A note on partial exact controllability for spherical membranes

We describe some of our (1997) recently obtained results. We consider the axially symmetric vibrations of a spherical membrane described by a pair of coupled partial differential equation in the meridional and radial displacement. Then we consider the equivalent problem of an integro-differential equation involving only the meriodional displacement, and state the partial exact controllability problem. To solve it, we follow the reachability Hilbert uniqueness method. As for the invertibility of the associate operator we derive a result concerning non-harmonic Fourier series. This result generalizes a theorem due to Ingham (1936) on trigonometric inequalities for almost period functions. The assumptions we made on this abstract theorem are motivated by the problem of partial controllability for spherical membranes. Indeed these results leads to solve the partial exact controllability problem for spherical membranes.