To improve the dynamic performance of a piezoelectric fast steering mirror in the space-telescope image-stabilization system, the dynamic hysteresis compensation and control of a piezoelectric actuator are investigated. According to the inversion complexity of the PI model based on the generalized Play operator and the asymmetry of hysteresis curves, a PI inverse model based on the generalized Stop operator is constructed to compensate the hysteresis nonlinearity. The Hammerstein model is applied to model the dynamic hysteresis of the piezoelectric actuator and to describe the static nonlinearity and rate-dependent properties of the Hammerstein hysteresis model using the generalized PI and auto-regressive exogenous models, respectively. A compound counter strategy that combines the feedforward compensation and linear quadratic Gauss (LQG) optimal control algorithm is proposed to solve the hysteresis rate dependent model uncertainty. The adaptive differential evolution algorithm is used to identify the model parameters and tune the controller parameters. The test results show that the dynamic hysteresis model can effectively describe the hysteresis curve of the piezoelectric actuator in the frequency range of 1—100 Hz, fitting tracking root mean square errors from 0.077 1 μm (at 1 Hz) to 0.512 3 μm (at 100 Hz), and relative errors from 0.003 1 (at 1 Hz) to 0.020 9 (at 100 Hz). The tracking accuracy of the LQG control algorithm increases by 48.6% and 27.02%, respectively, compared with the direct feedforward and PID controls, in the real-time tracking of the variable-frequency target displacement with an amplitude of 24.5 μm.