A note on a class of observability problems for PDEs

The question of observability arises naturally in the analysis of control problems. If the solution of a PDE initial-boundary value problem is known to be zero in a part of the domain, does this guarantee it is zero everywhere? The most popular techniques to establish such results are based on local unique continuation results (Holmgren’s theorem) or Carleman estimates. The purpose of this note is to draw attention to a class of problems where the observed region is bounded by characteristics, and local unique continuation fails. Nevertheless, observability may hold. A problem of this nature arose in recent work by the author on control of viscoelastic flows [M. Renardy, Are viscoelastic flows under control or out of control? System Control Lett. 54 (2005) 1183–1193]. In this note, we first analyze a simple example which shares the same essential features. Specifically, we consider the problem uxt=αuuxt=αu, for spatially periodic solutions. We show that observability holds for data given on the line x=0x=0. We shall show, however, that there is no observability estimate. We shall then show how the methods used in the more elementary example can be extended to the case of viscoelastic shear flows.