Regularity and Exact Controllability for a Beam with Piezoelectric Actuator

We consider an initial and boundary value problem modelling the vibrations of a Bernoulli–Euler beam with an attached piezoelectric actuator. We show that the Sobolev regularity of the solution is by $\frac{1}{2} + \epsilon $ higher than that one obtains by simply using the Sobolev regularity of the control term. The main results concern the dependence of the space of exactly controllable initial data on the location of the actuator. Our approach is based on the Hilbert uniqueness method introduced by Lions [Contrôlabilité exacte des systèmes distribués, Masson, Paris, 1988] combined with some results from the theory of diophantine approximation.

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