No abstract

Household savings are an important component of the financial system. An increase in the number of households following the savings and investment financial strategies leads to economic growth. The financial behavior of the Russian population can be described as passive without a long-term strategy. Therefore, the task of its activating requires careful state regulation. Special significance it acquires in the middle class, which has sufficient income and seeks to rationalize in all areas of their lives. The article explores the problem of managing of middle-class households savings using optimal control methods, correlation and regression analysis, mathematical and computer modeling. A mathematical model of the problem of managing savings by regulating the non-savers share is presented in the form of a boundary control problem for a distributed system with a parabolic equation. The methodology for obtaining quantitative estimates of the impact of economic factors on the household savings distribution by their size is described. An example of determining the optimal reduction of the share of non-savers households in the Perm Region is given and its quantitative relationship with the volume of investment is established. The proposed methodology and the results of the calculations can be used in the development of programs to stimulate savings and socio-economic development of the region in order to increase the validity of management decisions.

Monetary savings of the population are the subject of research of many specialists, since their detailed study in total with other various economic variables allows for a qualitative analysis of the current social and financial situation and make predictions. The definition of such an economic characteristic of society as the distribution of the population by money savings in practice is a very non-trivial task because of the lack of reliable information on the accumulation of citizens. Often, rather than a distributed value, a constant indicator is usedthe average level of household savings. In the case when the savings are distributed according to the normal law in society (as in the developed countries), the replacement of the distributed quantity by a constant is possible to simplify the tasks. It is shown that Perm krai has bimodal structure of society savings, what essentially differs from the normal distribution law and therefore from the mathematical point of view it is unacceptable to replace one with a constant value, but it is necessary to take it into account as a distributed one. This ensures the relevance of this study, whose goal is to identify and analyze the numerical characteristics of the economic structure of Perm krai. The main idea of the work is to apply the mathematical model of the accumulation spectrum of the society of D.S. Chernavskii for Perm krai, the numerical calculation of the model and the economic analysis of the obtained characteristics. Similar numerical calculations of the distribution of the population of Perm krai on savings based on official statistics have not been conducted previously. Methods of economic analysis, mathematical and computer modeling, methods of theories of ordinary differential equations, partial differential equations and stochastic differential equations, probability theory and mathematical statistics are applied. A numerical study of the mathematical model of Perm krai economic society structure was carried out, the most probable stationary levels of population savings in Perm krai were calculated, which approximately amounted to 10 and 63 subsistence minimums for 2016 year. Families seem to accumulate in the vicinity of these savings values, like elementary particles under Brownian motion (this is not a coincidence, since both processes are described by the stochastic differential Fokker-Planck equation in partial derivatives of the parabolic type). Stationary savings levels numerically characterize the consumption standards and the standard of living of a society and are formed on the basis of the value of the consumer basket. In future, it is planned to conduct research on the optimal control of the economic structure of society.

The active development of the optimal control theory of distributed systems in the second half of the last century was caused by a high demand for technical problems, however, this theory quickly showed invariance to other fields of application. Today, applied research aimed at improving the efficiency of complex socio-economic processes management, based on the using and development of methods of optimal control theory for systems with distributed parameters, is becoming increasingly relevant.The object of the study is a distributed system, described by an initial-boundary value problem for a partial differential equation of a parabolic type and modeling the population distribution of a region in terms of cash savings. For the system under study, an optimal control problem of the "distributed controlfinal observation" type has been posed. This means that it is necessary to approximate the state of the system at a fixed time to a certain pre-defined form by controlling the terms in the equation of state (ie, the influx of new members into the system or the outflow from it). The conclusion of the optimization system in a strong form and the law of optimal control, obtained in terms of the model, have been given.The methods of the theory of partial differential equations, the theory of optimal control of distributed systems, mathematical and computer modeling are applied.An example of the numerical study of the model according to the Perm region has been given. The Comsol Multiphysics package has been used for numerical implementations. The methodology proposed by the authors for carrying out such calculations, taking into account the interdependence of migration processes in the region and the financial condition of the population in it, can be applied in the development of effective measures to manage migration processes, which is an important task of both state and regional policy.

Sales volume is one of the most important characteristic of business efficiency. In a detailed analysis and planning of sales it is necessary to consider not only lumped, but also distributed indicators. For this purpose, it is proposed to use developed mathematical model of distribution of orders at their cost. The model constructed by stochastic dynamics methods and described by the initial-boundary value problem for a one-dimensional partial differential equation of parabolic type. The paper presents the model rationale and also numerical example made in Comsol Multiphysics.