Senio P. Methods of localization of functional uncertainties for analyzing systems

The thesis deals with construction and research of methods of system analysis by eliminating local uncertainties and localization of functional uncertainties of their mathematical models for making optimal decisions while managing objects under uncertainty conditions. Functional uncertainties arise in solving many problems of applied mathematics and, in particular, tasks of system analysis. It causes a need to construct fundamentally new methods for finding sufficiently narrow intervals that are guaranteed to contain the wanted solution of the problem, a majorant and a minorant of both known given functions and functions whose analytic expressions are unknown, in particular, functions-solutions of problems (for example, Cauchy tasks, boundary value problems, integral equations, forecasting problems, etc.). In the research paper the construction and research of such methods are carried out in three main directions. One of them is based on the pattern of behavior of the intermediate points of residual members in the form of Lagrange in Taylor formula under compression of the interval of decomposition of a given functional or mapping, respectively, into the point. The essential problem of methods for solving problems, mathematical models of which contain unknown functions, are functional uncertainties generated by such functions. In the present paper, we propose interval methods of localization of such uncertainties for solving certain classes of problems. Methods of localization of functional uncertainties under known analytical expressions of functions are based on the mathematics of functional intervals constructed in this research paper. This mathematics is a generalization of interval mathematics. We have set the conditions under which the sequence of linear functional intervals coincides. The metric space with the introduced metric is a complete metric space. Application of the metric makes this space as a topological space. In this case, the concept of coinciding and continuity can be used in the usual way. Algorithms are based on the fact that each functional interval defines a two-sided approximation of all functions at the same time, for which its bounding functions are a minorant and a majorant, respectively. Methods of localization of functional uncertainties with unknown analytic expressions of functions are based on the conclusions of a number of the theorems proved in this paper. Two-sided approximations of functions are constructed under known only functional intervals of some of their derivatives of higher orders and their individual values and the values of its derivatives of lower orders. In this research paper there are conditions defined and it is developed the method of narrowing the fields of uncertainty of functions, analytic expressions of which are unknown. On this basis, methods for localizing solutions of the Cauchy problem and boundary value problems for ordinary differential equations are constructed. The results obtained in the thesis are contributions to the development of methods for analyzing systems, eliminating and localizing various types of uncertainties with known and unknown functional dependencies.