VIBRATION CONTROL OF AN AXIALLY MOVING STRING BY BOUNDARY CONTROL

The stabilization of the transverse vibration of an axially moving str ing is implemented using time-varying control of either the boundary t ransverse motion or the external boundary forces. The total mechanical energy of the translating string is a Lyapunov functional and boundar y control laws are designed to dissipate the total vibration energy of the string at the left and/or right boundary. An optimal feedback gai n determined by minimizing the energy reflected from the boundaries, i s the ratio of tension to the propagation velocity of an incident wave to the boundary control Also the maximum time required to stabilize a ll vibration energy of the system for any initial disturbance is the r ime required for a wave to propagate the span of the string before hit ting boundary control. Asymptotic and exponential stability of the axi ally moving string under boundary control are verified analytically th rough the decay rare of the energy norm and the use of semigroup theor y. Simulations are used to verify the theoretically predicted, optimal boundary control for the stabilization of the translating string.