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

Bedouhene, Fazia; Boussaada, Islam; Niculescu, Silviu-Iulian

Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 1011-1032.

Ce travail exploite les propriétés structurelles d’une classe de matrices de Vandermonde fonctionnelles, pour mettre en évidence certaines propriétés qualitatives d’une classe d’équation différentielle d’ordre $n$, autonome linéaire avec un terme source dépendant de la variable retardée. Plus précisément, il traite de l’effet stabilisateur du paramètre de retard couplé à la coexistence du nombre maximal de valeurs spectrales réelles. Les conditions dérivées sont nécessaires et suffisantes et, à la connaissance des auteurs, représentent une nouveauté dans la littérature. Sous des conditions appropriées, une telle configuration caractérise l’abscisse spectrale correspondant à l’équation étudiée. Un nouveau critère de stabilité est proposé. Ce critère étend les résultats récents sur la factorisation de fonctions quasi-polynomiales. Le potentiel applicatif du procédé proposé est illustré par la stabilisation d’oscillateurs couplés.

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous $n^{\mathrm{th}}$ 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.

Classification : 34K20, 39B82, 70Q05, 47N70

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