On Remote State Estimation in the Presence of an Eavesdropper

This paper studies a remote state estimation problem in the presence of an eavesdropper. A sensor transmits local state estimates over a packet dropping link to a remote estimator, which at the same time can be overheard by an eavesdropper with a certain probability. The objective is to determine when the sensor should transmit, in order to minimize the estimation error covariance at the remote estimator, while trying to keep the eavesdropper error covariance above a certain level. This is done by solving an optimization problem that minimizes a linear combination of the expected estimation error covariance and the negative of the expected eavesdropper error covariance. Structural results on the optimal transmission policy are derived, and shown to exhibit thresholding behaviour in the estimation error covariances. In the infinite horizon situation, it is shown that with unstable systems one can keep the expected estimation error covariance bounded while the expected eavesdropper error covariance becomes unbounded.