A Logvinenko-Sereda theorem for vector-valued functions and application to control theory
Clemens Bombach, Martin Tautenhahn
TL;DR
The paper proves a vector-valued Logvinenko-Sereda type inequality for $f\in L^p(\mathbb{R}^d;X)$ with Fourier support in a parallelepiped $\Pi_\lambda$ and thick observation sets, allowing $X$ to be an arbitrary Banach space, potentially infinite-dimensional. This vector-valued spectral inequality is then integrated into a Lebeau-Robbiano observability framework for normally elliptic operators with operator-valued coefficients, yielding analytic semigroup representations via Fourier multipliers and robust dissipation estimates. Consequently, the authors derive observability results and demonstrate null-controllability (or approximate null-controllability) for a broad class of systems on Banach spaces, including coupled parabolic equations and parameter-dependent problems. The work thus provides a rigorous bridge between vector-valued harmonic analysis and infinite-dimensional control theory with explicit dependence on geometric and operator-analytic parameters.
Abstract
We prove a Logvinenko-Sereda Theorem for vector valued functions. That is, for an arbitrary Banach space $X$, all $p \in [1,\infty]$, all $λ\in (0,\infty)^d$, all $f \in L^p (\mathbb{R}^d ; X)$ with $\operatorname{supp} \mathcal{F} f \in \times_{i=1}^d (-λ_i/2 , λ_i /2)$, and all thick sets $E \subseteq \mathbb{R}^d$ we have \begin{equation*} \lVert \mathbf{1}_E f \rVert_{L^p (\mathbb{R}^d)} \geq C \lVert f \rVert_{L^p (\mathbb{R}^d)} . \end{equation*} The constant is explicitly known in dependence of the geometric parameters of the thick set and the parameter $λ$. As an application, we study control theory for normally elliptic operators on Banach spaces whose coefficients of their symbol are given by bounded linear operators. This includes systems of coupled parabolic equations or problems depending on a parameter.