Applied Mathematics and Control Sciences

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.

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.

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.

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.

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.