Summary: Let H be a real Hilbert space, A an unbounded, self-adjoint, positive and coercive linear operator on H, and B a bounded linear operator on H such that \(B=B*\geq 0\). We establish a logical equivalence between the exponential decay of solutions to the second order evolution equation \(y''+A\) \(y(t)+B\) \(y'(t)=0\), uniformly on bounded subsets of \(D(A^{1/2})\times H\) and a “\(B^{1/2}\)-controllability” property of the system governed by the undamped equation \(y''+Ay(t)=0\) on some time interval. This abstract remark is applied to various problems of hyperbolic type of well-posed in the sense of Petrowski.