Sufficient Criteria and Sharp Geometric Conditions for Observability in Banach Spaces

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 $C = {1}_{E}$ is the restriction onto a thick set ${E} \subset \mathbb{R}^d$. In this case, we show that the above system satisfies a final state observability estimate if and only if ${E} \subset \mathbb{R}^d$ 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.