Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

Nakonechnyi, Oleksandr; Podlipenko, Yuri; Shestopalov, Yury

doi:10.3390/math9243218

Open AccessArticle

Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

by

Oleksandr Nakonechnyi

^1,2,†,

Yuri Podlipenko

^1,† and

Yury Shestopalov

^2,*

¹

Faculty of Computer Science and Cybernetics, National Taras Shevchenko University of Kiev, 03680 Kiev, Ukraine

²

Faculty of Engineering and Sustainable Development, Academy of Technology and Environment, University of Gävle, 80176 Gävle, Sweden

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2021, 9(24), 3218; https://doi.org/10.3390/math9243218

Submission received: 10 November 2021 / Revised: 3 December 2021 / Accepted: 10 December 2021 / Published: 13 December 2021

(This article belongs to the Special Issue Decision Making and Its Applications)

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Abstract

The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.

Keywords: indirect noisy observations; estimation; Cauchy problems; impulsive ordinary differential equations; guaranteed mean square estimates; linear functionals

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MDPI and ACS Style

Nakonechnyi, O.; Podlipenko, Y.; Shestopalov, Y. Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed. Mathematics 2021, 9, 3218. https://doi.org/10.3390/math9243218

AMA Style

Nakonechnyi O, Podlipenko Y, Shestopalov Y. Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed. Mathematics. 2021; 9(24):3218. https://doi.org/10.3390/math9243218

Chicago/Turabian Style

Nakonechnyi, Oleksandr, Yuri Podlipenko, and Yury Shestopalov. 2021. "Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed" Mathematics 9, no. 24: 3218. https://doi.org/10.3390/math9243218

APA Style

Nakonechnyi, O., Podlipenko, Y., & Shestopalov, Y. (2021). Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed. Mathematics, 9(24), 3218. https://doi.org/10.3390/math9243218

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Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

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