Endpoint Asymptotics of Optimal Stabilization Prefactors

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 $$\widehat{C}(\delta ):=\underset{K\in {\mathbb{R}}^{m\times n}}{\inf }\underset{t\ge 0}{\sup }{e}^{\delta t}\mathrm{\parallel }{e}^{(A+BK)t}\mathrm{\parallel }\mathrm{.}$$ 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 $\widehat{C}(\delta )\asymp {\delta }^{\mu -1}$ as $\delta \to +\mathrm{\infty }$; whereas if $(A,B)$ is stabilizable but not controllable, then $\widehat{C}(\delta )\asymp ({\delta }_{\ast }-\delta {)}^{-(q-1)}$ as $\delta \uparrow {\delta }_{\ast }$, where ${\delta }_{\ast }$ 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 $\mathrm{Re}\lambda =-{\delta }_{\ast }$. Thus, in both cases, the endpoint asymptotic order of $\widehat{C}(\delta )$ 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.