Stabilizability and detectability for continuous and discrete linear systems

Let $F\in \{\mathbb{R},\mathbb{C}\}$, $n,m,p\ge 1$, $A\in F^{n\times n}$, $B\in F^{n\times m}$, and $C\in F^{p\times n}$. A matrix is Hurwitz when every complex eigenvalue has strictly negative real part, as in Hurwitz matrices and the continuous Lyapunov equation. It is Schur stable when every complex eigenvalue has modulus strictly less than one.

The pair $(A,B)$ is continuously stabilizable if some $K\in F^{m\times n}$ makes $A-BK$ Hurwitz. The pair $(A,C)$ is continuously detectable if some $L\in F^{n\times p}$ makes $A-LC$ Hurwitz.

The pair $(A,B)$ is discretely stabilizable if some $K\in F^{m\times n}$ makes $A-BK$ Schur stable. The pair $(A,C)$ is discretely detectable if some $L\in F^{n\times p}$ makes $A-LC$ Schur stable. These are exact existential properties of the given finite matrices. A residual size or an approximate spectral estimate does not itself establish any of them.