Admissible Controllability of Vibrating Systems with Constrained Controls

We study the control of a vibrating string when the controls $f(x,t)$ are applied only on a limited subset of the string and constrained by continuous functions $\alpha (x)$, $\beta (x)$ as $\alpha (x) \leqq f(x,t) \leqq \beta (x)$. We find that it is possible to transfer almost all states of vibration into rest by a constrained control. The higher space dimensional vibrating systems are also considered.