Controllability of polynomial systems is undecidable and not recursively enumerable
Jun Liu, Ivan O. Shevchenko · arXiv: 2610.26832v1 · Summary page
Why cover this story this week?
When control cannot be certified by an algorithm
Can a system be steered from any starting state to any desired destination? For systems governed by polynomial equations with rational coefficients, a new preprint proves that no algorithm can always answer that question. More surprisingly, there is not even an algorithm that could eventually list all the globally controllable systems, despite being allowed to run indefinitely. This confirms the undecidability part of Sontag’s conjecture while overturning its proposed escape route.
The proof translates a question about stability into a question about control. The authors explicitly construct a control system that is globally controllable exactly when the origin of an associated polynomial flow is not globally asymptotically stable. An existing result about the computational difficulty of stability then supplies the obstruction. The conclusion also holds when inputs enter through a constant matrix: even that familiar engineering form does not remove the fundamental barrier to universal certification.
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