# Observability for Non-Autonomous Systems
Interactive session at [SCINDIS 2020](https://www.fan.uni-wuppertal.de/de/scindis-2020.html) by [**Fabian Gabel** (Hamburg)](https://www.mat.tuhh.de/home/fgabel_en)
Joint work with: [**Clemens Bombach** (Chemnitz)](https://www.tu-chemnitz.de/mathematik/analysis/bombach/), [**Chrisitan Seifert** (Hamburg)](https://www.mat.tuhh.de/home/cseifert_en), [**Martin Tautenhahn** (Leipzig)](https://home.uni-leipzig.de/mtau/)
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# Motivation
:::danger
What kind of mathematical objects are involved when we talk about **Observability**?
What could be an interesting application?
Find out in the following 4 min video or just skim through this and the following sections.
:::
You find the full notes for the videos [here](https://cloud.tuhh.de/index.php/s/LNzfeYsPymkAwPT).
{%youtube aW-ly5cufjs %}
Let $X$ be a Banach spaces and consider the non-autonomous Cauchy problem
\begin{align*}
\dot{x}(t) = A(t)x(t), \; y(t) = C(t)x(t), \; t > 0, \quad x(0) = x_0 \in X \tag{na-ACP}
\end{align*} where
* $x(t)$ is a *state* in $X$
* $y(t)$ is an *observation* (of $x$) in $Y$
* $A(t)$ is the operator family that governs the evolution of $x$
* $C(t)$ is the operator family that describes the observation of $x$
:::info
**Motivating Question.**
If we measure $y(t)$ at times $E \subset [0,T]$, do we have an *Observability Estimate*?
\begin{align*}
\| x(T) \|_X \lesssim c \int_E \|y(t)\|_Y \; dt \tag{OBS}
\end{align*}
:::
:::success
**Outline**.
In the abstract setting of Banach spaces, there are conditions available that guarantee the existence of an observability estimate. We will also show that these conditions are met for an example of non-autonomous elliptic differential operators.
:::
# Evolution Families and Conditions for $(\mathrm{OBS})$
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Ideally, the solution to $(\text{na-ACP})$ with $A(t) : D(A) \to X$ and $C(t) \in \mathcal{L}(X,Y)$ is given via an *evolution family* (a generalization of semigroups to non-autonomous Cauchy problems)
\begin{align*}
x(t) := U(\cdot, 0) x_0 \in L^1(0,T; D) \cap W^{1,1}(0,T; X)
\end{align*}
**Theorem.**
The following ingredients altogether imply observability:
* exponential boundedness of $U(t,s)$
* uniform boundedness of $C(t)$
* existence of a family $(P_\lambda)_{\lambda > 0}$ of "Projectors" that fulfills
* a *dissipation estimate* $$\|(I - P_\lambda) U(t,s) \| \lesssim \exp(-\lambda(t - s))$$
* an *uncertainty estimate* $$\| P_\lambda U(t,0)\| \lesssim \exp(-\lambda) \; \|C(t) P_\lambda U(t,0)\|$$
**Remark.**
On the one hand, for a given non-autonomous Cauchy problem, the existence of a corresponding evolution family is highly non-trivial, cf. the works of Lunardi [L] or Pazy [P] for details. On the other hand, the above ingredients don't need $U(t,s)$ to be associated to a Cauchy problem at all.
# Differential Operators and Geometric Conditions
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The above ingredients can be verified in the following classical example:
* Non-autonomous strongly elliptic differential operator
\begin{align*}A(t) u := -\mathcal{F^{-1}}\Big(a(t,\cdot)\mathcal{F} u\Big)\end{align*}
* with associated exponentially bounded evolution family
\begin{align*}U(t,s) u := \mathcal{F^{-1}} \Big( \mathrm{e}^{-\int_s^t a(\tau,\cdot) d\tau} \mathcal{F} u \Big)\end{align*}
* observation operators given via multiplication with an indicator function
\begin{align*}
C(t) = \mathbf{1}_{\Omega(t)} \cdot
\end{align*}
* and "projectors" based on smooth cut-offs
\begin{align*}P_\lambda f := (\mathcal{F}^{-1} \chi_\lambda) \ast f \end{align*}
* with dissipation estimate and
* uncertainty estimate due to *Logvinenko-Sereda theorem* if $(\Omega_t)$ are **uniformly thick** (special geometric condition).
