Control of a network of Euler–Bernoulli beams

The aim is to study the boundary controllability of a system modeling the vibrations of a network of N Euler–Bernoulli beams connected by n vibrating point masses. Using the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds as soon as the roots of a function denoted by f∞ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a network of N=2 different beams, this assumption on the multiplicity of the roots of f∞ (denoted by (A)) is proved to be satisfied and controllability follows. For higher values of N, a numerical approach allows one to prove (A) in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open.