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