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Guaranteed Estimation Problems in the Theory of Linear Ordinary Differential Equations with Uncertain Data (Hardcover)
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Guaranteed Estimation Problems in the Theory of Linear Ordinary Differential Equations with Uncertain Data (Hardcover)
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This monograph is devoted to the construction of optimal estimates
of values of linear functionals on solutions to Cauchy and
two-point boundary value problems for systems of linear first-order
ordinary differential equations, from indirect observations which
are linear transformations of the same solutions perturbed by
additive random noises. It is assumed 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 the 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 or
guaranteed estimates. It is established that these estimates are
expressed explicitly via solutions to some uniquely solvable linear
systems of ordinary differential equations of the special type. The
authors apply these results for obtaining the optimal estimates of
solutions from indirect noisy observations. Similar estimation
problems for solutions of boundary value problems for linear
differential equations of order n with general boundary conditions
are considered. The authors also elaborate guaranteed estimation
methods under incomplete data of unknown right-hand sides of
equations and boundary data and obtain representations for the
corresponding guaranteed estimates. In all the cases estimation
errors are determined.
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