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Endpoint Asymptotics of Optimal Stabilization Prefactors

  • Changqin Quan
  • Gengsheng Wang
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Abstract

Consider the stabilizable finite-dimensional linear control system ẋ=Ax+Bu. For a prescribed decay rate δ>0, we define the optimal stabilization prefactor Ĉ(δ) := infK∈ℝm× n supt≥0eδt|e(A+BK)t|. We determine its asymptotic behavior as δ approaches the right endpoint of the achievable decay-rate range. If (A,B) is controllable and μ is its largest controllability index, then Ĉ(δ)asympδμ−1 as δ→+∞; whereas if (A,B) is stabilizable but not controllable, then Ĉ(δ)asymp(δ*−δ)−(q−1) as δ↑δ*, where δ* is the supremal achievable decay rate and q is the largest size of a Jordan block of the uncontrollable part associated with the spectral boundary Reλ=−δ*. Thus, in both cases, the endpoint asymptotic order of Ĉ(δ) is determined by the corresponding structural invariant---μ in the controllable case and q in the noncontrollable case---and conversely this order recovers that invariant.

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Cite

BibTeXRIS
BibTeX
@misc{quan2026endpoint,
  title = {{Endpoint Asymptotics of Optimal Stabilization Prefactors}},
  author = {Changqin Quan and Gengsheng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05863},
  eprint = {2609.05863},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05863},
}
RIS
TY  - UNPB
TI  - Endpoint Asymptotics of Optimal Stabilization Prefactors
AU  - Changqin Quan
AU  - Gengsheng Wang
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.05863
UR  - https://arxiv.org/abs/2609.05863
AB  - Consider the stabilizable finite-dimensional linear control system ẋ=Ax+Bu. For a prescribed decay rate δ>0, we define the optimal stabilization prefactor Ĉ(δ) := inf_K∈ℝ^m× n sup_t≥0e^δt|e^(A+BK)t|. We determine its asymptotic behavior as δ approaches the right endpoint of the achievable decay-rate range. If (A,B) is controllable and μ is its largest controllability index, then Ĉ(δ)asympδ^μ−1 as δ→+∞; whereas if (A,B) is stabilizable but not controllable, then Ĉ(δ)asymp(δ_*−δ)^−(q−1) as δ↑δ_*, where δ_* is the supremal achievable decay rate and q is the largest size of a Jordan block of the uncontrollable part associated with the spectral boundary Reλ=−δ_*. Thus, in both cases, the endpoint asymptotic order of Ĉ(δ) is determined by the corresponding structural invariant---μ in the controllable case and q in the noncontrollable case---and conversely this order recovers that invariant.
ER  -