The thesis presents a procedure of computing optimal and collision-free trajectories for low-dimensioned dynamical systems. In the context of the optimization, different objectives can additionally be taken into account; these, however, have to be described in the shape of desired resp. reference trajectories. The modelling is effected on the basis of a systematic discretization of the systems, which are basically continuous ones, and their environment, thus enabling the use of methods from the domain of combinatorial optimization. This approach allows estimation and scaling of the computational effort and is thus well suited for applications under real-time conditions. The procedure is introduced and evaluated using the example of the autonomous intersection management: For autonomous vehicles, modelled as linear dynamical systems of 2nd order, collision-free trajectories are computed in view of their crossing an intersection resp. any desired traffic junction. The computation takes into account target values for the optimization, such as time, comfort, and fuel consumption. In principle, the procedure can be transferred to any desired application of a similar type, e.g., the trajectory planning of robots sharing the same workspace or within an unknown environment.