I general, if you model a system (pendulum for example) and you do not include damping into the system (maybe caused by friction): Is it generally possible to tune a PI or PID controller for an second-order-dynamical system that is undamped? Because without friction/damping no energy can leave the system, and therefore I am wondering, if PID-control can generally work for these systems or if this will lead always to oscilations or unstable control?
Yes, as long as the plant is controllable, observable, and doesn't vary too much from the model.
In terms of pole placement, a PID controller with a settable bandwidth on the derivative adds two poles, and has four settable parameters. The system has four poles (two from the plant, two from the PID controller), so in theory, not only can you stabilize the system, you can put those poles wherever you want.
In practice, you'd want to use a more robust design technique.