A frequency domain approach to Kalman filtering on Hilbert spaces: Application to Sturm–Liouville systems with pointwise measurement

The problem of optimal state estimation via deterministic Kalman filtering in the time and in the frequency domains is considered. The frequency domain method based on spectral factorization, which was developed previously for linear quadratic optimal control, is extended here to Kalman filtering. For a class of Riesz-spectral systems, it is shown that the spectral factorization problem can be solved by symmetric extraction of poles and zeros, which leads to a tractable computational method in order to calculate the optimal output injection in the Kalman filter problem. Then the class of Sturm–Liouville operators is considered on the space of square integrable functions on a finite interval. According to the properties of such unbounded operators on that space, a set of interpolation Hilbert spaces is considered in a second time. Properties of Sturm–Liouville operators on these spaces are exhibited, together with properties of the $C_0-$semigroups that are generated by these operators. In addition, a characterization of approximate observability by means of point measurement operators is established for such systems. For the aforementioned Sturm–Liouville systems with pointwise measurement, the assumptions needed for applying the symmetric extraction method are shown to be satisfied, which entails that these systems are well-adapted for Kalman filtering with a pointwise measurement observation operator which is bounded on a well-chosen Hilbert state space. The great advantage of considering a new state space is pushed forward by this optimal state estimation problem, which would not make sense in the space of square integrable functions, notably in terms of Riccati equation. The main results are applied to the Kalman filtering of a diffusion system with mixed boundary conditions and pointwise measurement.