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 operator from $X$ to $Y$ and $T>0$. We consider systems of the form
and study the question whether one can reconstruct the final state $x(T)$ from the measurements $y(t)$ for $t \in (0,T)$. This relates to a final state observability estimate, that is, there exists $C_{\mathrm{obs}} > 0$ such that for all $x_0 \in X$ we have $\lVert x (T) \rVert_X \leq C_{\mathrm{obs}} \lVert y \rVert_{L_r ((0,T) ; Y)}$ for some $r\in [1,\infty]$.
The most studied example is the self-adjoint Schrödinger operator $A = \Delta - V$ in $L_2 (\Omega)$ with bounded potential $V$, and $C = \mathrm{1}_E$ for $E \subset \Omega \subset \mathbb{R}^d$. One possible approach to show an observability estimate is the so-called Lebeau-Robbiano method [1], that is, to prove a quantitative *uncertainty relation* for spectral projectors. This is an inequality of the type
\lVert \mathrm{1}_E P (\lambda) \psi \rVert_{L_2 (E)} ,
\end{equation*}
where $\gamma \in (0,1)$, $d_0,d_1 > 0$, and where $P (\lambda)$ denotes the projector to the spectral subspace of $-\Delta + V$ below $\lambda$. Subsequently, this strategy is generalized to semigroups in abstract Hilbert spaces. In particular the $P(\lambda)$ are allowed to be arbitrary projectors (onto semigroup invariant subspaces) by assuming additionally a so-called *dissipation estimate*, that is, a decay estimate of the semigroup on the orthogonal complement of the range of $P(\lambda)$.
A natural setup to ask for observability estimates is the context of Banach spaces and $C_0$-semigroups. In [2], we extend the above-mentioned stategy to the Banach space setting. In particular, we show in the general framework of Banach spaces that an uncertainty relation together with a dissipation estimate implies a final state observability estimate. Our observability constant $C_{\mathrm{obs}}$ is given explicitly with respect to the parameters coming from the uncertainty relation and the dissipation estimate and, in addition, is sharp in the dependence on $T$.
Using the well-known relation between observability and null-controllability of the predual system one can proof controllability results by the adapted Lebeau-Robbiano method. In [3] we consider parabolic control systems on $L_p(\mathbb{R}^d)$, $p\in [1,\infty)$, of the form
where $-A_p$ is a strongly elliptic differential operator with constant coefficients. Assuming that $E\subset \mathbb{R}^d$ is a so-called thick set, we proof (approximate) null-controllability, i.e. for all $x_0\in L_p(\mathbb{R}^d)$ there is $u \in L_r ((0,T);L_p (E))$ which steers the mild solution at time $T$ (approximately) to zero.
A weaker concept than null-controllability is stabilizability. In the ongoing work [4] the uncertainty relation and the dissipation estimate in the Lebeau-Robbiano method are weakend to proof stabilizabilty for systems which are not null-controllable.
## References
[1] G. Lebeau and L. Robbiano. Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20(1--2):335--356, 1995.
[2] D. Gallaun, C. Seifert, and M. Tautenhahn. Sufficient criteria and sharp geometric conditions for observability in Banach spaces, SIAM J. Control Optim., 58(4):2639--2657, 2020.
[3] C. Bombach, D. Gallaun, C. Seifert, and M. Tautenhahn. Observability and null-controllability for parabolic equations in $L_p$-spaces. [arXiv:2005.14503](https://arxiv.org/pdf/2005.14503.pdf)
[4] M. Egidi, D. Gallaun, C. Seifert, and M. Tautenhahn. Sufficient criteria for stabilization properties in Banach spaces. In preparation.
