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The optimal mean‐square filtering problem for linear hereditary dynamic systems with *nonzero* initial conditions is considered. In contrast to the resolved classical problem with zero initial conditions, the filtering problem under consideration reduces to a non‐classical linear‐quadratic optimal problem. The solution of the latter problem is unknown. We replace this non‐classical problem by a simplified classical one, which has a known solution. The solution of this approximate problem is used instead of the solution for the exact original problem. By applying the duality theory we obtain an estimate for the nonoptimality level of the simplified solution. It is essential that this estimate can be computed *without exactly solving* the initial complex filtering problem.

*Keywords: *
linear hereditary dynamic systems; stochastic mean‐square filtering problem; duality theory

*Journal Article.*
*0 words.*

*Subjects: *
Mathematics

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