School of Mathematical Sciences, Shenzhen University

Liyuan Distinguished Scholar Lecture Series Session 43

Lecture Title: Endpoint Asymptotics of Optimal Stabilization Prefactors

Speaker: Professor Gengsheng Wang (Hetao Institute of Mathematics and Interdisciplinary Sciences)

Time: 16:30 - 17:30 , Sept.16, 2026

Venue: Lecture Hall 1, Huixing Building, Yuehai Campus, Shenzhen University

Abstract: Consider the stabilizable finite-dimensional linear control system $\dot x=Ax+Bu$. For a prescribed decay rate $\delta>0$, we define the optimal stabilization prefactor

$$\Chat(\delta):=\inf_{K\in\R^{m\times n}}\sup_{t\ge0}e^{\delta t}\norm{e^{(A+BK)t}}.$$

We determine its asymptotic behavior as $\delta$ approaches the right endpoint of the achievable decay-rate range. If $(A,B)$ is controllable and $\mu$ is its largest controllability index, then $\Chat(\delta)\asymp\delta^{\mu-1}$ as $\delta\to+\infty$; whereas if $(A,B)$ is stabilizable but not controllable, then $\Chat(\delta)\asymp(\delta_*-\delta)^{-(q-1)}$ as $\delta\uparrow\delta_*$, where $\delta_*$ 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$\operatorname{Re}\lambda=-\delta_*$. Thus, in both cases, the endpoint asymptotic order of $\Chat(\delta)$ is determined by the corresponding structural invariant---$\mu$ in the controllable case and $q$ in the noncontrollable case---and conversely this order recovers that invariant.

Speaker Profile: Gengsheng Wang has long been engaged in research on control theory for differential equations, with a particular focus on time-optimal control, the geometric characteristics of the region of control, and stability problems, and has produced several notable works in these areas. In 2026, he will deliver a 45-minute lecture on control and optimization at the International Congress of Mathematicians (ICM).

All faculty and students are welcome to attend!

School of Mathematical Sciences

September 14, 2026