A General Necessary Condition for Exact Observability

Abstract

Suppose A generates an exponentially stable strongly continuous semigroup on the Hilbert space X, Y is another Hilbert space, and $C:D(A) \to Y$ is an admissible observation operator for this semigroup. (This means that to any initial state in X we can associate an output function in $L^2 ([0,\infty ),Y)$.) This paper gives a necessary condition for the exact observability of the system defined by A and C. This condition, called $({\bf E})$, is related to the Hautus Lemma from finite dimensional systems theory. It is an estimate in terms of the operators A and C alone (in particular, it makes no reference to the semigroup). This paper shows that $({\bf E})$ implies approximate observability and, if A is bounded, it implies exact observability. This paper conjectures that $({\bf E})$ is in fact equivalent to exact observability. The paper also shows that for diagonal semigroups, $({\bf E})$ takes on a very simple form, and applies the results to sequences of complex exponential functions.