Stabilizability with bounded feedback for analytic linear control systems
Yaxing Ma, Emmanuel Trélat, Lijuan Wang, Huaiqiang Yu
TL;DR
This work addresses the stabilization of analytic linear control systems with unbounded inputs by introducing parallel state-space frameworks $X$ and $\mathcal{X}$ connected through an almost exponential decay condition $\text{(AEDC)}$. It proves that stabilizability with bounded feedback in the $X$-space is equivalent to stabilizability in $\mathcal{X}$ and to stabilizability of $[\mathcal{A},B]$ in $\mathcal{X}$, characterized by a Hautus-type frequency-domain inequality. The results extend classical finite- and infinite-dimensional stabilizability theories to non-compact semigroups and non-admissible control operators, and provide a constructive bounded-feedback design via projection onto unstable subspaces. Applications to parabolic PDEs illustrate practical stabilization scenarios, including boundary and differential-type controls, with rapid stabilization possible under thick-set conditions. Overall, the paper unifies competing stabilization frameworks and delivers explicit, bounded-feedback strategies for a broad class of analytic control systems.
Abstract
In this paper, we give sufficient conditions under which linear abstract control systems for which the semigroup is analytic are stabilizable with a bounded feedback. We obtain various characterizations of that property, which extend some earlier works. We illustrate our findings with several examples.