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We consider the stabilization of a discrete-time linear system in the presence of continuous multiplicative observation noise. Previous work has explored time-varying periodic non-linear control approaches for this problem. To understand the information-gathering role of control in this problem, this report explicitly computes how the conditional density of the state of the system evolves given the observations. The calculations suggest a novel control strategy that chooses the control equal to the maximum a-posteriori estimate for the state. We show that as n goes to infinity this control strategy does indeed drive the system state to 0 almost surely.

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