On Approximate Solving The Mean-Square Filtering Problem In Hereditary Systems

The stochastic mean-square filtering problem for dynamic systems with delay is considered. The state-space representation for optimal filter is obtained. The optimal algorithm contains a system of functional partial differential equations. Simplified algorithms for solving the filtering problem are proposed. With the help of the theory of duality for convex variational problems the estimates of the nonoptimality levels of the simplified algorithms are constructed. These estimates can be effectively computed without solving the original full filtering problem. Key words: linear hereditary dynamic systems, stochastic mean-square filtering problem, theory of duality AMS Subject Classification: 34K35, 93C30, 93C41 1 Introduction A lot of processes in technology, medicine, economics, etc. are described by differential equations with delay. The theory of systems with delay is being intensively developed (see, e.g., [1] -- [4]). Filtering problems take an important place in this theory. Th..