Mobile wireless channels change persistently and rapidly, so power control needs to adapt similarly in order to save battery power and mitigate interference. Training signals in every data packet are used to achieve this in current systems, albeit in a suboptimal fashion from a power consumption perspective. This problem is used as the basis for posing and solving in detail an optimal dual adaptive control problem and then deriving heuristic controllers from this optimal solution. Dual refers to the joint requirements of the control signal to probe the system for parameter estimation and to regulate the total energy use; these are conflicting requirements which reveal the complexity of optimal stochastic control in general. The information state is defined, computed and explicitly incorporated into the optimization. Performance and computational load comparisons are made between: the optimal control, the certainty equivalence control, and a simplified heuristic approach. The contribution of the paper is the explicit solution of a classically hard optimal stochastic control problem to expose the role of the information state. (C) 2016 Elsevier Ltd. All rights reserved.