Applied Mathematics and Control Sciences

In recent years, the attention of researchers has been attracted by an active fluid that includes elements (cells, macromolecules, bacteria) capable of self-motion. The behavior of such a fluid is determined by the ability of the elements to transform the energy of the medium into mechanical work and create new medium states. The use of programmable microbots opens up opportunities to achieve such states that are not observed in natural conditions. In this paper, we assume that freely floating microbots have the property of thermotaxis, i.e. they exhibit a motor response to a temperature gradient. Since the density of the bots themselves can be set during their production, the swarm can locally create a density that differs from the density of the pure medium. Thus, the collective actions of bots to redistribute the swarm concentration in the liquid can potentially compensate for changes in the density of a critically overheated liquid in real time.In this paper, we theoretically study the possibility of a swarm to actively control a physical system considering a toroidal thermosyphon, which is a narrow closed channel with a circular cross-section, under the action of gravity and a given heat flow through the boundaries. We develop a mathematical model of the phenomenon, which includes equations of fluid motion, heat transfer, and microbots concentration. Then we apply the Galerkin method to obtain a finite-dimensional dynamic model of the 7th order, in which the first equation describes the fluid velocity in the channel, two equations describe the dynamics of thermal modes, and remaining equations determine the dynamics of the swarm of bots. Nonlinear analysis of the resulting model ODEs shows that under certain conditions, a swarm of microbots is able to switch stationary thermal convection to periodic and chaotic regimes. We demonstrate that the control critically depends on the speed of the swarm's reaction to external changes and the density of microbots.

One of the most important problems in the economy, the educational process and other areas is the task of rational allocation of limited resources. The relevance of solving this problem is determined by the increase in the cost of resources and an increase in their contribution to the final product. There are tasks in the educational process that require the allocation of resources for their implementation. Such tasks are, for example, study assignments, projects, and jobs. The resources can be hours of training sessions, the number of events, and information support.The purpose of the article is to develop a method for optimal allocation of resources in the educational process under conditions of uncertainty. To achieve the goal, a weighted sum of the probabilities of completing all the tasks of this work is selected as an efficiency indicator; restrictions are set based on available resources.A new analytical method for solving the problem of resource allocation has been developed. The method is based on the use of indeterminate Lagrange multipliers. A study and justification of the necessary and sufficient conditions for the existence of the extremum of the objective function is carried out. To account for the fuzziness of the information, the initial data of the problem is given in the form of fuzzy triangular numbers. The developed method identifies three optimization problems of nonlinear programming for the best, average and worst conditions. The solution of the problem for the distribution of homogeneous and heterogeneous resources is considered.The result of the research is a developed new way of distributing homogeneous and heterogeneous resources under conditions of uncertainty. The method proposed in the article can be used not only in the educational process, but also in other fields, for example, in economics, agriculture.

The study describes the results of using dynamic and canonical analysis in forecasting the attendance of outpatient clinics in the Samara region. The healthcare sector is currently in a difficult state. This can be considered both from the point of view of the medical institutions themselves and their visitors. Outpatient clinics play a key role in the healthcare system. Therefore, the development and implementation of innovative approaches to management, improving the quality of medical care and increasing the availability of medical services in outpatient clinics is one of the main tasks of the modern healthcare system. There are many statistical methods for analyzing the attendance of medical institutions, including dynamic and canonical analysis. It is in the integrated approach to analysis, using the listed methods, that the scientific novelty of the work lies. This allows us to identify not only direct but also indirect relationships between the number of visitors and the state of the environment. The practical significance of the study is expressed in the possibility of using the obtained results to adjust management strategies and development of outpatient clinics, improve the quality of medical care and increase the efficiency of using healthcare resources. The purpose of this work is to assess the predicted number of visits to outpatient clinics in the Samara region for the period from 2004 to 2021, as well as to study the degree of relationship with environmental factors.

Classical methods of solving bimatrix games assume the fulfilment of the common knowledge clause, according to which the game with all rules is known to the players and each of them knows that all participants are informed about what is known to the other partners in the game, and such a position is preserved until the end of the game, and the results of the decisions made by the players are represented by point, numerical values. There are quite a lot of situations requiring decision making, formalised as a bimatrix game, in which subjective representations of participants about game parameters – values of elements of the payment matrix – are not known to the other party. Besides, these values are approximate due to incompleteness of the information available at the moment of decision making. Thus, two types of non-statistical uncertainties arise: the first is due to ignorance of the specific strategy of the other participant, and the second is due to the inaccurate determination of the values of the elements of the payment matrices, destroying the position of common knowledge. Such situations can be represented as a fuzzy bimatrix game.The paper shows that in such a game, in the general case, players will not be able to find equilibrium strategies, and because of the vagueness of the values of the elements of the payment matrices, there are no conditions for the correct definition of mixed strategies. As a solution, it is proposed to determine the strategies that provide a compromise result that best suits both participants. For this purpose the fuzzy results of possible strategies of a player are represented by an integral fuzzy evaluation of the whole set of strategies of another participant in the form of an equivalent fuzzy set with a triangular belonging function, and the best compromise solution is determined by analysing the areas of intersection of equivalent fuzzy sets.