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

2024/01/17 by Clemens Bombach, Bombach, Clemens, Martin Tautenhahn +1 · 1 citation
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2401.09159

Abstract

We prove a Logvinenko-Sereda Theorem for vector valued functions. That is, for an arbitrary Banach space X, all p ∈ [1,∞], all λ∈ (0,∞)^d, all f ∈ L^p (ℝ^d ; X) with supp F f ∈ ×_i=1^d (-λ_i/2 , λ_i /2), and all thick sets E ⊆ ℝ^d we have ‖ 1_E f ‖_{L^p (ℝ^d)} ≥ C ‖ f ‖_{L^p (ℝ^d)} . 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.

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