This paper deals with the well-posedness of a linear closed-loop system with an explicit feedback law previously introduced by Komornik. His method covers the case of boundary control systems and leads to arbitrarily large decay rates. We define a mild solution of the closed-loop problem using a dual problem and we prove that the original operator perturbed by the feedback is (up to the use of an extension) the infinitesimal generator of a strongly continuous group. We also give a justification of the exponential decay of the solutions. Our proofs do not use the optimal control theory through the minimization of a cost functional but exploit directly the algebraic Riccati equation satisfied by an operator involved in the feedback.