Not All Feedback Stabilized Hyperbolic Systems are Robust with Respect to Small Time Delays in Their Feedbacks

We present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls. We show that in a general case, when the controls are distributed, stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks. We also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems.