Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous $n^{\rm th}$ order differential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing effect of delay parameter coupled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and sufficient and, to the best of the authors’ knowledge, represent a novelty in the literature. Under appropriate conditions, such a configuration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed. This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators.