considers the issues of presenting the results of simulation experiments with models based on Petri nets for a researcher. The need for a qualitative review of the applied methods for visualizing Petri nets of both the processes themselves and the modeling results has been identified and justified. An analysis of publications on the subject of Petri nets allowed us to conclude that the visualization of such models and the results of their modeling is a known weak point. It is this factor that significantly limits the use of Petri nets as a tool for solving practical problems. The characteristic features of the calendar planning problem and the applied forms of presenting the results of its solution are analyzed. Based on the analysis of literary sources and empirical data, the feasibility of solving the planning problem on the basis of preliminary simulation modeling by Petri nets is substantiated. The method of automatic formation of Gantt diagrams is formalized. The mechanisms for identifying the state of system resources by transitions of the network model are determined. In developing the approach, the attributes and characteristics of its elements already existing in the simulation network model were used to the maximum. An example of using the proposed approach based on the model of a typical robotic technological complex is given. According to the results of the experiment, it was revealed that the software implementation of the described formal rules does not significantly affect the speed of model simulation for the presented robotic system. The proposed approach has the same versatility as the mathematical apparatus used timed Petri nets. This allows us to describe the results of modeling systems and processes of different types, for example, transport or information. An important advantage of the implemented mechanism for the connection of Petri nets and Gantt diagrams is the transfer of the dynamic nature of networks to the process analysis procedures based on diagrams. There is also the possibility of introducing stochastic characteristics of the processes both in the simulation model and in the generated Gantt diagram. The direction of further research on the topic of the article may be the application and corresponding refinement of the obtained formal rules for simulating a network with identification of resources for other classes of Petri nets often used in practice, for example, color Petri nets or E-nets.

**Applied Mathematics and Control Sciences**. 2020;(2):29-40

SELECTION AN OPTIMAL NUMBER OF VARIABLES IN REGRESSION MODELS USING ADJUSTED COEFFICIENT OF DETERMINATION AS A MIXED INTEGER LINEAR PROGRAMMING PROBLEM

#### Abstract

When constructing a regression model, the primary problem faced by the researcher is that it is not clear what the equation of connection between the explained and explanatory variables should be. This initial stage of construction the selection of the model structural specification is called. When choosing a regression specification in parallel, the question arises of which explanatory variables should be included in the equation. This is the problem of variables selection in regression models. Its essence is to single out from the set of “candidates” for inclusion a subset of the most informative of them based on some quality criterion. The article is devoted to the problem of variables selection in regression models estimated using the ordinary least squares. The previously proposed approach to selection a given number of variables based on mixed 0-1 linear programming is considered. The unknown parameters in this problem are the beta coefficients of standardized regression and Boolean variables that are responsible for the occurrence of factors in the model. The optimal values of unknown parameters are found on the basis of maximizing the value of the coefficient of determination of regression. Unfortunately, to solve the problem under consideration, it is required to manually set the number of selected factors, which is often impossible to determine in advance. Therefore, the goal was to formalize the problem so that as a result of its solution the optimal number of selected regressors was also determined. For this purpose, the adjusted determination coefficient, depending on the number of model factors, was used as the objective function. As a result, the problem of mixed integer linear programming was formulated. The unknown parameters in it are still beta coefficients and Boolean variables, as well as an integer variable - the number of regressors. Based on data on prices and characteristics of sedans and hatchbacks of the American automobile industry, a computational experiment was carried out confirming the correctness of the developed mathematical apparatus. The problem formalized in this work in the form of a mixed integer linear programming looks more preferable from a computational point of view than the same problem formalized in modern scientific literature as a mixed quadratic linear programming.

**Applied Mathematics and Control Sciences**. 2020;(2):41-54

A “BILLIARDS” SIMULATION OF AN UNIVERSAL LOGIC MODULES BASED ON THE FREDKIN GATES FOR THE QUANT-UTING

#### Abstract

In the wave of the green computing trend, research has recently intensified on the so-called adiabatic logic, reversible computing, which is supposed to be the basis of quantum computers and bring to a new level of computing power, combined with low power consumption. The basis of this logic is special reversible gates, for example, Fredkin’s gates. Reversibility is a one-to-one correspondence (bijection) between the inputs and outputs of circuits, which means, on the one hand, the possibility of total control of the results of calculations, and on the other hand, the possibility of returning the obtained "energy" quanta for the perform calculations to their source. This approach can significantly reduce the power consumption of computers, as well as increase the reliability of calculations. There are a lot of publications on this topic, however, the development of universal logic modules on such a basis has not been fully considered. The aim of the study is the development and modeling of universal logic modules based on the Fredkin element. In this case, the methods of logical synthesis of a reversible scheme based on a binary Fredkin element, modeling and analysis of billiard calculations are used. The article presents the proposed schemes of the decoder and multiplexer based on the Fredkin element, the "billiard" simulation. The practical significance of the study lies in the fact that the obtained universal logic modules can be used in the synthesis of binary reversible circuits, for example, FPGAs. The performed simulation can be used as examples in practical exercises in the discrete mathematics, mathematical logic, mathematical modeling, and circuitry disciplines.

