Abstract: | The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal
Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A
similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer
operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control
theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems.
In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which
its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of
finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem
for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography:
54 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187. |