Filtering boundary value continuous time invariant linear systems

We develop the optimal filter for a linear time invariant boundary value system. A linear time invariant boundary value system is a standard linear system, A,B,C,D with the initial conditions replaced by well-posed boundary conditions. We assume that the system is driven by and observed in white Gaussian noise and the boundary value is an independent Gaussian random vector. The boundary value filter reconstructs the current state from the past observations and the mean and variance of the boundary value. It minimizes the error variance over all possible filters. We show by example that knowledge of the boundary conditions can lead to an order of magnitude reduction in the standard error as compared with a Kalman filter. The boundary value filter for an n dimensional system can be realized by a 2n dimensional system.