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This paper deals with feedback stabilization of second order equations of the form y tt + A 0 y + u (t) B 0 y (t) = 0, t ∈ [0, +∞[, where A 0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B 0 is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨ y t , B 0 y ⟩ H implies the strong stabilization. This result is derived from a general compactness theorem for semigroup with compact resolvent and solves several open problems.

Strong stabilization of controlled vibrating systems