Observability of Compactly Perturbed Systems

We propose a general and constructive method for the study of observability of weakly coupled or, more generally, compactly perturbed linear distributed systems. This approach allows us to avoid the usual indirect compactness]uniqueness arguments in many cases. Our method is based on two observations. First, in many cases there exists a Riesz basis formed by ordinary and generalized eigenfunctions of the underlying operator of the evolutionary equation. Second, in the usual estimates some constants depend on the initial data in a rather particular way. This allows us to write the solution as a trigonometric sum with vector coefficients and then to apply a generalization of an estimation method introduced by Haraux for the study of scalar nonharmonic series. Applying this approach we improve and generalize some former theorems of Lions, Haraux, and Jaffard