  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.
