  This paper deals with the stabilization of an anti-stable string equation
with Dirichlet actuation where the instability appears because of the
uncontrolled boundary condition. Then, infinitely many unstable poles are
generated and an infinite dimensional control law is therefore proposed to
exponentially stabilize the system. The idea behind the choice of the
controller is to extend the domain of the PDE so that the anti-damping term is
compensated by a damping at the other boundary condition. Additionally, notice
that the system can then be exponentially stabilized with a chosen decay-rate
and is robust to uncertainties on the wave speed and the anti-damped
coefficient of the wave equation, with the only use of a point-wise boundary
measurement. The efficiency of this new control strategy is then compared to
the backstepping approach.
