Systems and methods to pump fracturing fluid into a wellhead may include a gas turbine engine including a compressor turbine shaft connected to a compressor, and a power turbine output shaft connected to a power turbine. The compressor turbine shaft and the power turbine output shaft may be rotatable at different rotational speeds. The systems may also include a transmission including a transmission input shaft connected to the power turbine output shaft and a transmission output shaft connected to a hydraulic fracturing pump. The systems may also include a fracturing unit controller configured to control one or more of the rotational speeds of the compressor turbine shaft, the power turbine output shaft, or the transmission output shaft based at least in part on target signals and fluid flow signals indicative of one or more of pressure or flow rate associated with fracturing fluid pumped into the wellhead.