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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.
We study the boundary controllability of a nonlinear Korteweg–de Vries equation with the Dirichlet boundary condition on an interval with a critical length for which it has been shown by Rosier that the linearized control system around the origin is not controllable. We prove that the nonlinear term gives the local controllability around the origin.
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