Математическое и компьютерное моделирование эколого-экономического состояния региона: задачи идентификации, прогнозирования, достижимости и управления
The main results obtained during the project № 10-01-96054 “Mathematical and computer
modeling of ecological and economic state for a region: problems of identification, forecasting,
attainability and control” are described in this paper in a popular form. The aim of the project was to
develop mathematical and software bases for construction of mathematical models complex of
ecological and economic state of the region taking into account the specifications, diversity and
relationship processes occurring in the region. The created complex is focused on studying the
stability of dynamic models, forecasting ecological and economic state of the region and solving
control problems with finding the control variables and corresponding trajectory of development.
For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problems, an important class of solutions are soliton solutions. The study of soliton solutions is based on the existence of a one-to-one correspondence between soliton solutions for initial systems and solutions of induced functional- differential equations of pointwise type (FDEPT). The existence and uniqueness theorem for an induced FDEPT guarantees the existence and uniqueness of a soliton solution with given initial values for systems with a quasilinear potential. For systems with a quasilinear potential, one can also formulate the conditions for the existence of a periodic solution. A system with a polynomial potential can be redefined so that the resulting potential turns out to be quasilinear. If a guaranteed periodic soliton solution for such an overdetermined system lies in a sphere, outside which the potential is redefined, then we obtain the conditions for the existence of a periodic soliton solution for the initial system with a polynomial potential. An important task is the numerical realization of periodic soliton solutions for systems with a polynomial potential, which has been successfully solved.
The controller design for a high-order object prone to oscillations under sufficient of interest in automation and robotics is being solved. For the first time, the square root in the error growth detector used as a component for the cost function calculating unit is proposed. Justification for this proposal is given, and the effectiveness of such modification of the cost function is demonstrated on example of object. Software SimInTech was used for some calculations. It was shown that strong procedures of optimization are wide necessary for controller design.
For a functional differential system with continuous and discrete times, the general linear boundary value problem and the problem of control with respect to an on-target vector-functional are considered. Conditions for the solvability of the problems are obtained. Questions of computer-aided techniques for studying these problems are discussed.
One of the sections of economic and mathematical modeling is a section dedicated to the modeling of economic dynamics of large systems. The model in this section is a system of equations and inequalities, describing a closed production cycle - from the formation of the main production resources to replenish their next production cycle due to the distribution of results for the resumption of production and growth of resources. Such a closed system of equations and inequalities reflects the fundamental relationship of real economic production systems, and therefore can be used in a variety of economic experiments. With their help it is possible to determine the results of the regulatory impact on the economy, to assess the significance of various measures of state regulation - from changes in the tax system to a variety of protectionist measures. This book is a new model of economic dynamics, based on the principles of complex-economy. Using models of complex variables allows us to describe these economic processes and relationships that are either difficult or impossible to describe using models of real variables.
Dynamic models under consideration cover a wide class of models in mathematical Economics and Ecology taking into account some aftereffects and including equations with both continuous and discrete times. Control problems are considered in a general case when the aimes of control are given by a system of linear functionals with an arbitrary number of functionals. Complete description of all control actions that solve the control problem is given for the case when only discrete control is applied.
For a functional differential system with continuous and discrete times, the problem of control with respect to an on-target vector-functional is considered. Conditions for the solvability of the problem are obtained.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.