arXiv subjects

Changqin Quan

Publications and source records attributed to Changqin Quan.

3 recordsLinked to original sources

arXiv2026

Asymptotic Growth of Optimal Stabilization Prefactors for the Schrödinger Equations

We study the asymptotic behavior of the optimal stabilization prefactor for the free Schrödinger equation, optimized over all bounded stationary linear feedbacks. For a prescribed decay rate $δ>0$, let $\widehat{C}_S(δ)$ denote this optimal prefactor. Assuming that the uncontrolled region contains a nonempty open set and that the observability cost satisfies $C(T)\le C_0e^{C_0/T}$ for small $T>0$, we prove $$ e^{c\sqrtδ}\le \widehat{C}_S(δ)\le e^{C\sqrtδ}, \qquad δ\gg1. $$ The lower bound is based on the uncontrolled open hole and a quantitative cutoff construction adapted to the Laplacian, while the upper bound combines an exponentially weighted Gramian with the small-time observability cost $C(T)\lesssim e^{C/T}$. The results apply to controlled Schrödinger equations in three settings: flat tori, including a strip-control example beyond the geometric control condition; the whole space $\mathbb{R}^n$ with exterior-ball control; and bounded smooth domains satisfying the geometric control condition for generalized geodesics.

math.OC↗

arXiv2026

Exact Truncation and Radial Rigidity in Time-Optimal Control

We consider minimum-time control for the linear system $$ \dot{z}(t)=Az(t)+Bu(t),\qquad \|u\|_{L^\infty(0,\infty;\mathbb{R}^m)}\leq 1, $$ with $A\in\mathbb R^{n\times n}$ and $B\in\mathbb R^{n\times m}$. 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 $\varepsilon>0$, the optimal control for the tolerance ball $\overline{B}_\varepsilon(0)$ is precisely the restriction of the point-target optimal control? We prove that this {\it{exact truncation}} property is equivalent to the rigidity conditions $$ B B^\top=βI_n,\qquad A+A^\top=2aI_n, \qquad β>0,\; a\leq 0, $$ 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 $0<\varepsilon<|x|$, and the resulting optimal times and point-target optimal feedback are both explicit.

math.OC↗

arXiv2026

Endpoint Asymptotics of Optimal Stabilization Prefactors

Consider the stabilizable finite-dimensional linear control system $\dot x=Ax+Bu$. For a prescribed decay rate $δ>0$, we define the optimal stabilization prefactor $$ \hat{C}(δ) := \inf_{K\in\mathbb{R}^{m\times n}} \sup_{t\ge0}e^{δt}\|e^{(A+BK)t}\|. $$ 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 $\hat{C}(δ)\asympδ^{μ-1}$ as $δ\to+\infty$; whereas if $(A,B)$ is stabilizable but not controllable, then $\hat{C}(δ)\asymp(δ_*-δ)^{-(q-1)}$ as $δ\uparrowδ_*$, 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 $\operatorname{Re}λ=-δ_*$. Thus, in both cases, the endpoint asymptotic order of $\hat{C}(δ)$ 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.

math.OC↗