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We prove a Logvinenke-Sereda Theorem for vector valued functions. That is,for an arbitrary Banach space $X$, all $p \in [1,\infty]$, all $\lambda \in(0,\infty)^d$, all $f \in L^p (\mathbb{R}^d ; X)$ with $\operatorname{supp}\mathcal{F} f \in \times_{i=1}^d (-\lambda_i/2 , \lambda_i /2)$, and all thicksets $E \subseteq \mathbb{R}^d$ 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 geometricparameters of the thick set and the parameter $\lambda$. As an application, westudy control theory for normally elliptic operators on Banach spaces whosecoefficients of their symbol are given by bounded linear operators. Thisincludes systems of coupled parabolic equations or problems depending on aparameter.

A Logvinenko-Sereda theorem for vector-valued functions and application to control theory