Exact Truncation and Radial Rigidity in Time-Optimal Control

Abstract

We consider minimum-time control for the linear system ż(t)=Az(t)+Bu(t), |u|_{L^∞(0,∞;ℝ^m)}≤ 1, with A∈ℝ^{n× n} and B∈ℝ^{n× m}. While the individual point-target and ball-target problems are classical, we study a different question: when are their optimal controls exactly compatible, in the sense that, for every nonzero initial state x and all sufficiently small ε>0, the optimal control for the tolerance ball B̅_ε(0) is precisely the restriction of the point-target optimal control? We prove that this {\it{exact truncation}} property is equivalent to the rigidity conditions B B^⊤=βI_n, A+A^⊤=2aI_n, β>0, a≤ 0, and also to Euclidean radiality of the point-target minimum-time function. Thus exact truncation holds precisely when the sublevel sets of the point-target minimum-time function are Euclidean balls centered at the origin, matching the geometry of the tolerance targets. Moreover, the local property automatically extends to every 0<ε<|x|, and the resulting optimal times and point-target optimal feedback are both explicit.

On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems
Lida Lamakani, Efstratios N. Pistikopoulos
arXiv23 Oct 2026PreprintarXiv licenceFull textOptimisation & controlarXiv
Optimal Time-Dependent Jump Truncation for Time-Singular Lévy Processes
Victoria Knopova, Denis Platonov
arXiv14 Sep 2026PreprintarXiv licenceFull textProbabilityarXiv
Radial Projectively Induced Canonical Kähler Metrics: Rigidity and Classification
Claudio Arezzo, Andrea Loi, Giovanni Placini +1
arXiv8 Sep 2026PreprintCC BY-NC-ND 4.0Full textGeometry & topologyarXiv
Rigidity of Kähler-Finsler Metrics with Constant Radial Curvature
Jinling Li, Longjie Li, Chunhui Qiu
arXiv29 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Radial Marchenko-Pastur laws: projection characterizations and rigidity
Xiaohui Xie
arXiv8 Sep 2026PreprintarXiv licenceFull textProbability StatisticsarXiv

Cite

BibTeX RIS

BibTeX

@misc{quan2026exact,
  title = {{Exact Truncation and Radial Rigidity in Time-Optimal Control}},
  author = {Changqin Quan and Gengsheng Wang and Lijuan Wang and Qishu Yan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22802},
  eprint = {2609.22802},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22802},
}

RIS

TY  - UNPB
TI  - Exact Truncation and Radial Rigidity in Time-Optimal Control
AU  - Changqin Quan
AU  - Gengsheng Wang
AU  - Lijuan Wang
AU  - Qishu Yan
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22802
UR  - https://arxiv.org/abs/2609.22802
AB  - We consider minimum-time control for the linear system ż(t)=Az(t)+Bu(t), |u|_L^∞(0,∞;ℝ^m)≤ 1, with A∈ℝ^n× n and B∈ℝ^n× m. While the individual point-target and ball-target problems are classical, we study a different question: when are their optimal controls exactly compatible, in the sense that, for every nonzero initial state x and all sufficiently small ε>0, the optimal control for the tolerance ball B̅_ε(0) is precisely the restriction of the point-target optimal control? We prove that this {\it{exact truncation}} property is equivalent to the rigidity conditions B B^⊤=βI_n, A+A^⊤=2aI_n, β>0, a≤ 0, and also to Euclidean radiality of the point-target minimum-time function. Thus exact truncation holds precisely when the sublevel sets of the point-target minimum-time function are Euclidean balls centered at the origin, matching the geometry of the tolerance targets. Moreover, the local property automatically extends to every 0<ε<|x|, and the resulting optimal times and point-target optimal feedback are both explicit.
ER  -

Abstract

Related

Cite