In this note, I review a function analysis perspective for the controllability–observability duality. This is often stated just as a notational operation as below, despite the interesting depth:

$$ (A,C)\text{ observable} \quad\Longleftrightarrow\quad (A^\top,C^\top)\text{ controllable}. $$

In fact, this comes from the famous relationship between the null space of a linear operator and the range of its adjoint on a Hilbert space, together with the existence of a time reversal procedure that is sometimes overlooked. Let’s see the interesting relation!

(See also my previous note )

Observability from the operator perspective

Consider an autonomous system

$$ \dot x = A x,\qquad x(0)=x_0,\qquad y=Cx,\qquad t\in[0,T]. $$

For this system, define the observation map from the initial point:

$$ \Phi:\mathbb{R}^n \to L_2[0,T], \qquad (\Phi x_0)(t) = C e^{At} x_0. $$

This operator captures the relation between the initial point and the output. Now, recall the definition of observability: This then means we can determine $x_0$ from the output. In other words, different initial states produce different outputs (injective):

$$ \Phi x_0 = \Phi \tilde x_0 \;\Rightarrow\; x_0=\tilde x_0, $$

i.e. the null space $\mathcal{N}(\Phi)=\{x\mid\Phi x=0\}$ of $\Phi$ is $\mathcal{N}(\Phi)=\{0\}.$

So, it turns out that observability $\iff$ $\mathcal{N}(\Phi)=\{0\}.$

The adjoint $\Phi^*$ and controllability of the adjoint system

We can compute the adjoint operator $\Phi^$ , i.e., $\langle \Phi x,\eta\rangle=\langle x,\Phi^ \eta \rangle$. In particular, the adjoint operator is reduced to

$$ \Phi^* v \;=\; \int_0^T e^{A^\top \tau} C^\top v(\tau)\,d\tau. $$

This has a dynamical interpretation: $\Phi^* v$ is the value $\xi(0)$ obtained by solving the backward system

$$ -\dot \xi = A^\top \xi + C^\top v, \qquad \xi(T)=0. $$