I'm looking for references about a particular derivation of the reachability/controllability Gramian. Consider a linear system given by $$ \dot{\textbf{x}} = A\textbf{x} + B\textbf{u} $$ In linear systems theory, the reachability set from the $0$ inital state to time $t$ is given by the set:
$$ R(\textbf{0}, 0, t) = \{ \textbf{x} \in \mathbb{R}^{n} : \textbf{x} = \int_{0}^{t}{ e^{A(t-\tau)}Bu(\tau) d\tau } \hspace{0.5cm} \forall u \in \mathcal{U} \} $$ Where $\mathcal{U}$ is the set of the input functions $u(\cdot) : [0, t] \rightarrow \mathbb{R}^m $, with a inner product space structure given by: $$ <u, v> = \int_{0}^{t} { u(\tau)^{T} v(\tau) d\tau } $$
Now, define the function $\phi_f(u(\cdot)) : \mathcal{U} \rightarrow \mathbb{R}^n$ that associates an input function with the state $x(t)$ (result of the forced evolution of the system): $$ \phi_f(u(\cdot)) = \int_{0}^{t}{ e^{A(t-\tau)}Bu(\tau) d\tau } $$
$\phi_f$ is a linear operator and the reachability set can be rewritten as its image: $ R(\textbf{0}, 0, t) = \textit{im} \{ \phi_f \} $.
Now, $\phi_f$ is a bit cumbersome to use because the domain is an infinite dimensional vector space and no matrix for it can be written (as far as I understand).
Here comes the theory for which I'm looking for: define the adjoint operator of $\phi_f$, $\phi_f^* : \mathbb{R}^n \rightarrow \mathcal{U}$, as the unique operator such that:
$$ {\langle x, \phi_f(u(\cdot)) \rangle} = {\langle \phi_f^*(x), u \rangle} $$
for every $u$ and $x$. Developing the inner product, one finds that the adjoint has a closed form: $$ \phi_f^*(\textbf{x}) = t \rightarrow B^T e^{A^T(t-\tau)} \textbf{x} $$
The key result is that $$ \textit{im} \{ \phi_f \} = \textit{im} \{ \phi_f \phi_f^* \} $$ It's an easy result to prove but it would not have occured to me (an analogous thing can be done with the kernel of the composite, which can be used to prove the form of the observability Gramian).
Exploiting this fact, we have:
$$ R(\textbf{0}, 0, t) = \textit{im} \{ \phi_f \phi_f^* \} = \textit{im} \{ \int_{0}^{t}{ e^{A(t-\tau)}BB^T e^{A^T(t-\tau)} d\tau } \} = \textit{im} \{ G_R(0, t) \} $$
Where $G_R(0, t)$ is the reachability Gramian, which is a square matrix.
Another important fact is that $\mathbb{R}^n$ can be expressed as an orthogonal sum: $$ \mathbb{R}^n = \textit{im} \{ \phi_f^* \} \bigoplus \textit{ker} \{ \phi_f \} $$
And $u(t) = \phi_t^*(G_R^{-1}\textbf{x})$ is fully contained in the first subspace, therefore it is the least-energy control to force the system from $x(0)=\textbf{0}$ to $x(t)=\textbf{x}$.
Tracing back on the various reference, I found this concepts in the book by Desoer and Callier "Linear System Theory", which cites back to the original Kalman papers which, AFAICT, don't have a similar treatment of the reachability Gramian.
My question is:
- Can I find the first place where this kind of derivation was introduced?
- Is there any systematic treatement about linear algebra/linear operators that also gives the results about $\textit{im} \{ \phi_f \} = \textit{im} \{ \phi_f \phi_f^* \}$? (So, not strictly related to control/linear systems theory)