Abstract
This paper treats the fixed-interval smoothing problems for a linear stochastic distributed parameter system with a noisy observation at discrete points on the spatial domain or its smooth boundary. The new feature of this paper is that the derivation of the optimal fixed-interval smoothing estimator is based only on the Wiener-Hopf theory. Combining the Wiener-Hopf equation for the optimal smoothing problem with that for the optimal filtering problem, we construct the optimal fixed-interval smoothing estimator which was derived before by using the Kalman's limiting procedure. Furthermore, we derive the optimal estimation error covariance function for the fixed-interval smoothing problems by using the properties of the fundamental solution for the differential operator. Finally, for the convenience of the numerical computation, we rewrite the results obtained here for the distributed parameter system by using the Fourier expansion method.
