Optimal filtering in linear distributed-parameter systems†

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Abstract
The optimal filter of Kalman type is derived for linear distributed-parameter systems, which are subjected to both white gaussian distributed noise and boundary noise. The measurement data are also corrupted by white gaussian noise. It is assumed that the number of measuring instruments is finite; therefore, the measurement data are taken at several points of spatial domain. The Wiener—Hopf equation is obtained by applying the calculus of variations technique, from which the filter partial differential equations for the optimal estimate and its covariance matrix can be derived. The optimal estimate and its covariance are expanded into the series of eigenfunctions of the homogeneous partial differential equation with the homogeneous boundary condition. Thus a system of ordinary differential equations for coefficient functions of the series is derived

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