J. Evol. Equ. 9 (2009), 103–121 © 2009 Birkhäuser Verlag Basel/Switzerland 1424-3199/09/010103-19, published online February 25, 2009 DOI 10.1007/s00028-009-0004-z Journal of Evolution Equations Feedback stabilization of a class of evolution equations with delay E. M. Ait Benhassi, K. Ammari ∗ , S. Boulite and L. Maniar Abstract. In this paper, we characterize the stabilization of some delay systems. The proof of the main result uses the method introduced in Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the correspond- ing uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system. 1. Introduction and main results Controllability and stabilizability of second order infinite dimensional systems has attracted a lot of interest in recent years. Direct methods, as multiplier and spectral analysis techniques, have been applied [8, 12, 13, 17, 19–24, 30]. Ammari and Tucsnak [1] introduced a method where the exponential stability for the closed loop second order system is reduced to an observability inequality for the corresponding uncon- trolled system combined with a boundedness property of the transfer function of the associated open loop system. This method has been used later for the stabilization of various partial differential equations, see, e.g., [1–6, 11, 26, 28]. Recently, Nicaise–Pignotti adopted in [25, 26] the same method for the wave equa- tion with boundary or interior delay. More precisely, they showed that if the coefficient of the delayed damping term is smaller than the one of the undelayed damping term, the observability implies the exponential decay of some energy function. In the opposite case, it has been shown that some delays can destabilize these systems. These results have been obtained in the case of one dimensional systems in [29] by a direct spectral method. The robustness of the time delay in the boundary or the internal stabilization of the wave and other equation, has been studied in [7, 8, 14–16]. In this paper, using the method of [1], we characterize the stabilization of a class of abstract second order evolution equations with delay, generalizing previous works. Let H be a Hilbert space equipped with the norm ||.|| H , and let A : D( A) → H be a self-adjoint, positive and invertible operator. We introduce the scale of Hilbert spaces Mathematics Subject Classification (2000): 34K06, 34K20, 35L05, 35L10, 47D06, 93D15 Keywords: Systems with delay, Feedback stabilization, Observability inequality, Exponential decay, Semigroup. ∗ The author would like to thank the Institute of Research for Development of France (IRD) and LMDP, the Laboratory of Mathematics and Dynamic of Populations in Marrakesh, for supporting his visit.