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We consider data transmission through a time-selective, correlated (first-order Markov) Rayleigh fading channel subject to an average power constraint. The channel is estimated at the receiver with a pilot signal, and the estimate is fed back to the transmitter. The estimate is used for coherent demodulation, and to adapt the data and pilot powers. We derive the Hamilton--Jacobi--Bellman (HJB) equation for the optimal policy in a continuous-time limit where the channel state evolves as an Ornstein--Uhlenbeck diffusion process, and is estimated by a Kalman filter at the receiver. Finding an explicit solution to the HJB equation, as well as proving that a (twice-differentiable) solution exists, appears to be quite challenging. However, assuming that such a solution does exist, we explicitly determine the optimal pilot and data power control policies. The optimal pilot policy switches between zero and the maximum (peak-constrained) value (“bang-bang” control), and approximates the optimal discrete-time policy at low signal-to-noise ratios (SNRs) (equivalently, large bandwidths). The switching boundary is defined in terms of the system state (estimated channel mean and associated error variance), and can be explicitly computed. Under the optimal policy, the transmitter conserves power by decreasing the training power when the channel is faded, thereby increasing the data rate. Numerical results show a significant increase in achievable rate due to the adaptive training scheme with feedback, relative to constant (nonadaptive) training, which does not require feedback. The gain is more pronounced at relatively low SNRs and with fast fading. Results are further verified through Monte Carlo simulations. [ABSTRACT FROM PUBLISHER]