Strong Stability of an Unstable Wave Equation by Boundary Feedback With Only Displacement Observation

The stabilization of a 1-D wave equation that contains instability at its free end and control at the other end is considered. The controller is designed through the estimated state that is designed in the case that only displacement is available. The method of “backstepping” is adopted in investigation. The
C
₀
-semigroup theory and Lyapunov method are used to show that the closed-loop system is asymptotically stable.