Filtering and identification of stochastic diffusion systems with unknown boundary conditions

This paper treats the filtering and parameter identification for the stochastic diffusion systems with unknown boundary conditions. The physical situation of the unknown boundary conditions can be found in many industrial problems, i.g., the salt concentration model of the river Rhine is a typical example. After formulating the diffusion systems by regarding the noisy observation data near the systems boundary region as the system's boundary inputs, we derive the Kalman filter and the related likelihood function. The consistency property of the maximum likelihood estimate for the systems parameters is also investigated. Some numerical examples are demonstrated.