TITLE:
Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data
AUTHORS:
Yury Shestopalov, Yury Podlipenko, Olexandr Nakonechnyi
KEYWORDS:
Optimal Minimax Mean Square Estimates, Uncertain Data, Two-Point Boundary Value Problems, Random Noises, Observations
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.4 No.4,
April
23,
2014
ABSTRACT: In this
paper we construct optimal, in certain sense, estimates of values of linear
functionals on solutions to two-point boundary value problems (BVPs) for
systems of linear first-order ordinary differential equations from observations
which are linear transformations of the same solutions perturbed by additive
random noises. It is assumed here 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 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 mean square or guaranteed estimates. We establish
that the minimax mean square estimates are expressed via solutions of some
systems of differential equations of special type and determine estimation
errors.