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.
Highlights
The theory of finding estimates of solutions to stochastic differential equations has been intensively developingHow to cite this paper: Shestopalov, Y., et al (2014) Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data
We study estimation of solutions of boundary value problems (BVPs) for ordinary differential equations at fixed points of interval from additional data about their solutions
For a system described by a one-dimensional two-point BVP with decoupling boundary conditions at the endpoints of the interval and quadratic restrictions imposed on the unknown deterministic data and the second moments of observation noise, we have obtained guaranteed mean square estimates of inner product (a,φ ( s)), where φ (s) is the unknown solution of the BVP at a point s ∈ (0,T ) and a ∈ n
Summary
How to cite this paper: Shestopalov, Y., et al (2014) Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data. Under different constraints imposed on function v2 (t ) and for known function v1 (t ) he proposed various methods of estimating inner products (a, x (T )) , where a ∈ n , in the class of operations linear with respect to observations that minimize the maximal error Later these estimates were called minimax a priori or guaranteed estimates (see [3] [4]). We study estimation of solutions of boundary value problems (BVPs) for ordinary differential equations at fixed points of interval from additional data about their solutions. Such settings may be considered as inverse problems when additional data are given with errors. It is shown that optimal guaranteed estimates are expressed via solutions to special BVPs for ordinary differential equations
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