# Sufficient criteria and sharp geometric conditions for observability in Banach spaces

## TL;DR

This paper establishes new sufficient conditions for observability in Banach space systems using uncertainty and dissipation estimates, extending previous Hilbert space results and applying to elliptic operators with thick set observations.

## Contribution

It introduces a unified framework for observability criteria in Banach spaces, generalizing earlier Hilbert space results and characterizing observability with geometric conditions.

## Key findings

- Final state observability holds under a thick set condition.
- The approach unifies and extends previous results.
- Bounds on control costs are derived for the dual system.

## Abstract

Let $X,Y$ be Banach spaces, $(S_t)_{t \geq 0}$ a $C_0$-semigroup on $X$, $-A$ the corresponding infinitesimal generator on $X$, $C$ a bounded linear operator from $X$ to $Y$, and $T > 0$. We consider the system \[ \dot{x}(t) = -Ax(t), \quad y(t) = Cx(t) \quad t\in (0,T], \quad x(0) = x_0 \in X. \] We provide sufficient conditions such that this system satisfies a final state observability estimate in $L_r ((0,T) ; Y)$, $r \in [1,\infty]$. These sufficient conditions are given by an uncertainty relation and a dissipation estimate. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider the example where $A$ is an elliptic operator in $L_p(\mathbb{R}^d)$ for $1<p<\infty$, and where $C = \mathbf{1}_\omega$ is the restriction onto a thick set $\omega \subset \mathbb{R}^d$. In this case, we show that the above system satisfies a final state observability estimate if and only if $\omega \subset \mathbb{R}^d$ is a thick set. Finally, we make use of the well-known relation between observability and null-controllability of the predual system, and investigate bounds on the corresponding control costs.