Applied Mathematics and Control Sciences

In class measurable (in Lebesgue sense) and bounded control vector functions, we consider one non-smooth optimal control problem of the Goursat - Darboux system with a multipoint quality functional, which is a generalization terminal type functional. Applying one modified version of the increment method, and assuming that the right side of the equation and the functional qualities in a vector state have derivatives in any direction, the necessary optimality condition in derivative terms in the direction flax general is proved. The case of a quasidifferentiable quality functional is considered. In particular, the minimax problem is studied. Under the assumption that the control region is convex, taking into account the properties of non-differentiable functions, the necessary optimality condition is established, which is an analog of the linearized integral principle of maximin, which is constructive in nature and generalizes the point wise linearized (differential) maximum principle.

In this article you can find some mathematical models of the smart management mechanisms that can be used in the organizational system of management of production corporations. The following mechanisms can be used to improve the management effectiveness of this system: the resource-allocating mechanism (the mechanism of direct priorities, the reverse priorities, the competitive mechanism, the mechanism of open management); the mechanism of active expertise (the mechanism of averaging experts' opinions, mechanisms based on median schemes); the mechanism of domestic prices; cost-cutting pricing and taxation mechanisms; the supply chain optimization mechanism; the assortment selection mechanism; incentive mechanisms (incentives for individual results, collective results, and the brigade payment mechanism); integrated mechanisms.

The article uses the analytic hierarchy process as an example of a management decision support system in the field of approving an enterprise budget. The process of solving the problem by the above method is described. The strengths and weaknesses of this method are considered, based on the most important problems, the following studies are carried out: the dependence of the matrix consistency is studied and the sensitivity of priorities is examined on the total number of criteria. The conclusions obtained because of the research are formulated, recommendations are given on the application of the method.

The control problem of a multi-criteria object is considered. Controlled object that has several criteria that are significant for a decision maker. Each criterion characterizes a control object in terms of a particular result of activity or an efficiency indicator. To evaluate the effectiveness of the functioning of the managed facility as a whole, the rating matrix mechanism is used, taking into account all the criteria in the complex. The optimal control problem is formulated as a search for the values of aggregated criteria that provide a given value of a complex indicator with minimal costs for providing values of particular criteria. The generalized cost function was reduced to an equation with one variable. The analytical equation of the level line of the indicator aggregated as a result of the convolution of two criteria is obtained. The line equation is found for an arbitrary binary convolution matrix, including the elements of which are given continuous values. It is shown that the objective function is reduced to a fourth-order polynomial, which can be analytically solved using the Ferrari or Descartes-Euler methods. It is shown that the task of searching for the values of two particular criteria describing the state of the control object for which the complex indicator calculated using the additive-multiplicative approach to complex assessment is equal to the given value and the costs for their provision are minimal, has a solution in general form for arbitrary nondecreasing convolution matrix of two criteria. Particular solutions to the control problem are found using costly functions, which are the inverse function of the Cobb-Douglas production function. It was shown that the cost function of the aggregate indicator has additional terms and is described by an algebraic equation with nonzero coefficients for variables and an additional constant. Based on what it was concluded that the cost functions, which are the inverse function e of the Cobb-Douglas production function, can be applied to control objects that have only two criteria. A similar formulation of the control problem for an arbitrary non-decreasing convolution matrix of two criteria is considered when using the additive-multiplicative approach to aggregation and when using cost functions described by a second-order algebraic equation in general form. As a result of the study, it is shown that the form of the cost function for the aggregated indicator is preserved. Thus, using cost functions in the form of second-order equations, the control problem has a solution in the general form for any number of criteria.

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 their search. The review covers the period 2010-2019. 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 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.