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This paper deals with the stabilization of an anti-stable string equationwith Dirichlet actuation where the instability appears because of theuncontrolled boundary condition. Then, infinitely many unstable poles aregenerated and an infinite dimensional control law is therefore proposed toexponentially stabilize the system. The idea behind the choice of thecontroller is to extend the domain of the PDE so that the anti-damping term iscompensated by a damping at the other boundary condition. Additionally, noticethat the system can then be exponentially stabilized with a chosen decay-rateand is robust to uncertainties on the wave speed and the anti-dampedcoefficient of the wave equation, with the only use of a point-wise boundarymeasurement. The efficiency of this new control strategy is then compared tothe backstepping approach.

Stabilization of an unstable wave equation using an infinite dimensional dynamic controller