On the Stabilization of A Flexible Cable with Boundary Feedback
In this paper we study the stability of a flexible cable that is clamped at one end and free at the other. To stabilize this system we apply a control force in position and velocity at the free end of the cable. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system. Using a method due to Shkalikov we prove that the spectrum determines the exponential decay rate of the energy under certain conditions.
Hyperbolic boundary value problem; Exponential stability; Asymptotic behavior; Semigroup theory
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