Some Remarks on the Pendulum Stabilization. Multiple Delays & Decay Rate Assignment

A series of recent works have highlighted the interest of multiplicity varieties in the characterization of the exponential decay rate for the solution of linear dynamical systems represented by delayed differential equations. In fact, it has been shown in the case of a single delay that a sufficiently high multiplicity spectral value tends to be dominant, in what is now known as the multiplicity-induced-dominance property (MID). Despite the many existing results in such a configuration, only a few concern the case of multiple delays. Through this work, we propose a first characterization of the MID property for second-order systems controlled by a two-delay ”block”. As an application of the results obtained, we consider the problem of stabilization of the classical pendulum with exclusive access to the delayed position.