Endpoint Asymptotics of Optimal Stabilization Prefactors

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 $$ \hat{C}(\delta) := \inf_{K\in\mathbb{R}^{m\times n}} \sup_{t\ge0}e^{\delta t}\|e^{(A+BK)t}\|. $$ 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 $\hat{C}(\delta)\asymp\delta^{\mu-1}$ as $\delta\to+\infty$; whereas if $(A,B)$ is stabilizable but not controllable, then $\hat{C}(\delta)\asymp(\delta_*-\delta)^{-(q-1)}$ 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 $\hat{C}(\delta)$ 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.