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We consider the scenario in which a continuous-time Gauss-Markov process is estimated by the Kalman-Bucy filter over a Gaussian channel (sensor) with a variable sensor gain. The problem of scheduling the sensor gain over a finite time interval to minimize the weighted sum of the data rate (the mutual information between the sensor output and the underlying Gauss-Markov process) and the distortion (the mean-square estimation error) is formulated as an optimal control problem. A necessary optimality condition for a scheduled sensor gain is derived based on Pontryagin’s minimum principle. For a scalar problem, we show that an optimal sensor gain control is of bang-bang type, except the possibility of taking an intermediate value when there exists a stationary point on the switching surface in the phase space of canonical dynamics. Furthermore, we show that the number of switches is at most two and the time instants at which the optimal gain must be switched can be computed from the analytical solutions to the canonical equations.