One of the most important results of control theory for linear systems is the so called "Frequency Response Theorem" (here the references):
Briefly, it says that under the hypotesis of stability and linearity, if the input signal is sinusoidal, the output signal will be the original sine with phase and amplitude variations respectively equal to phase and amplitude of the transfer function of that system.
Now, let's consider for instance a RC low pass filter:
The output signal will be like this:
So, it is not sinusoidal, although both R and C are linear components. Where is the mistake?