Stability analysis of neutral type systems in Hilbert space

Rabah Rabah (1) , Grigory M. Sklyar (2) , Aleksandr V. Rezounenko (3)

1 IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes
2 Institute of Mathematics
3 Department of Differential Equations and Control

The asymtoptic stability properties of neutral type systems are studied mainly in the critical case when the exponential stability is not possible. We consider an operator model of the system in Hilbert space and use recent results on the existence of a Riesz basis of invariant finite-dimensional subspaces in order to verify its dissipativity. The main results concern the conditions of asymptotic non exponential stability. We show that the property of asymptotic stability is not determinated only by the spectrum of the system but essentially depends on the geometric spectral characteristic of its main neutral term. Moreover, we present an example of two systems of neutral type which have both the same spectrum in the open left-half plane and the main neutral term but one of them is asymptotically stable while the other is unstable.

Mots clés: Neutral type systems; Exponential stability; Strong stability; Infinite dimensional systems.; Infinite dimensional systems

Domaines: Systèmes dynamiques [math.DS]; Optimisation et contrôle [math.OC]

Rabah Rabah, Grigory M. Sklyar, Aleksandr V. Rezounenko. Stability analysis of neutral type systems in Hilbert space. Journal of Differential Equations, 2005, 214 (2), pp. 391--428. ⟨10.1016/j.jde.2004.08.001⟩. ⟨hal-00819133⟩