# Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data

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In thispaper we construct optimal, in certain sense, estimates of values of linearfunctionals on solutions to two-point boundary value problems BVPs forsystems of linear first-order ordinary differential equations from observationswhich are linear transformations of the same solutions perturbed by additiverandom noises. It is assumed here that right-hand sides of equations andboundary data as well as statistical characteristics of random noises inobservations are not known and belong to certain given sets in correspondingfunctional spaces. This leads to the necessity of introducing minimax statementof an estimation problem when optimal estimates are defined as linear, withrespect to observations, estimates for which the maximum of mean square errorof estimation taken over the above-mentioned sets attains minimal value. Suchestimates are called minimax mean square or guaranteed estimates. We establishthat the minimax mean square estimates are expressed via solutions of somesystems of differential equations of special type and determine estimationerrors.

KEYWORDS

Optimal Minimax Mean Square Estimates, Uncertain Data, Two-Point Boundary Value Problems, Random Noises, Observations

Cite this paper

Shestopalov, Y. , Podlipenko, Y. and Nakonechnyi, O. 2014 Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data. Advances in Pure Mathematics, 4, 118-146. doi: 10.4236-apm.2014.44019.

Author: Yury Shestopalov, Yury Podlipenko, Olexandr Nakonechnyi

Source: http://www.scirp.org/

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