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PreprintCC BY 4.0arXiv

The Global Asymptotic Stability Problem for Linear MPC Is Undecidable

  • Johan Löfberg
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Abstract

We prove that deciding global asymptotic stability for constrained finite-horizon linear model predictive control is undecidable. This holds at horizon one with identity state, input, and terminal weights, unique optimizers, and global feasibility. Separate reductions cover predicted-state boxes, hard input boxes, and quadratically softened input boxes. A fourth reduction fixes the state and input dimensions to three and six. Hence undecidability is not caused by long horizons, growing dimensions, failures of recursive feasibility, or nonuniqueness.

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Cite

BibTeXRIS
BibTeX
@misc{lofberg2026global,
  title = {{The Global Asymptotic Stability Problem for Linear MPC Is Undecidable}},
  author = {Johan Löfberg},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09930},
  eprint = {2609.09930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09930},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Global Asymptotic Stability Problem for Linear MPC Is Undecidable
AU  - Johan Löfberg
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09930
UR  - https://arxiv.org/abs/2609.09930
AB  - We prove that deciding global asymptotic stability for constrained finite-horizon linear model predictive control is undecidable. This holds at horizon one with identity state, input, and terminal weights, unique optimizers, and global feasibility. Separate reductions cover predicted-state boxes, hard input boxes, and quadratically softened input boxes. A fourth reduction fixes the state and input dimensions to three and six. Hence undecidability is not caused by long horizons, growing dimensions, failures of recursive feasibility, or nonuniqueness.
ER  -