**Applied Mathematics and Control Sciences**. 2020;(2):55-72

SOLVING THE PROBLEM OF INCOMPLETE INFORMATION ABOUT AN AUTOMATIC CONTROL OBJECT BASED ON REAL-TIME VIRTUAL SENSORS

#### Abstract

Modern automatic control systems use built-in mathematical models for estimation of unmeasured by the direct methods parameters such as NO X emission in aeroengine low-emission combustion chamber. The two models of NO X emissions virtual sensor built into the controller are proposed. A stochastic nonlinear mathematical model is based on the Zeldovich equation. It applies the superposition principle of NO X production in diffusion and homogeneous flames. Probability density distribution functions of the air-fuel mixture concentration in these flames take into account both of a spatial non-uniformity of the mixture composition and a harmonic component of the acoustic waves generated by the heat release. The concept of integral relations models has been developed with the use of numerical modeling of spatial and temporal non-uniformities of the air-fuel mixture concentration (4D-metamodeling) and available experimental data. Another virtual sensor model is based on the neural network predicting NO X emission in gas turbine combustion chamber. The example of a neural network and results of its training on a real combustion chamber is presented. It is shown that the two or three-layer neural network having 20-30 neurons provides an acceptable error (not exceeding 10 %) of the NO X emission display and can be used as a virtual emission sensor in an engine control system. The normalized level of NO x emission per take-off and landing cycle is considered as a target function of the automatic control of low-emission combustion. To estimate the level of NO X emission a built-in virtual sensor is proposed.

**Applied Mathematics and Control Sciences**. 2020;(2):75-95

AIR SUPPLY REGULATOR FOR THE WORKING SPACE HEAT INSTALLATION BASED ON FUZZY LOGIC

#### Abstract

The combustion process in the working space of a heat installation is simulated. The static characteristic of the dependence of fuel consumption on air consumption in the conditions of stabilization of heat generated as a result of the combustion process is shown, therefore, the possibility of optimizing this process is proved. A block diagram, mathematical model and algorithm for calculating energy-saving fuzzy control of the combustion process in the working space of a thermal installation are developed. Combustion Membership functions of fuzzy sets for input linguistic variables are compiled. The rule bases are formulated taking into account the static characteristics of the control object. Examples of technological and linguistic justification of fuzzy controller rules are given. The dynamics of changing the controlled parameter is determined by solving two differential equations. The search for levels of "cut-offs" for the prerequisites of each of the rules using the "minimum" operation in accordance with the compiled database of rules is performed. The defuzzification procedure was performed (bringing it to clarity). The advantages of this system over traditional stabilizing control systems under the action of a set of random factors and a significantly nonlinear relationship between input and output values are shown. Optimal values of scaling factors for the developed automatic control system are determined. Let's consider an example of the formation of a control effect when implementing fuzzy energy-saving control of the combustion process in the working space of the boiler unit No. 3 of the GKP-1A UGPU of Gazprom dobycha Urengoy LLC. The calculations of the amount of inleakage of atmospheric air, excess air flow, the sectional area of the pipe supplying air, the total air flow in the working space of the thermal installation, the volume of natural gas participating in the combustion process, heat resulting from the combustion, the amount of air that did not participate in the combustion process, the amount of heat required to increase the temperature of excess air until the average temperature of the flue gas remaining heat of the combustion reaction, changes in the oxygen concentration in the exhaust flue gases depending on the air flow to the burner. The functional dependence of the flue gas temperature on the natural gas consumption is constructed by approximating the initial data of the regime map of the steam boiler. The dynamic properties of an inertia-delayed controlled process are taken into account when using the Euler method. The combustion control mode influence on the specific fuel consumption is estimated. The effects of using a system with fuzzy control in terms of saving natural gas and electricity over the entire range of performance of a thermal installation are estimated.

