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On the basis of H_∞ control approach instead of using the conventional optimization method, we present a distributed power allocation algorithm for a cognitive radio network (CRN), where underlying secondary users (SUs) share same licensed spectrum with primary users (PUs) and the channel gain is timevarying between two time slots. Based on the target signal-to-interference-plus-noise ratio (target-SINR) tracking power control (TPC) algorithm in the conventional network and the dynamic description of the channel gain fluctuation in the CRN as a first-order Markov model, we formulate the power allocation in the CRN into a state-space system model with exogenous input. In this time, the core for us becomes to design a H_∞ state feedback controller obtained by solving a linear matrix inequality (LMI) for this system to realize power allocation for SUs. The SINR requirement of SUs and the interference temperature (IT) constraint of all PUs can be guaranteed. According to this controller design principle, we also give a H_∞ delay-independent state feedback controller to treat time-delay for the protection of the communication performance. Simulation results demonstrate the validity, effectiveness, and advantages of this approach compared with the algorithms obtained by the optimization theory for the power allocation in CRNs.