A Note on Controllability and non-Controllability for a Rayleigh Beam with Piezoelectric Actuator^*

In this paper, the exact controllability problem for a Rayleigh beam with piezoelectric actuator is considered. The main contributions of this work are to give the exact controllability results and to give the minimal controllability time. Controllability results show that the space of controllable initial data depends on the location of the actuator. The approach to prove controllability results is based on Hilbert Uniqueness Method and some results on the theory of Diophantine approximation. Due to the rotary inertia term in the Rayleigh beam equation, Rayleigh beam equation possesses finite propagation speed, and consequently the controllability results hold when the control time surpasses a critical time. This critical time is proved to be the minimal controllability time by using the Riesz basis property of exponential family in L2(0,T). The controllability in critical time is still an open problem.*The full version of the paper can be accessed at https://hal.science/hal-03964846