## No 1 (2024)

**Year:**2024**Articles:**10**URL:**https://ered.pstu.ru/index.php/amcs/issue/view/412**DOI:**https://doi.org/10.15593/2499-9873/2024.1

On asymptotic properties of the Cauchy function for autonomous functional differential equation of neutral type with distributed delay

#### Abstract

In this paper we investigate stability with respect to the initial function of a linear autonomous functional differential equation of neutral type. Lyapunov stability, asymptotic and strong asymptotic as well as exponential stability of the equation and their interrelation are studied. The definitions of all types of stability are formulated in terms of the Cauchy function, a function that allows the general solution of the equation to be written down explicitly. The main attention is paid to the study of stability with respect to the initial function from the spaces of integrable functions. We use a well-known representation of the solution of a functional differential equation as an integral operator with Cauchy function as a kernel. Stability issues are considered for an equation with commensurable delays at the derivative and distributed delay at the unknown function. It is shown that for such an equation all the properties previously proved for the equation with commensurable delays at the unknown function are preserved. Namely, it is shown that the strong asymptotic stability of the considered equation with an initial function from the space L1 is equivalent to the exponential evaluation of the Cauchy function, and moreover, any of these properties entails exponential stability on the initial function in any of the spaces Lp for 1≤p≤∞ . In this case, as for the equation with commensurable delays, strong asymptotic stability in the Lp space for some p>1 may not be equivalent to exponential stability.

**Applied Mathematics and Control Sciences**. 2024;(1):8-28

Development of software tools for automatisation of the process of the fabric tensor parameters determination for porous materials

#### Abstract

The idea of constructing a special tensor to describe the parameters of structurally inhomogeneous materials arose from a number of attempts to quantitatively characterize the microstructure of an elastic porous material. The using of special tensor quantities to describe the stereometric characteristics of structurally anisotropic materials makes it possible to express in a compact form the significant structural parameters of the studying objects. The predominant orientation of pores inside the sample is well described by means of the fabric tensor and the anisotropy tensor which algebraically related to the fabric tensor. The mathematical calculations presented in the work made it possible to formalize the process of calculating all the parameters necessary for constructing the structure tensor. Algorithmization of the method for determining the mean intercept length formed the basis for the developed specialized software for calculating the components of the fabric tensor. Verification of the software module was carried out by conducting a stereological study of a number of idealized test structures and a sample of porous material for which the fabric tensor was known in advance. The obtained results did not contradict natural reality, coincided with previously data and described the anisotropy degree of the studied structures with a high degree of accuracy. To demonstrate the practical using of the developed software package, the work presents the results of studying of the human femoral neck trabecular bone tissue sample and the autoclaved aerated concrete sample. Calculations of all necessary parameters were carried out and images of the fabric ellipse of the studied porous materials were presented. From the results it is clear that the structure tensor is capable to describe the stereometric characteristics of natural and artificial porous structures, and the software package allows us to automate the process of determining all necessary parameters.

**Applied Mathematics and Control Sciences**. 2024;(1):29-40

Generation of levels of a single-player 3D game based on BSP trees

#### Abstract

The paper examines the problems of level generation for single-player 3D games and proposes a method aimed at speeding up and reducing the cost of the process of developing game levels, revealing the approaches and technologies used to solve the problems of level generation, implementing algorithms for generating rooms and paths between them. After analyzing existing methods for generating levels, the BSP-tree generation method was selected, which can create unique levels based on input variables, allowing to reduce the development time of game levels. Creating an endless level is a difficult task, but with some helpful tips and techniques it becomes much easier. The first step to organizing an infinite level is to create an empty object that serves as the base for the level, then you can add various environmental elements. To achieve the effect of an endless level, it is proposed to use the “scrolling” technique. This means that when the player moves in one direction, objects in the level move in the opposite direction. This creates the illusion of infinity and allows the player to continue exploring new areas of the level.

**Applied Mathematics and Control Sciences**. 2024;(1):41-54

The use of maximum entropy principle to construct robust estimators under point Bayesian contamination. Part I