**Applied Mathematics and Control Sciences**. 2020;(2):99-121

APPLICATION OF IDEF0 METHODOLOGY FOR CREATION OF A FUNCTIONAL MODEL OF MANAGEMENT OF EDUCATIONAL ACTIVITIES OF A HIGHER EDUCATIONAL INSTITUTION

#### Abstract

New information technologies penetrate all areas of the modern economy, including organizations of higher education. Ways to improve the management process of educational activities of a higher educational institution considered. It is advisable to reorganize the educational process at the management system of the university to improve the quality of specialist education. The urgent task is to build a holistic model of the educational institution management system. In the framework of the consideration of theoretical aspects, definitions of the concepts “system” and “system model” given, and it also indicated why the system model developed and what it describes. The general structure of the model of the university’s educational activities management system is presented, which includes the main subsystems: “Educational process planning”, “Educational process management”, “Monitoring the educational process,” including processes and subprocesses. The results of this study presented in table form. The theoretical aspects of the IDEF0 notation of the SADT methodology considered. The feasibility of applying this methodology to create a functional model of the educational process of a higher educational institution is substantiated. The rationality of using IDEF0 modelling for conducting a complex (structural-functional) analysis of the educational process management system of an educational institution argued based on the main advantages of the named methodology. The presented structure of the system allows obtaining a general idea of its elements and relationships between them. Functional charts of the main business processes of managing the educational activities of the university are given. The presented functional models reflect the structure and functions of the system, as well as the flows of information and material objects converted by these functions. The functional model for managing educational activity is constructed from the perspective of the administration because the model purpose is to analyze the business processes of a higher educational institution, that are part of the complex of managerial tasks facing the university’s leadership. This paper presents the developed functional model of the subsystem “Planning the educational process”, which includes IDEF0 charts of several levels. The top-level chart - the context chart - is detailed by child charts containing subprocesses, functions and specific procedures, as well as incoming and outgoing data flows, control information and resources. The article presents an integrated approach to the study of the system "Management of educational activities of the university". The given study allows us to select the main ways of improving key management processes of the educational activities of the institution, in particular, to optimize the transition from “patchwork” university informatization to integrated management of the entire system as a whole.

**Applied Mathematics and Control Sciences**. 2020;(2):125-143

AGREEMENT CONCEPT FOR DECISION-MAKING IN MANAGING PROJECT OF RESIDENTIAL REPAIR BASED ON THE MULTI-MODEL ASSESSMENT MECHANISMS AND FORECASTING PARAMETERS

#### Abstract

Decision-making concept managing facilities tasks in field of housing and communal services is discussed. This concept characterized by the human factor significant consideration. This approach can affect the class socio-economic systems effectiveness level. The optimal project selection is based on the coordination and consideration the preferences of all interested parties for changing property state. Mechanism application for the comprehensive assessment of characteristics in a multifactorial space, which are subject to optimization in the presence of restrictions is based by the selection process, The operating procedure of the organizational system for managing housing repair projects is considered. A conceptual model for coordinating decision-making in the managing housing repair projects tasks is formulated, its main provisions are presented. The concept use original mechanisms for reconciling the interests of key decision-makers in various stages of repair and restoration work project management. Estimating and predicting mechanisms of the estate object parameters, due to in fact they form the information basis of all targeted choice subsequent steps. In the second part of the article they are in more detail presented. Matrix of direct and transitive relations of coordination between the active elements of the organizational system under consideration has been developed for systematize the required coordination mechanisms. Transitive relationships can be multivariate in depending on the contextual complexity of the organizational system, it raises the question of target choice on a variety of alternatives, that differ in the degree of influence on the organizational system and the complexity of implementation. Possible options are formulated for the problem of estimating the characteristics parameters of real estate building structures. Fuzzy active examination procedure is formulated for cases where the statements of experts may differ from each other in a subjective attitude to the quantitative assessment of the results of visual observation, based on generalized median schemes, characterized by a more accurate establishment of the agreed results of characteristics of building structures assessing in conducting technical expertise process. Possible options are formulated for the problem of the forecasting characteristics parameters of real estate building structures. For each of the three options for setting the problem is illustrated the procedure for building structures parameters predicting changes. Modification of the extrapolation method is proposed, implemented by Steve Jobs taking combining creativity and manufacturability according into accounts the logistic physical wear curves.

