Applied Mathematics and Control Sciences

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.

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.

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.

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.