Exact controllability for a Rayleigh beam with piezoelectric actuator

In this paper, exact controllability problem for a Rayleigh beam with piezoelectric actuator is considered. Controllability results show that the space of controllable initial data depends on the regularity of the control and the location of the actuator. Two different spaces of control, L2(0,T) and (H1(0,T))′, correspond to two different controllability properties, L2-controllability and (H1)′-controllability, respectively. The approach to prove controllability results is based on the Hilbert Uniqueness Method. Some non-controllability results are also obtained. In particular, non-controllability in short control time is studied by using Riesz basis property of exponential family in L2(0,T). Finally, minimal control time for exact controllability is obtained.