Control of a Clamped-free Beam by a Piezoelectric Actuator

We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations. Mathematics Subject Classification. 93C20, 35B75, 35B60. Received September 27, 2004. Revised May 23, 2005. Introduction There exists now a large literature concerning the study of flexible structures in interaction with piezoelectric actuators. Two directions of research can be exhibited. The modeling problem and the controllability problem. A large number of models have been proposed and many dynamical systems modelizing a flexible structure with a piezoelectric actuator can be found in the literature; e.g. model based on finite elements (see [2, 12]), finite-dimensional approximations (see [11, 21, 22]) and also partial differential equations (see [6, 7]). Concerning controllability results, there exists a wide literature. See e.g. [18] where the eigenvalue specification problem is studied for the Euler-Bernoulli beam (with the same boundary condition but with different control than in this paper). See also [8, 20] where flatness technics are used for approximate controllability problems. To the best to our knowledge, the only exact controllability result for a beam with a piezoelectric controller is due to [23], where a beam with hinged boundary conditions is considered. The main purpose of our paper is to study the exact controllability of a beam in a more physical configuration: the clamped-free boundary conditions, i.e. a beam clamped at one end and free at the other end. There exist technological and industrial (see [15]) motivations to study the control of beams with piezoelectric actuators with such boundary conditions. In this paper we use the Hilbert Uniqueness Method (HUM) which is a classical approach to look for a controllability result (see [16]) combined with new results of the theory of diophantine approximations. With such physical conditions, the controllability problem is a completely different issue from [23] although the method used in this paper and in [23] is the same one (namely HUM). Indeed the computations and the diophantine approximations used in [23] and in this paper are slightly different and here we use some new results in the theory of diophantine approximations.