Robustness to Incorrect Priors and Controlled Filter Stability in Partially Observed Stochastic Control

We study controlled filter stability and its effects on the robustness properties of optimal control policies designed for systems with incorrect priors applied to a true system. Filter stability refers to the correction of an incorrectly initialized filter for a partially observed stochastic dynamical system (controlled or control-free) with increasing measurements. This problem has been studied extensively in the control-free context, and except for the standard machinery for linear Gaussian systems involving the Kalman Filter, few studies exist for the controlled setup. One of the main differences between control-free and controlled partially observed Markov chains is that the filter is always Markovian under the former, whereas under a controlled model the filter process may not be Markovian since the control policy may depend on past measurements in an arbitrary (measurable) fashion. This complicates the dependency structure and therefore results from the control-free literature do not directly apply to the controlled setup. In this paper, we study the filter stability problem for controlled stochastic dynamical systems, and provide sufficient conditions for when a falsely initialized filter merges with the correctly initialized filter over time. These stability results are applied to robust stochastic control problems: under filter stability, we bound the difference in the expected cost incurred for implementing an incorrectly designed control policy compared to an optimal policy. A conclusion is that filter stability leads to stronger robustness results to incorrect priors (compared with setups without controlled filter stability). Furthermore, if the optimum cost is that same for each prior, the cost of mismatch between the true prior and the assumed prior is zero.