Generic static-output stabilizability

Definitions and mathematical statement

Fix positive integers $m,p$ and put $n=mp$. Consider real matrices $A\in\mathbb R^{n\times n}$, $B\in\mathbb R^{n\times p}$, and $C\in\mathbb R^{m\times n}$ defining \[ \dot x=Ax+Bu,\qquad y=Cx. \] Here $x\in\mathbb R^n$ is the state, $u\in\mathbb R^p$ the input, and $y\in\mathbb R^m$ the measured output. A static output feedback is a constant real matrix $K\in\mathbb R^{p\times m}$ with $u=Ky$, producing the closed-loop matrix $A+BKC$.

The system is stabilizable by this feedback if some such $K$ makes every eigenvalue of $A+BKC$ have strictly negative real part. Equivalently, all roots of $\det(zI-A-BKC)$ lie in the open left half-plane. There is no bound on the size of $K$, but dynamic or time-dependent feedback is not permitted.

Determine for which pairs $(m,p)$ this stabilizability property holds generically for triples $(A,B,C)$ of the indicated sizes. Here generically means on an open dense subset of their real Euclidean parameter space. A single unstabilizable exceptional triple does not refute generic stabilizability; an open set of such triples does. The target is existence of some stable closed-loop spectrum, not assignment of arbitrary prescribed poles. The equality $n=mp$ is fixed throughout and is the critical dimension condition in this problem.

Short English statement

At the critical state dimension, which input and output dimensions make a generic real system stabilizable by static output feedback?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700044 (AMR-036-0044), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Static output feedback question. Original source indexed by AMR-036-0044. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Stabilizability by static output feedback. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/gpolep.pdf.