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

Yury  Shestopalov; Yury  Podlipenko; Olexandr  Nakonechnyi

doi:10.4236/apm.2014.44019

Yury Shestopalov^*, Yury Podlipenko, Olexandr Nakonechnyi

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.

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.

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