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

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 $\dot{x}(t)=-A_{p}x(t)+1_{E}u(t)$, $x(0)=x_0\in L_p({\mathbb{R}}^d)$, 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 $p=1$.

Cited by in corpus (3)

• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups
• Sufficient criteria for stabilization properties in Banach spaces