Observability on Noncompact Symmetric Spaces

The \classical case" is the case in which X is a compact riemannian manifold and D is the (positive de nite) Laplacian. Then (1.1) is the heat equation on X . In this paper we'll look at the special case where X is a riemannian symmetric space of noncompact type. Thus X is a noncompact riemannian manifold with a very large symmetry group G, harmonic analysis on X is understood in terms of the structure of G, and the operator D is G{invariant. The idea is to use some geometry and group structure to guide methods of observation, control and quadrature.