Exact controllability of a Rayleigh beam with a single boundary control

We prove exact boundary controllability for the Rayleigh beam equation $${\varphi_{tt} -\alpha\varphi_{ttxx} + A\varphi_{xxxx} = 0, 0 < x < l, t > 0}$$ 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 $${\varphi_{xx}(l, t)}$$ or the rotation angle $${\varphi_{x}(l, t)}$$ at an end of the beam. In each case, exact controllability is obtained on the space of optimal regularity for L2(0, T) controls for $${T > 2l\sqrt{\frac{\alpha}{A}}}$$. 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.