#### Abstract

The paper develops the theory of robust parameter estimation of statistical models using the apparatus of information theory. The approach of A. M. Shurygin based on the model of a series of samples with random point contamination (point Bayesian contamination model) is considered. Perhaps, stoyki (steady) estimators have the most interesting properties among the estimators proposed by A. M. Shurygin. Although this approach can be associated with F. Hampel's approach to robust estimation, the need to postulate a parametric form of the contamination point distribution when finding stoyki estimators does not allow this to be considered a robust procedure. In the first part of our work, a non-parametric method of selecting said distribution is proposed - by maximizing Shannon entropy in the neighborhood of the model distribution limited by the value of Kulbak - Leibler divergence. This way of finding the distribution density of the contamination point allows us to consider the resulting estimators as robust, and, moreover, having the optimality property. We call the obtained estimators generalized radical, since their special case is the radical estimators of A. M. Shurygin. Generalized radical estimators are widely known in foreign publications as estimators of the minimum logarithmic density power divergence (gamma-divergence), while the question of their optimality is not investigated there. Generalized radical estimators include some popular estimators of the location: the estimators of Meshalkin (Welsh), Andrews, Smith, Bernoulli, Tukey's biweight estimator, Huber-type skipped mean and the generalized Charbonnier estimators. Also in the first part of the work it is proposed to use the cross entropy functional. Cross entropy, used as an optimized functional instead of Shannon entropy, allows us to get a family of estimators with the widest range of values of the parameter specifying this family. Since the problem of maximizing the expectation of the loss function of the maximum likelihood estimators in the point Bayesian contamination model is reduced to the problem of maximizing the cross entropy, the generalized radical estimators can be interpreted as being defended against malicious modifying of the maximum likelihood estimates. In the second part of the work, another optimal solution is obtained on the basis of the formalism of A. Rényi (or the formalism of C. Tsallis equivalent in terms of our problem) that gives a new family of estimators, the special cases of which are also some well-known estimators. To select one estimate from a family defined by different divergence constraints, an optimization approach is proposed that is similar to the approach leading to stoyki estimators, but, unlike the latter, remaining non-parametric. The main theoretical results obtained in the paper are illustrated in its second part by the example of location estimating for the cosine distribution.

**Applied Mathematics and Control Sciences**. 2024;(1):55-72

Development and validation of a thermophysical numerical model of an inert chamber for pyrochemical conversion of spent nuclear fuel

#### Abstract

The object of the study is an inert chamber with in-chamber technological equipment of the pyrochemical conversion process - a stage of the closed nuclear fuel cycle technology, designed to develop the technological process of processing spent nuclear fuel of the pyrochemical conversion for Module of the Experimental and Demonstration Energy Complex. The article describes the data obtained during bench tests of inert chamber experimental sample, the design of an experimental sample and calculated mathematical models of its thermal state. The analysis of the initial geometry of the camera is carried out and the accepted simplifications are justified. The principle of modeling structural elements of enclosing structures based on the use of shells of zero thickness is substantiated. In this case, the wall thickness and thermal properties are set mathematically, taking into account the properties both across and along the shell. A methodology for developing a numerical computational model of an inert chamber based on the finite volume method using the ANSYS Fluent software package is described. The results of calculations of the thermal state of the chamber under study, both in stationary and non-stationary settings, as well as validation of the developed numerical model based on the results of bench experimental studies are presented. It is shown that the developed numerical model has high accuracy (deviation from experimental data is no more than 5 %).

**Applied Mathematics and Control Sciences**. 2024;(1):73-93

The hybrid adaptation loop synthesis for control system with an implicit reference model

#### Abstract

A scheme for constructing direct adaptive control systems with an implicit reference model for linear continuous objects is being developed. The key condition for the convergence of the tunable parameters to their ideal values is the constancy of the "rich" harmonic excitation of the system. In real systems, this requirement is difficult to fulfill because the setting effects must be worked out by the system over a limited time interval. The desire to obtain fast parametric convergence in adaptive control systems is the engine of research in the field of hybrid adaptive systems. In this paper, hybridity is understood in the sense that the control object is continuous, and the controller, or any part of it, is discrete. A scheme for the synthesis of a hybrid adaptation circuit is proposed, in which the continuous part of the algorithm uses as a parameter a discrete estimate of the ideal adjustment matrix of the regulator. The task is to obtain a parametric error model in the form of a linear regression equation with respect to the parameters of the ideal setting of the regulator. A quadratic criterion of parametric mismatch is introduced. The corresponding minimization problem according to the desired estimates leads to the least squares algorithm, which can be used both in one-step and recurrent form. It is shown that the recurrent least squares algorithm can be used in this part. Stability analysis is carried out by the Lyapunov function method. The conditions under which a closed system is dissipative are determined. The purpose of this work is to show the advantage of the hybrid adaptation circuit over the traditional one. To do this, the theoretical results are supplemented with examples of numerical modeling. The dynamics of the system is compared with the traditional continuous adaptation algorithm and with the hybrid algorithm. In both cases, the Lyapunov function under study is non-increasing. This reflects stability of the system. However, in a traditional system, the convergence rate is significantly lower, which leads to a larger coordinate error. In a system with a hybrid adaptation contour, there is a relatively fast both coordinate and parametric convergence into an admittedly small dissipation zone.

