Taylor & Francis Guaranteed Estimation Problems in the Theory of Linear Ordinary Differential Equations with Uncertain Data (River Publishers Series in Mathematical and Engineering Sciences)

This monograph is devoted to the construction of optimal estimates of values of linear functionals on solutions to Cauchy and two-point boundary value problems for systems of linear first-order ordinary differential equations, from indirect observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing the minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax or guaranteed estimates. It is established that these estimates are expressed explicitly via solutions to some uniquely solvable linear systems of ordinary differential equations of the special type. The authors apply these results for obtaining the optimal estimates of solutions from indirect noisy observations.Similar estimation problems for solutions of boundary value problems for linear differential equations of order n with general boundary conditions are considered. The authors also elaborate guaranteed estimation methods under incomplete data of unknown right-hand sides of equations and boundary data and obtain representations for the corresponding guaranteed estimates. In all the cases estimation errors are determined. About the Author In 1969 Prof. Oleksandr Nakonechnyi graduated from the Mechanics and Mathematics Faculty of T.G. Shevchenko Kyiv State University in the Ukraine. In 1973 defended a candidate dissertation (PhD) in the specialty theory of probability and mathematical statistics. In 1982 defended a doctor of science dissertation in the specialty mathematical cybernetics.Head of Department and Professor of Systems Analysis and Decision Theory, Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv. Honorary Professor of Lankaran State University (Azerbaijan) and Honorary Doctor of the Ukrainian-American Concordia University. Honorary member of the International Academy of Sciences of Informatization of Education (Georgia). Laureate of the State Prize of Ukraine in the field of science and technology. President of the Higher School Academy of Sciences of Ukraine.Author of 7 monographs, more than 250 scientific articles on the problems of estimating the parameters of equations with ordinary and partial derivatives under uncertainty, system analysis of processes described by equations of population dynamics and their application to solving applied problems.Prof. Yuri Podlipenko Graduated from the Faculty of Mathematics and Mechanics, Moscow Lomonosov State University (specializing in mathematics).In 1977, defended a candidate dissertation (PhD) in the specialty mathematical analysis. In 1993, defended a doctor of science dissertation in the specialtymathematical physics. Both degrees were received at the Institute of Mathematics, Academy of Sciences of Ukraine.In 1977-1988, a senior researcher of the Kyiv PolytechnicInstitute. From 1988, a senior researcher, from 1994, a leading researcher of the research section, Taras Shevchenko National University of Kyiv. In 2003, obtained the academic title of a professor, Department of System Analysis and Decision Making. Supervised 10 Candidate dissertations.