Optimal filtering and smoothing algorithms for linear distributed-parameter systems with pointwise observation

Sequential algorithms for filtering, fixed-interval, fixed-point and fixed-lag smoothing are developed for a class of linear distributed-parameter systems. The class of systems concerned is that involving noisy measurement data which are obtained from a finite number subdomain in the spatial domain, i.e. pointwise sensor. The basic tools of the development are the Wiener-Hopf equation, innovation theory and the fundamental value theory for the differential operator. The fixed-lag smoother obtained is new. Finally, for the numerical solution, the expansion method of eigen-functions in L²(D) and L²(D × D) is asserted, and it is shown that the results obtained here are general forms, which contain the results of the finite state space reported earlier.