Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs
Abstract
Prescribed-time controllers drive the state to the origin at a time T chosen in advance, with gains that grow without bound as t approaches T. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with |ż(t)|≤ L(t)|z(t)| satisfies |z(t)|≥|z(0)|e−∫0t L. From this: a gain profile L supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over [0,T) is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget ℬ=∫0T L the best worst-case contraction over the class is exactly e−ℬ, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes |ż|≤ L(t) ω(|z|), first arrival at the origin requires budget at least ∫0|z0|du/ω(u); arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.
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BibTeX
@misc{aal2026fundamental,
title = {{Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs}},
author = {Osama F. Abdel Aal and Jiale Chen and Jairo Viola and YangQuan Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17484},
eprint = {2610.17484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17484},
}
RIS
TY - UNPB
TI - Fundamental Limits of Prescribed-Time Regulation: Gain Budgets, Accuracy Frontiers, and Arrival Costs
AU - Osama F. Abdel Aal
AU - Jiale Chen
AU - Jairo Viola
AU - YangQuan Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17484
UR - https://arxiv.org/abs/2610.17484
AB - Prescribed-time controllers drive the state to the origin at a time T chosen in advance, with gains that grow without bound as t approaches T. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with |ż(t)|≤ L(t)|z(t)| satisfies |z(t)|≥|z(0)|e^−∫_0^t L. From this: a gain profile L supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over [0,T) is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget ℬ=∫_0^T L the best worst-case contraction over the class is exactly e^−ℬ, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes |ż|≤ L(t) ω(|z|), first arrival at the origin requires budget at least ∫_0^|z_0|du/ω(u); arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.
ER -