A Dual Characterization of Observability for Stochastic Systems

This paper is concerned with the definition and characterization of the observability for a continuous-time hidden Markov model where the state evolves as a continuous-time Markov process on a compact state space and the observation process is modeled as nonlinear function of the state corrupted by a Gaussian measurement noise. The main technical tool is based on the recently discovered duality relationship between minimum variance estimation and stochastic optimal control: The observability is defined as a dual of the controllability for a certain backward stochastic differential equation. Based on the dual formulation, a test for observability is presented and related to literature. The proposed duality-based framework allows one to easily relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table 1).