$\textbf{Introduction}$: I study linear control theory. Among strategies, we begin with vector field $Ax + Bu$, $A \in M_{n^2}(\mathbb{R})$, $B \in M_{n \times m}(\mathbb{R})$, and synthesize a controller $u$ given by product $F x$ such that matrix $A+BF$ has all eigenvalues on the left complex plane. For that, take a look at a brief explanation I write on [1] at the end of this post.
$\textbf{Explanation}$: My supervisor asked me if I could stabilize the same linear system with dynamic feedback $\dot{u}$ of kind $Fx + Gu$ i.e. synthesize matrices $F$ and $G$. My progress so far is to recognize matrix $\bar{A} = \begin{bmatrix} A & B \\ F & G\end{bmatrix}$ needs to have eigenvalues on the left complex plane OR there are positive symmetric matrices $\bar{P}$ and $\bar{Q}$ such that we satisfy equality $\bar{A}^\intercal \bar{P} + \bar{P} \bar{A} = -\bar{Q}$. He suggested finding such matrix $\bar{P}$, since it may be possible to extend to nonlinear systems.
$\textbf{Question}$: How to synthetize matrices $F$ and $G$ to stabilize linear system $\begin{bmatrix} \dot{x} \\ \dot{u} \end{bmatrix} = \begin{bmatrix} A & B \\ F & G\end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix}$
$\textbf{Appendix}$ Hello :). This is a summary of feedback stabilization $F x$. Matrix $F$ is given by formula $-P V^{-1}$, such that matrix P represents $n$ vectorial degrees of freedom for eigenvectors to follow and matrix of eigenvectors $V$ has column $v_i$ given as product $(\lambda_i I_n - A)^{-1} \, B \, p_i$, for eigenvalue $\lambda_i \not \in \lambda(A)$, or respective eigenvector associated to eigenvalue $\lambda_i$ i.e. it satisfies equality $(\lambda_i I - A) v_i = 0$. There are other corner cases such as not-controllable eigenvalues i.e. not-shiftable, whose eigenvectors $v_i$ and parameter $p_i$ are still related by the formula $(\lambda_i I - A) v_i - B p_i$, we conclude vector $p_i$ is 0, eigenvalue $\lambda_i$ must be on the left complex plane