**Applied Mathematics and Control Sciences**. 2020;(2):144-161

A REVIEW OF THE METHODS OF ECONOMIC AND MATHEMATICAL MODELING BASED ON THE PRINCIPLES OF ECONOPHYSICS. PART 2

#### Abstract

A review of theoretical and applied results obtained in the framework of the scientific direction in econophysics at the Department of information systems and mathematical methods in economics is given. The first part gives the concept of a financial bubble and methods for finding them. At the beginning of the article, the development of econophysics is given. Therefore, using the research of physicists as a model, econophysics should begin its research not from the upper floors of an economic building (in the form of financial markets, distribution of returns on financial assets, etc.), but from its fundamental foundations or, in the words of physicists, from elementary economic objects and forms of their movement (labor, its productivity, etc.). Only in this way can econophysics find its subject of study and become a "new form of economic theory". Further, the main prerequisites of financial bubble models in the market are considered: the principle of the absence of arbitrage opportunities, the existence of rational agents, a risk-driven model, and a price-driven model. A well-known nonlinear LPPL model (log periodic power law model) was proposed. In the works of V.O. Arbuzov, it was proposed to use procedures for selecting models. Namely, basic selection, "stationarity" filtering, and spectral analysis were introduced. The results of the model were presented in the works of D. Sornette and his students. The second part gives the concept of percolation and its application in Economics. We will consider a mathematical model proposed by J.P. Bouchaud, D. Stauffer, and D. Sornette that recreates the behavior of an agent in the market and their interaction, geometrically describing a phase transition of the second kind. In this model, the price of an asset in a single time interval changes in proportion to the difference between supply and demand in this market. The results are published in the works of A.A. Byachkova, B.I. Myznikova and A.A. Simonov. There are two types of phase transition: the first and second kind. During the phase transition of the first kind, the most important, primary extensive parameters change abruptly: the specific volume, the amount of stored internal energy, the concentration of components, and other indicators. It should be noted that this refers to an abrupt change in these values with changes in temperature, pressure, and not a sudden change in time. The most common examples of phase transitions of the first kind are: melting and crystallization, evaporation and condensation. During the second kind of phase transition, the density and internal energy do not change. The jump is experienced by their temperature and pressure derivatives: heat capacity, coefficient of thermal expansion, or various susceptibilities. Phase transitions of the second kind occur when the symmetry of the structure of a substance changes: it can completely disappear or decrease. For quantitative characterization of symmetry in a second-order phase transition, an order parameter is introduced that runs through non-zero values in a phase with greater symmetry, and is identically equal to zero in an unordered phase. Thus, we can consider percolation as a phase transition of the second kind, by analogy with the transition of paramagnets to the state of ferromagnets. The percolation threshold or critical concentration separates two phases of the percolation grid: in one phase there are finite clusters, in the other phase there is one infinite cluster. The key situation to study is the moment of formation of an infinite cluster on the percolation grid, since this means the collapse of the market, when the overwhelming part of agents for this market has a similar opinion about their actions to buy or sell an asset. The main characteristics of the process are the threshold probability of market collapse, as well as the empirical distribution function of price changes in this market.

**Applied Mathematics and Control Sciences**. 2020;(2):165-190

COMPENSATION PLAN SIMULATION MODELING CONSIDERING DIFFERENT RANDOM QUANTITY DISTRIBUTIONS

#### Abstract

The article reveals one of the decision support system (DSS) aspects in compensation plan (CP) problems. In particular, in the research simulation modeling is used to establish relations between functioning results of different CP. These results are total production and wage fund. The author claims, that formulas of different CP may be estimated in a different conditions by using simulation modeling, thus specific patterns may be reviled. Having those patterns, DSS in CP problems may be developed. And its predictions will come true with a high probability. The article demonstrates simulation modeling of time-bonus and piecework CP`s. Patterns are described. According to the research piecework CP always has higher total production and wage fund results then time-bonus CP. Moreover, in the article other patterns after more complicated simulation modeling are shown. The main research results are follows. First, models and conditions for simulation. Second, statistics gathered. Third, conclusions about independence of the CP results according to distribution random quantities and preset different constant parameters, The simulation models include several principles. Models formulas are based on well known formulas, common sense, and “economic person” conception which was developed by S. Mill over 200 years ago and widely used in modern world. Moreover, formulas used in simulation models includes increasing/decreasing coefficients in order to consider randomicity of real economy. This random coefficient generated with a different distributions. These are normal, chi square, exponential, gamma. Simulation parameters were set according to economics logic to fit in common sense. Furthermore, different constants of CP`s are taken into account. Thus, over 2,5 billions combinations were considered. After overall simulations, statistics was gained and specific patterns are shown.

**Applied Mathematics and Control Sciences**. 2020;(2):191-210