Observability and null-controllability for parabolic equations in $L_p$-spaces

TL;DR

This paper investigates the null-controllability of parabolic equations in various Lp-spaces, providing explicit control bounds and unifying previous results across different functional settings.

Contribution

It establishes null-controllability criteria for parabolic systems in Lp-spaces using duality, uncertainty principles, and dissipation estimates, extending prior work to include p=1.

Findings

Null-controllability is proven for systems in Lp-spaces.

Explicit bounds on control costs are derived.

Results unify and extend earlier Hilbert and Banach space findings.

Abstract

We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems of the form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) = x_0\in L_p (\mathbb{R}^d)$, with interior control on a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, and where $A$ is an elliptic operator of order $m \in \mathbb{N}$ in $L_p(\mathbb{R}^d)$. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case $p=1$.