A General Necessary Condition for Exact Observability

• Authors: David L. Russell;George Weiss
• Venue: SIAM Journal on Control and Optimization
• Year: 1994

Suppose $A$ generates an exponentially stable strongly continuous semigroup on the Hilbert space $X,Y$ is another Hilbert space, and $C : D(A) \rightarrow Y$ 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.