Optimal partitioned filter of stochastic distributed parameter dynamical systems with unknown initial state

A partitioned filtering approach for stochastic distributed parameter systems with unknown initial states is presented. The observation system is considered here is of a practical pointwise-type. Under the assumption that the initial state can be partitioned into two independent Gaussian random variables in a function space, the optimal partitioned filter is obtained, which can be processed in a parallel fashion and has some effective filtering initializations. Further, some relationships between this filter type and the well-known Kalman-type filter are revealed in terms of the Meditch-type fixed-point smoother in Hilbert spaces. Then, with the aid of these results, two numerical algorithms are proposed to solve the unsteady-and steady-state solutions for the operator Riccati equations. Finally, some examples are given of the application of the two numerical methods to engineering problems.