How can a system reach its steady-state if the I-action of a PID controller, u(t), doesn't go to zero as the system approaches its desired value?
Imagine a real system such as an oven. You want to maintain 350°F so that your cookies will be properly baked. 

The input to your controller is the temperature sensor. 

The output to the oven is the percentage the heaters should remain on vs. off. 

In order to maintain a constant temperature, the output must be non-zero, and ideally (without you opening the door or line voltage changes etc.) will be constant. 



Overshoot happens because of the closed-loop system dynamics (underdamped), and it can also happen because of integral windup, which is a nonlinear effect.