Energy decay problems in the design of a point stabilizer for coupled string vibrating systems

This paper solves completely the following problem posed by Chen, Coleman, and West [SIAM J. Appl. Math., 47, pp. 751–780]: When can we achieve the desirable uniform exponential decay property of the vibration energy for two coupled vibrating strings with a stabilizer or damper installed at the coupling point? An approach using abstract semigroups is shown. The paper proves the following: For the cases when the stabilizers are “symmetrically placed,” the system’s energy does not decay to zero with respect to time for some initial states. For all the other cases, if the proportion of the wave speeds on the two vibrating strings ${{c_2 } / {c_1 }}$, is a rational number, applying the formula of the growth order of a $C_0$-semigroup on a Hilbgrt space, derived by Huang [2], proves that the semigroup satisfies the Spectrum-Determined Growth Assumption, so the uniform exponential decay property depends completely on the representative form of the ratio ${{c_2 } / {c_1 }}$. Finally, the paper proves that if ${{...