Multiple-point internal observability of membranes and plates

It is well known that by observing a vibrating beam at a generic given interior point during an arbitrarily small time period we may determine the whole behaviour of the string. This result is expressed as an observability inequality involving some Sobolev norms of the initial data. We show that by increasing the number of observation points we may increase the order of these Sobolev norms. Analogous results are obtained for vibrating strings, and the results are generalized for arbitrary space dimension if the underlying domain is a product of intervals. We also investigate the optimality of the Sobolev norms.