Mechanisms behind the stability properties of oscillatory systems with large delays

The behavior of the rightmost characteristic roots of linear time-delay systems as a function of the delay parameter is studied for small values of a gain parameter. The results explain the qualitative behavior of the stability regions in the delay parameter space of oscillatory systems subjected to large delays, and shed a new light on the corresponding stabilization problem. In particular, for a chain of multiple oscillators the determination of a stabilizing value of the delay parameter is interpreted as a phase synchronization problem. The results are illustrated with the output feedback stabilization of an oscillator and the stability analysis of a gyroscopic system.