Guaranteed Estimation of Solutions of First Order Linear Systems of Ordinary Differential Periodic Equations with Inexact Data from Their Indirect Noisy Observations

Nakonechnyi, Oleksandr; Podlipenko, Yuri

doi:10.1109/saic.2018.8516776

Cited by 6 publications

(6 citation statements)

References 6 publications

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“…Proof. Applying the same reasoning as in the proof of Theorem 1 from [8] and taking into account estimate (1.21) from [15], one can verify that the functional I(u) is strictly convex and lower semicontinuous on U. Since…”

Section: Representations For Minimax Estimates and Estimation Errorsmentioning

confidence: 84%

“…This paper continues our research cycle accomplished in [6,7], where the guaranteed (minimax) estimation method has been worked out for estimating linear functionals over the set of unknown solutions and data under the condition that unknown right-hand sides of the equations and initial conditions entering the statement of the Cauchy problems belong to a certain set in the corresponding Hilbert space (for details, see [8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 91%

“…l (•)) T are observation errors in ( 7) and (8), respectively, that are realizations of random vectors ξ i = ξ i (ω) ∈ R m and random vector-functions ξ j (t) = ξ j (ω, t) ∈ R l and G 2 denotes the set of random elements ξ whose components ξ i and ξ j (•) are uncorrelated; that is, it is assumed that…”

Section: Setting Of the Minimax Estimation Problemmentioning

confidence: 99%

“…In this section, we will define an optimal, in a certain sense, estimate of unknown solution x(t) of the Cauchy problem (3) and ( 4) at the moment T that is linear with respect to observations (7) and (8) and show that this estimate of x(T) coincides with the function x(t) obtained from the solution to problems (37)-(40) at the moment T.…”

Section: σ 1 -Optimal Estimates Of Unknown Solution Of the Cauchy Pro...mentioning

confidence: 99%

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

2021

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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.

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Section: Representations For Minimax Estimates and Estimation Errorsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 91%

Section: Setting Of the Minimax Estimation Problemmentioning

confidence: 99%

Section: σ 1 -Optimal Estimates Of Unknown Solution Of the Cauchy Pro...mentioning

confidence: 99%

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

2021

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“…Notice that this work is a continuation of our earlier studies set forth in [1,2], where we elaborate the guaranteed (minimax) estimation method for the case of the problem of estimation of linear functionals from unknown solutions and right-hand sides of first order linear periodic systems of ordinary differential equations.…”

Section: Introductionmentioning

confidence: 95%

Optimal estimation of unknown data of periodic boundary value problems for first order linear impulsive systems of ordinary differential equations from indirect noisy observations of their solutions

Nakonechnyi¹,

Podlipenko²

2021

Math. Model. Comput.

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We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

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Guaranteed Estimation of Solutions to First Order Compatible Linear Systems of Periodic Ordinary Differential Equations with Unknown Right-Hand Sides

Nakonechnyi

Podlipenko

2022

Studies in Computational Intelligence

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Guaranteed Estimation of Solutions of First Order Linear Systems of Ordinary Differential Periodic Equations with Inexact Data from Their Indirect Noisy Observations

Cited by 6 publications

References 6 publications

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

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

Optimal estimation of unknown data of periodic boundary value problems for first order linear impulsive systems of ordinary differential equations from indirect noisy observations of their solutions

Guaranteed Estimation of Solutions to First Order Compatible Linear Systems of Periodic Ordinary Differential Equations with Unknown Right-Hand Sides

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