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

Fazia Bedouhene (1) , Islam Boussaada (2, 3, 4) , Silviu-Iulian Niculescu (2, 4)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées [Tizi-Ouzou]
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 IPSA - Institut Polytechnique des Sciences Avancées
4 L2S - Laboratoire des signaux et systèmes

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous n−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 represent a novelty in the litterature. 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.

Mots clés: control design; Time-delay; asymptotic stability; exponential stability; exponential decay rate; Vandermonde matrix; quasipolynomial factorization

Domaines: Automatique / Robotique; Systèmes dynamiques [math.DS]

Fazia Bedouhene, Islam Boussaada, Silviu-Iulian Niculescu. Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay. Comptes Rendus. Mathématique, 2020, 358 (9-10), pp.1011-1032. ⟨10.5802/crmath.112⟩. ⟨hal-02476403⟩