**Applied Mathematics and Control Sciences**. 2024;(1):94-110

Exploring color dependency of convolutional neural networks for computer vision promlems

#### Abstract

The article presents some problems with convolutional neural networks concern-ing their dependency on image color parameters. The hypotheses stated that training of neural networks for computer vision tasks is dependent on the colors of the objects in the training dataset. The special experiments on the synthetic images are made for proving the hypotheses. The results of the experiments shows the presence of dependency of the trained networks on the colors of the objects. Methods of solving some of the proven problems suggested. The future directions of the research outlined.

**Applied Mathematics and Control Sciences**. 2024;(1):111-122

Technology for constructing quite interpretable quasilinear regression models

#### Abstract

This article is devoted to the current problem of searching for patterns in large volumes of statistical data. The tool for data analysis is regression analysis. When constructing regression models, researchers often strive only for their high quality of approximation. But, as noted in modern scientific works, such a metric alone is not enough. Therefore, interpretable machine learning is actively developing today. Previously, the author proposed a definition of a quite interpretable linear regression, and the problem of its construction was formalized as a mixed integer 0-1 linear programming problem. Research has revealed the high efficiency of the developed mathematical apparatus in solving problems of big data processing. Therefore, it was decided to expand the proposed technology for constructing quasilinear regressions. The article gives a definition of a quite interpretable quasilinear regression, which includes 6 conditions. An algorithm has been developed for interpreting the influence in the estimated quasilinear regression of monotonically transformed explanatory variables on the dependent variable. The problem of constructing a quite interpretable quasilinear regression is formalized as a mixed integer 0-1 linear programming problem. It is shown how to select the acceptable limits of the parameter M in this problem. To demonstrate the performance of the proposed mathematical apparatus, the problem of modeling the compressive strength of concrete using data containing more than 1000 observations was solved. For this purpose, the VInter-2 program was used. The constructed model included the following transformed variables: cement-water ratio, blast furnace slag, plasticizer and concrete age. The constructed regression turned out to be better in terms of the quality of approximation and simpler in the structure of the existing model. An interpretation of the constructed quasilinear regression is given. The influence of explanatory variables on the strength of concrete in it is consistent both with the substantive meaning of the problem and with other existing mathematical models. The technology proposed in the article for constructing quite interpretable quasilinear regressions has high potential for solving problems of big data processing in various subject areas.

**Applied Mathematics and Control Sciences**. 2024;(1):123-138

Application of optimization methods in the problem of information dissemination in mass media

#### Abstract

The use of mathematical methods in various applied areas plays an important role in making management decisions. Optimization models are an integral part of the mathematical apparatus used by both various government institutions and businesses to help decision makers in difficult conditions in order to conduct a complete and objective analysis of subject activity. This paper discusses an optimization approach to solving the problem of determining the list of platforms for disseminating information in the media. The article formulates new formulations of integer linear programming and multicriteria optimization problems for modeling information dissemination. An algorithm for processing statistical data has been implemented to generate a matrix of object-features. Methods for solving the formulated optimization problems in the problem of determining the list of information dissemination sites have been implemented and tested. A sensitivity analysis was carried out in a multicriteria optimization problem, the results of numerical modeling for various input parameters were considered, and appropriate conclusions and comments were made. The relevance is dictated by the growing role of information platforms and the need to optimize the process of making decisions in the field of information management.

**Applied Mathematics and Control Sciences**. 2024;(1):139-159

Simulation model of a two-channel queuing system with exchange and retry requests

#### Abstract

Non-stationary effects in the operation of a two-channel RQ system with the ex-change of requests during the service process are considered. The study is aimed to identify the features of the functioning a hybrid call center with parallel service by a human dispatcher and an intelligent voice bot. Requests that do not obtained service are transferred to orbits, from where retry calls are made. After a random time, determined by the patience of clients, requests leave the orbit. An intelligent bot has a higher service rate compared to a human dispatcher. The flow of input requests is distributed randomly between service channels. It is taken into account that some clients during the course of service may prefer to switch to an alternative dispatcher. During the operation process, the human dispatcher has short breaks for rest and a long break for lunch. At these time periods, maintenance is performed entirely by a bot. The intermittency of service modes leads to non-stationary effects in the form of a current modification in the throughput of the QS. Incoming requests are described by Poisson processe when the service is open. Numerical calculations are based on discrete event simulation modeling of the system. At each step, the model time is increased by a fixed step amount. The state of the system changes randomly over a time step, as a Markov process, i.e. the transition probability depends only on the current state of the QS and does not take into account memory effects. A simulation modeling algorithm is proposed, based on dividing the entire period of operation of the QS into small intervals, when the probabilities of changes are small. The presence of alternatives in the operation of the QS leads to branching of the process. The statistical nature of the functioning of the QS is taken into account in the ensemble of realizations of the computer model and is characterized by the dispersion of the results. As an example, a simulation of a call center for a housing management company was carried out and showed the prospects of increasing the share of intelligent bots in servicing calls.

**Applied Mathematics and Control Sciences**. 2024;(1):160-172