Generic stabilization by static output feedback
For positive integers $m, p$ put $n = mp$, and let $$\mathcal{A}_{m,p} = \mathbb{R}^{n \times n} \times \mathbb{R}^{n \times m} \times \mathbb{R}^{p \times n},$$ the space of linear systems $\dot x = Ax + Bu$, $y = Cx$ with $n$ states, $m$ inputs and $p$ outputs. A system $(A, B, C)$ is stabilizable by static output feedback if $A + BKC$ is Hurwitz for some real $m \times p$ matrix $K$; let $\mathcal{S}_{m,p} \subset \mathcal{A}_{m,p}$ be the set of such systems, which is open. For $n < mp$, Wang proved that real pole placement by static output feedback is possible for generic systems. For $n = mp$, Brockett and Byrnes proved that complex pole placement is generically possible, with degree $d(m,p)$ equal to the number of $m$-planes meeting $mp$ general $p$-planes in $\mathbb{C}^{m+p}$; so real pole placement, and hence stabilization, is generically possible when $d(m,p)$ is odd, that is, when $\min\{m,p\} = 1$, or $\min\{m,p\} = 2$ and $\max\{m,p\} + 1$ is a power of $2$. Byrnes and Anderson showed that for $m = p = 2$ a nonempty open set of systems is not stabilizable. Eremenko and Gabrielov proved that when $m$ and $p$ are both even, real pole placement fails on a nonempty open set of systems.
Problem. Determine exactly which pairs $(m, p)$ of positive integers make $\mathcal{S}_{m,p}$ dense in $\mathcal{A}_{m,p}$, in the Euclidean topology.
Source: Byrnes and Anderson (1984), generic stabilizability by static output feedback; for n = mp, Eremenko and Gabrielov (2002), §1, for even m and p, and Eremenko (2010), Question. Stated as a classification of the pairs (m, p) by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AG
- Keywords
- AIM Open Applied Problems, linear control systems, pole placement, static output feedback
References
@article{byrnes1984,
author = {Byrnes, Christopher I. and Anderson, Brian D. O.},
title = {Output feedback and generic stabilizability},
journal = {SIAM J. Control Optim.},
volume = {22},
number = {3},
year = {1984},
pages = {362--380},
doi = {10.1137/0322024},
}
@article{eremenko2002,
author = {Eremenko, Alexandre and Gabrielov, Andrei},
title = {Pole placement by static output feedback for generic linear systems},
journal = {SIAM J. Control Optim.},
volume = {41},
number = {1},
year = {2002},
pages = {303--312},
doi = {10.1137/S0363012901391913},
}
@misc{eremenko2010,
author = {Eremenko, Alexandre},
title = {Stabilizability by static output feedback},
howpublished = {Note on the author's web page of unsolved problems},
year = {2010},
url = {https://www.math.purdue.edu/~eremenko/dvi/gpolep.pdf},
}
@article{brockett1981,
author = {Brockett, Roger W. and Byrnes, Christopher I.},
title = {Multivariable {N}yquist criteria, root loci, and pole placement: a geometric viewpoint},
journal = {IEEE Trans. Automat. Control},
volume = {26},
number = {1},
year = {1981},
pages = {271--284},
doi = {10.1109/TAC.1981.1102571},
}
@article{wang1992,
author = {Wang, Xiaochang},
title = {Pole placement by static output feedback},
journal = {J. Math. Systems Estim. Control},
volume = {2},
year = {1992},
pages = {205--218},
}
@misc{aim-generic-static-output-feedback-stabilization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Generic stabilization by static output feedback}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/319-generic-static-output-feedback-stabilization.md},
note = {Entry 319 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00525, "Generic stabilization by static output feedback", clopen.solutions.