Closed-region PBH rank tests characterize continuous and discrete stabilizability and detectability

Necessity in all four assertions is exactly [F2]. This proves the forward implications from stabilizability or detectability to the stated rank tests, including every point on each closed boundary.

Fix either forbidden closed region $\mathrm{\Omega }=\{\lambda :\mathrm{Re}\lambda \ge 0\}$ or $\mathrm{\Omega }=\{\lambda :\mathrm{\mid }\lambda \mathrm{\mid }\ge 1\}$. Suppose the horizontal rank test holds for all $\lambda \in \mathrm{\Omega }$, and apply [F3] to $(A,B)$ over the original field $F$. The uncontrollable block $A_u$ has no eigenvalue in $\mathrm{\Omega }$, so its eigenvalues lie in the open left half-plane or open unit disk, respectively. Assign the reachable block repeated pole $-1$ or $0$ by [F3], obtaining $K_r$. With its decomposition matrix $P$, set $K=[K_{r}0]P^{-1}$; if $r=0$ take $K=0$, and if $r=n$ omit the empty trailing block. Then $P^{-1}(A-BK)P=\left[\begin{array}{cc}{A}_r-B_r{K}_r& D\\ 0& A_u\end{array}\right]$. By [F4] its eigenvalues are the assigned stable poles and those of $A_u$, with multiplicity; hence $A-BK$ is Hurwitz or Schur stable, and $K\in F^{m\times n}$.

For every $\lambda \in \mathbb{C}$, ${\mathrm{rank}}_{\mathbb{C}}[\bar{\lambda }I-A^{\ast }{C}^{\ast }]={\mathrm{rank}}_{\mathbb{C}}\left[\begin{array}{c}\lambda I-A\\ C\end{array}\right]$ because these matrices are adjoints and [F5] applies. Each forbidden region is invariant under conjugation, so its vertical test for $(A,C)$ is the horizontal test for $(A^{\ast },C^{\ast })$. Step 1.2 supplies $J\in F^{p\times n}$ with $A^{\ast }-C^{\ast }J$ Hurwitz or Schur stable. Its adjoint $A-J^{\ast }C$ has conjugate eigenvalues with the same real parts and moduli, so $L=J^{\ast }\in F^{n\times p}$ is the required injection. This proves both detectability converses, including their boundary cases.