Problems heating up pans on an induction hob

Any electrical device that is used to maintain a constant temperature (ovens, induction hobs, immersion cookers, even thermostat-based space heaters) does so by first sensing whether the current temperature is below the set temperature. If so, the heating element/induction coil/etc... is turned on. Once the temperature reaches some upper threshold, the element is turned off again, and the cycle repeats.

Different devices have different ranges of thresholds. For instance, the temperature in an oven may vary by as much as 30°C, while an immersion cooker will keep the temperature range withing 1°C or so.

The reasons your hob overshoots the set temperature, then takes some time to stabilise, can be:

1. The hob is built to operate on a much wider range of temperature thresholds than you thought.
2. The hob is built to keep a pan with contents at a constant temperature, but heats up too powerfully for an empty pan (i.e., the hob makes the 'assumption' that there is something in the pan, and delivers power based on that assumption).
3. There is some distance between the hob's temperature sensor (often in the center) and the induction coil (a ring some distance from the center), and your pan does not conduct heat instantly. Then, while the pan has reached target temperature at the spots closest to the induction coil, it is still below the target temperature at the hob's sensor, causing the hob to add more power.

Using an empty pan will make points 1 and 3 worse as well. You can check if any of these are occuring in your case by heating a pan of water, and taking the water temperature (points 1 and 2), or by moving your surface thermometer to different spots in the pan (point 3). If the issue turns out to be 1, there is not much you can do. If it's either 2 or 3, you can mitigate the problem by starting to cook (almost) immediately.

Alternatively, you could force a slower heating curve by starting the hob at its lowest temperature and manually changing the temperature setting every minute or so.

You say, probably paraphrasing the manual, that "temperature and power can be combined independently*. But they can't. There may be separate controls, and you may be able to set T, P or both with predictable effects, but they're not independent. The temperature (or rate of temperature rise) will always depend on the power input and what you're heating, by simple physics. Control is achieved by turning the coil on and off and/or by actually driving it at lower power.

I suspect that you can set a target temperature, and an output power that's effectively a maximum used to bring the pan up to temperature. By necessity the temperature is sensed using a sensor in the hob. That will take some time to respond during which the pan will keep getting hotter - a (thin) carbon steel pan could get very hot. The temperature could be sensed in several ways; the fastest, and how I'd design it, is an IR sensor built in to the hob; that could still take a few seconds to settle down. It's also a little inaccurate due to making assumptions about the pan material, but it should be close enough for normal cooking.

The only time you'd really want to heat a completely empty pan is for dry frying/toasting, and then a heavy base works better. The worst case is probably a pan with a very little oil; this would start to smoke quite suddenly, and you'll need to get used to how long it takes to avoid it.

I have the same problem. Empty or full my cast iron pan will overshoot the set temp by a bunch at first. It's worse with thinner cookware. I think the thermostat takes awhile to figure out the temp and shutting off. My best solution is to preheat the pan by slowly creeping up the temp setting or boiling off a little water. You want the pan to be evenly heated anyway so that should not be a problem. The other option is to use only the power setting which bypasses the thermostat.