In the present article we study the stabilization of first-order linear-differential hyperbolic equations. For such equations we prove that the in finite time is equivalent to the exact controllability. The proof relies on a Fredholm transformation that maps the original into a finite-time stable target system. The controllability assumption used to prove the invertibility of such a transformation. Finally, using the of moments, we show in a particular case that the controllability is to the criterion of Fattorini.
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Coron et al. (2015) studied this question.