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 $δ$ app