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

Oleksandr Nakonechnyi, Yuri Podlipenko and Yury Shestopalov
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Oleksandr Nakonechnyi: Faculty of Computer Science and Cybernetics, National Taras Shevchenko University of Kiev, 03680 Kiev, Ukraine
Yuri Podlipenko: Faculty of Computer Science and Cybernetics, National Taras Shevchenko University of Kiev, 03680 Kiev, Ukraine
Yury Shestopalov: Faculty of Engineering and Sustainable Development, Academy of Technology and Environment, University of Gävle, 80176 Gävle, Sweden

Mathematics, 2021, vol. 9, issue 24, 1-17

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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