Sequential estimation for continuous-time linear dynamic systems with completely unknown inputs and duality with singular control

Completely unknown but deterministic (or Fisher-nnknown) inputs often occur in practice and are approximated by Gauss-Markov processes in applying Kalman-Bucy filtering results. In this paper, the exact filtering equations for such inputs are presented in terms of an information-form filter. The equations are obtained by a limit procedure from the corresponding equations for discrete-time systems. Using duality arguments, the necessary and sufficient conditions for singular control problems are shown to follow from these results.