Sufficient Criteria for Stabilization Properties in Banach Spaces

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in \(L_p(\mathbb {R}^d)\) $L_p({\mathbb{R}}^d)$ for \(p\in [1,\infty )\) $p\in [1,\infty )$ and thick control sets.