  It is known that Kalman-Bucy filter is stable with respect to initial
conditions under the conditions of uniform complete controllability and uniform
complete observability (Bishop et. al 2017, Ocone et. al 1996). In this paper,
we prove the stability of Kalman-Bucy filter for the case of noise free
dynamical system. The earlier stability results cannot be applied for this
case, as the system is not controllable at all. We further show that the
optimal linear filter for certain class of non-Gaussian initial conditions is
asymptotically proximal to Kalman-Bucy filter. It is also shown that the filter
corresponding to non-zero system noise in the limit of small system noise
approaches the filter corresponding to zero system noise in the case of
Gaussian initial conditions.
