  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 estimates. We establish that the minimax estimates
are expressed via solutions of some systems of differential equations of
special type. Similar estimation problems for solutions of BVPs for linear
differential equations of order n with general boundary conditions are
considered. We also elaborate minimax estimation methods under incomplete data
of unknown right-hand sides of equations and boundary data and obtain
representations for the corresponding minimax estimates. In all the cases
estimation errors are determined.
