Stability of time-periodic and delayed systems — a route to act-and-wait control

The history of the linear time-delayed and time-periodic oscillator demonstrates how stability theory has developed from the damped oscillator to the delayed Mathieu equation. Based on these results, it is a natural idea to apply time-periodic control gains when large time delay in the feedback loop tends to destabilize the system. By formulating the act-and-wait control concept as a special case of periodic controllers, a time delayed version of the Brockett problem is posed. Examples demonstrate the efficiency of the kind of time periodic control where the feedback is switched off for a waiting interval longer than the delay.