**Definition (not formal).**
* A family $(\Omega(t))$ is called *uniformly thick* if for each cube $Q^d$
\begin{align*}
|\Omega(t) \cap Q^d| \gtrsim |Q^d|
\end{align*}
* A family $(\Omega(t))$ is called *mean thick* if for each cube $Q^d$
\begin{align*}
\int_0^T | \Omega(t) \cap Q^d| \gtrsim |Q^d|
\end{align*} This notion of builds on a notion used in [BEP-S].
One can show that the existence of an observability estimate $(\text{OBS})$ in the above example implies that $(\Omega(t))$ is **mean thick**. In the autonomous case $\Omega(t) = \Omega$, the above notions of thickness coincide and thickness of $\Omega$ is also necessary for observability, cf. the works of Bombach, Gallaun, Seifert and Tautenhahn [BGalST], [GST].
:::danger
**Open Question.**
Is *uniform thickness* necessary for observability in the non-autonomous case?
:::
# References and Further Reading
## Observability (and Null-Controllability)
[BEP-S] M. Beauchard, M. Egidi, K. Pravda-Starov. *Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports*, preprint, 2019. [arXiv:1908.10603](https://arxiv.org/abs/1908.10603)
[BGalST] C. Bombach, D. Gallaun, C. Seifert, M. Tautenhahn. *Observability and null-controllability for parabolic equations in $L_p$-spaces*, preprint, 2020. [arXiv:2005.14503](https://arxiv.org/abs/2005.14503)
[BGabST] C. Bombach, F. Gabel, C. Seifert, M. Tautenhahn. *Observability for Non-Autonomous Systems*, in preparation. [arXiv:2203.08469](http://arxiv.org/abs/2203.08469)
[GST] D. Gallaun, C. Seifert, M. Tautenhahn, *Sufficient Criteria and Sharp Geometric Conditions for Observability in Banach Spaces*,SIAM J. Control Optim., **58**, 2639--2657. [doi:10.1137/19M1266769](https://doi.org/10.1137/19M1266769), [arXiv:1905.10285](https://arxiv.org/abs/1905.10285)
## Evolution Families and Regularity
[GM] C. Gallarati, M. Veraar, *Evolution Families and Maximal Regularity for Systems
of Parabolic Equations*, Adv. Differential Equations **22**(2017), 169--190. [doi:10.1090/spmj/1653](https://doi.org/10.1090/spmj/1653), [arXiv:1510.07643](https://arxiv.org/abs/1510.07643),
>[name=Fabian Gabel] Our definition of *evolution family for the operator $A$* bases on the one used in this article which is in turn based on the one used in [P].
[L] A. Lunardi, *Analytic Semigroups and Optimal Regularity in Parabolic Problems*, BirkhĂ¤user, 1995. [doi:10.1007/978-3-0348-0557-5](https://doi.org/10.1007/978-3-0348-0557-5)
>[name=Fabian Gabel] See Chapter 6 *Linear nonautonomous equations*. Section 6.4 *Bibliographical Remarks* gives an overview of further relevant articles wrt. the subject of evolution families.
[P] A. Pazy, *Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer, 1983. [doi:10.1007/978-1-4612-5561-1](https://doi.org/10.1007/978-1-4612-5561-1)
>[name=Fabian Gabel] Chapter 5 *Evolution Equations* provides a framework for non-autonomous Cauchy problems.