Normally those glass top stoves have a red neon lamp in the glass top, which turns on when the plates reach around 50C, telling you the glass is hot. If it is not working then the most common suspect is the thermal switch failing, which is cheap to replace, or the neon lemp sputtering itself black, also cheap.
Those controls are panastats, which are basically a form of relaxation oscillator, having a snap action switch and a heating coil, where the heater coil is wired on the output, and heats a bimetal strip to eventually break the output contact, which then disconnects the plate and the heater, till it cools down and the bimetal strip allows the switch to close again. control is via the knob adjusting the amount of tension on a spring that biases the bimetal strip, allowing you to control the power setting ( the PWM of the oscillator) from a low setting of around 10% to full power. There is also a contact on there connected to the pilot light per plate, which closes when the setting knob is turned from the off position.
The power output is typically a 30- 60 second pulse of power, spaced any time from 20 seconds to a minute apart, at low settings, gradually changing to near full time at high, and then permanently on at high, and is only approximate, varying with mains voltage, temperature of the control area and pretty much phase of the moon, but is over a 10 minute period going to be within 10%, good enough to cook with.
There is no real temperature control, merely a variation of the pulse width, except for certain special controls that are only on solid plate non glass top stoves, where they have a sensor bulb and a switch in the actual solid plate, and you need a pot on them to power the plate, otherwise it remains cold. this is there for simmering things like soup and such, which requires lower power, and you need to keep it below boiling temperature, so the plate power is limited to around 500W, and the switch disconnects power when the pot is lifted. those are rather inaccurate, and fail regularly, plus are expensive to replace, as opposed to the simple spiral element under the glass and the Panastat, which are under $20 each as service spares, and which in many cases are identical from different manufacturers in stoves, as they all use the same suppliers of them.