The motor is a permanent magnet brushless motor with sinusoidal or trapezoidal back-EMF.
There is some confusion about the terms, however cheap permanent magnet brushless motors are often called "Brushless DC" (BLDC), which is also the way they are controlled.
The simplest control method is brushless DC, where two phases are conducting at each time, resulting in a phase current which is a squared + -> zero -> - pattern. The commutation between phases is synchronized with rotor position, and the right instant for commutation is usually estimated by reading the voltage (back-EMF) on the non-conducting phase.
A rough speed regulation can be done simply imposing a certain PWM duty-cycle when the phase is conducting.
This kind of control requires a 3-phase inverter and a simple microcontroller capable of generating 3 complementary PWM signals (as someone already said).
At zero or low speed, commutation instant cannot be detected by means of back-EMF, so an open-loop arbitrary commutation is imposed (with a high current, and then with low efficiency), relying on the fact that the rotor will align to the phase having positive current. This could explain why speed is limited to >35%.