In control theory, there are two different, but related, definitions about Hankel Operator, depends on the system the definition for.
For stable and minimum realization system (G = (A,B,C)), the Hankel operator is defined by
where (G|_{L_2(-infty,0]}) denotes the restriction of (G) to the subspace (L_2(-infty,0]), and (P_+) is the operator that projects a signal in (L_2(-infty,infty)) to (L_2[0,infty)) by truncation. Correspondingly, define the controllability operator (Psi_c:L_2(-infty,0] o mathbb{C}^n) by
and define the observability operator (Psi_o:mathbb{C}^n o L_2[0,infty)) by
Then it holds that
An alternative definition of Hankel operator is for unstable system. That is,
where (P_-) is the operator that projects a signal in (L_2(-infty,infty)) to (L_2(-infty,0]) by truncation. The corresponding controllability operator (ar{Psi}_c:L_2[0,infty) o mathbb{C}^n) is defined by
and observability operator (ar{Psi}_o:mathbb{C}^n o L_2(-infty,0]) is defined by
Then it also holds that
There is a systemic interpretation for controllability and observability operators. For "stable" definition, (Psi_c) just maps the input (u in L_2(-infty,0]) supported in the past to (x(0)), and (Psi_o) maps (x(0)) to the system output (y(t),tge 0), which no input applied for (tge 0).
While for "unstable" definition, it is not so intuitive. Note that (A) is anti-stable and
Then (e^{-At}x(t) = x_0 + int_{0}^{infty} e^{-A au}Bu( au)d au) and letting (t o infty) obtains
This can be obtained from another point of view that consider (x(t)) as the "initial state" of the system and (x(0)=x_0) as the "final state" such that
For sufficiently large (t), (e^{-At}x(t)) is small enough such that (x_0 approx int_{t}^{0} e^{-A au}Bu( au)d au).