We consider the Cauchy problem for the first-order linear systems of ordinary differential equations with unknown right-hand sides and initial conditions that are supposed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of points and intervals, we obtain the linear guaranteed mean square estimates of linear functionals on unknown data of the above-mentioned problems. It is established that if the correlation functions of observational errors are not known and belong to special sets, such estimates are expressed via solutions to some boundary value problems for linear systems of impulsive ordinary differential equations.

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