The presented diploma thesis is focussed on the thermal modelling of a GaN bridge leg using a non-linear controller and observer structure for determination and control of the transistor junction temperature. The thesis starts with a description of the characteristics and basic properties of GaN transistors. Subsequently a test set-up is developed based on a USB-connected PC-AD interface which allows the determination of multiple surface temperatures of the bridge leg sensed by NTC thermistors. The main part of the thesis is the mathematical modelling of the colling system as well as the calculation of the losses generated by the GaN transistors of the bridge leg. For this purpose, the cooling system is formulated as a concentrated parametric heat transfer problem. The Cauer-type thermal equivalent circuit diagrams consider the static and dynamic behaviour of the cooling system, based on the thermal losses as input variables and the van supply voltage representing the control quantity. To control the transistor's junction temperature in case of load and working point variations, a non-linear controller and observer structure with trajectory planning is developed. The simulation and measurement results demonstrate that the developed model in connection with the proposed controller and observer structure achieves a stable and well-damped determination of the junction temperature, even in case the identified system parameters are known merely with uncertainties as being typical for practical converter systems.