On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings

We consider a tripod as an exemplaric network of strings. We know that such a network is exactly controllable in the natural finite energy space, if, e.g., the simple nodes are controlled by Dirichlet controls in H^l (0, T ). Assume that we want to calculate the corresponding norm-minimal controls using semi-discretization in space. We then obtain a system of coupled second-order-in-time ordinary differential equations with three control inputs. Controllability of the latter system can easily been checked by Kalman's rank condition on each space discretization level h. One expects, as h tends to zero, that the exact controllability of the continuous system is revealed. This expectation is frustrated, as has been shown by Infante and Zuazua (1998) for a single string and by Zuazua (1999) for a membrane. Indeed, it was shown there that uniformity of observability estimates is lost in the limit. On the other hand, spectral filtering allows to cure this pathology. We show in this paper that similar results hold for our string network. The generalization to arbitrary networks of strings in the out-of-the-plane as well as in the in-plane or 3-d-setup is then a technical matter. Therefore, this paper essentially extends the existing results to semidiscretizations of wave equations on arbitrary irregular computational grids.