  We consider the problem of estimating a continuous-time Gauss-Markov source
process observed through a vector Gaussian channel with an adjustable channel
gain matrix. For a given (generally time-varying) channel gain matrix, we
provide formulas to compute (i) the mean-square estimation error attainable by
the classical Kalman-Bucy filter, and (ii) the mutual information between the
source process and its Kalman-Bucy estimate. We then formulate a novel "optimal
channel gain control problem" where the objective is to control the channel
gain matrix strategically to minimize the weighted sum of these two performance
metrics. To develop insights into the optimal solution, we first consider the
problem of controlling a time-varying channel gain over a finite time interval.
A necessary optimality condition is derived based on Pontryagin's minimum
principle. For a scalar system, we show that the optimal channel gain is a
piece-wise constant signal with at most two switches. We also consider the
problem of designing the optimal time-invariant gain to minimize the average
cost over an infinite time horizon. A novel semidefinite programming (SDP)
heuristic is proposed and the exactness of the solution is discussed.
