Exact controllability of a Rayleigh beam with a single boundary control

We prove exact boundary controllability for the Rayleigh beam equation phi(tt) - alpha phi(ttxx) + A phi(xxxx) = 0, 0 < x <l, t > 0 with a single boundary control active at one end of the beam. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment phi(xx) (l,t) or the rotation angle phi(x) (l,t)at an end of the beam. In each case, exact controllability is obtained on the space of optimal regularity for L (2)(0, T) controls for L-2(0, T) controls for T > 2l root alpha/a). In certain cases, e.g., the clamped case, the optimal regularity space involves a quotient in the velocity component. In other cases, where the regularity for the observed problem is below the energy level, a quotient space may arise in solutions of the observed problem.