# Final state observability estimates and cost-uniform approximate null-
controllability for bi-continuous semigroups

Karsten Kruse  0000-0003-1864-4915 Hamburg University of Technology
Institute of Mathematics
Am Schwarzenberg-Campus 3
21073 Hamburg
Germany<EMAIL_ADDRESS>and  Christian Seifert  0000-0001-9182-8687
<EMAIL_ADDRESS>
(Date: August 30, 2024)

###### Abstract.

We consider final state observability estimates for bi-continuous semigroups
on Banach spaces, i.e. for every initial value, estimating the state at a
final time $T>0$ by taking into account the orbit of the initial value under
the semigroup for $t\in[0,T]$, measured in a suitable norm. We state a
sufficient criterion based on an uncertainty relation and a dissipation
estimate and provide two examples of bi-continuous semigroups which share a
final state observability estimate, namely the Gauß–Weierstraß semigroup and
the Ornstein–Uhlenbeck semigroup on the space of bounded continuous functions.
Moreover, we generalise the duality between cost-uniform approximate null-
controllability and final state observability estimates to the setting of
locally convex spaces for the case of bounded and continuous control
functions, which seems to be new even for the case of Banach spaces.

###### Key words and phrases:

final state observability estimate, bi-continuous semigroups, cost-uniform
approximate null-controllability, Saks space, mixed topology

###### 2020 Mathematics Subject Classification:

Primary 93C20, 93C25, 47N70, Secondary 47D06, 46A70

##  1\. Introduction

Let $X$ be a Banach space and $(S_{t})_{t\geq 0}$ a semigroup on $X$, i.e.
$S_{0}=I$ and $S_{t+s}=S_{t}S_{s}$ for all $t,s\geq 0$. Moreover, let $Y$ be
another Banach space and $C\in\mathcal{L}(X;Y)$, a so-called observation
operator, and $T>0$. Then $(S_{t})_{t\geq 0}$ satisfies a _final state
observability estimate_ w.r.t. some (Banach) space $\mathcal{Z}$ of functions
on $[0,T]$ with values in $Y$, if there exists $C_{\mathrm{obs}}\geq 0$ such
that

$\left\lVert S_{T}x\right\rVert_{X}\leq C_{\mathrm{obs}}\left\lVert
CS_{(\cdot)}x\right\rVert_{\mathcal{Z}}\qquad(x\in X).$

Put differently, we can estimate the norm of the final state $S_{T}x$ by just
taking into account the observations $CS_{t}x$ for $t\in[0,T]$. Typical
applications stem from evolution equations on some function space over (a
subset of) $\mathds{R}^{d}$, where $C$ is a restriction operator to a suitable
subset $\Omega$ of $\mathds{R}^{d}$ (or of the subset of $\mathds{R}^{d}$ the
functions are defined on) such that we want to control the final state on all
of $\mathds{R}^{d}$ by just measuring the evolution on the subset $\Omega$.
Final state observability estimates have been studied in various contexts due
to its relation to null-controllability, see e.g. [4, 18, 25, 29, 21, 23, 9,
1, 10, 2] and references therein.

Classically, the space $\mathcal{Z}$ is some $L_{r}$-space with
$r\in[1,\infty]$ (when working in Hilbert spaces, one usually chooses $r=2$),
and then the final state observability estimate yields the form

$\left\lVert
S_{T}x\right\rVert_{X}\leq\begin{cases}C_{\mathrm{obs}}\left(\int_{0}^{T}\left\lVert
CS_{t}x\right\rVert_{Y}^{r}\mathrm{d}t\right)^{1/r}&\text{if
}r\in[1,\infty),\\\
C_{\mathrm{obs}}\operatorname*{ess\,sup}_{t\in[0,T]}\left\lVert
CS_{t}x\right\rVert_{Y}&\text{if }r=\infty,\end{cases}\qquad(x\in X).$

Clearly, in order to formulate this final state observability estimate (i.e.
to have a well-defined right-hand side) we need some regularity of the
semigroup. Indeed, we require measurability of $t\mapsto\left\lVert
CS_{t}x\right\rVert_{Y}$ for all $x\in X$. Of course, strong continuity of
$(S_{t})_{t\geq 0}$ yields continuity of these maps and is therefore
sufficient, but also weaker regularities are suitable. In [1], dual semigroups
of strongly continuous semigroups were considered which yield sufficient
regularity.

In this short note we aim at two types of results. First, we consider final
state observability estimates for so-called bi-continuous semigroups, see e.g.
[16, 17], which are not strongly continuous for the norm-topology on $X$ but
only for a weaker topology. Note that dual semigroups are a special case of
bi-continuous ones when considering the weak∗ topology. There are classical
examples of bi-continuous semigroups such as the Gauß–Weierstraß semigroup on
$C_{b}(\mathds{R}^{d})$, the space of bounded continuous functions (on
$\mathds{R}^{d}$), as well as the Ornstein–Uhlenbeck semigroup on
$C_{b}(\mathds{R}^{d})$. Second, we relate cost-uniform approximate null-
controllability of a control system sharing only weak continuity properties
such as bi-continuity with a final state observability estimate for the dual
system, thus generalising the well-known duality in Hilbert and Banach spaces
[6, 4, 25, 29]. Since this demands to work in Hausdorff locally convex spaces,
we here focus on continuous control functions which results in the space
$\mathcal{Z}$ above being a space of vector measures.

The paper is organised as follows. In Section 2 we review bi-continuous
semigroups and then turn to final-state observability estimates in Section 3
together with two examples in Section 4. Final state observability estimates
are then related with cost-uniform approximate null-controllability via
duality, which we will exploit in our context in Section 5.

##  2\. Bi-Continuous Semigroups

In this short section we recall some notation and definitions from the theory
of bi-continuous semigroups that we need in subsequent sections. For a vector
space $X$ over the field $\mathds{R}$ or $\mathds{C}$ with a Hausdorff locally
convex topology $\tau_{X}$ we denote by $(X,\tau_{X})^{\prime}$ the
topological linear dual space and just write
$X^{\prime}:=(X,\tau_{X})^{\prime}$ if $(X,\tau_{X})$ is a Banach space. We
use the symbol
$\mathcal{L}(X;Y):=\mathcal{L}((X,\left\lVert\cdot\right\rVert_{X});(Y,\left\lVert\cdot\right\rVert_{Y}))$
for the space of continuous linear operators from a Banach space
$(X,\left\lVert\cdot\right\rVert_{X})$ to a Banach space
$(Y,\left\lVert\cdot\right\rVert_{Y})$ and denote by
$\left\lVert\cdot\right\rVert_{\mathcal{L}(X;Y)}$ the operator norm on
$\mathcal{L}(X;Y)$. If $X=Y$, we set $\mathcal{L}(X):=\mathcal{L}(X;X)$.

######  Definition 2.1 ([5, I.3.2 Definition]).

Let $(X,\left\lVert\cdot\right\rVert_{X})$ be a normed space and $\tau_{X}$ a
Hausdorff locally convex topology on $X$.

  1. (a)

The triple $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ is called a _Saks
space_ if $(X,\left\lVert\cdot\right\rVert_{X})$ is a Banach space,
$\tau_{X}\subseteq\tau_{\left\lVert\cdot\right\rVert_{X}}$ and
$(X,\tau_{X})^{\prime}$ is norming for $X$ where
$\tau_{\left\lVert\cdot\right\rVert_{X}}$ denotes the
$\left\lVert\cdot\right\rVert_{X}$-topology.

  2. (b)

The _mixed topology_
$\gamma_{X}\coloneq\gamma(\left\lVert\cdot\right\rVert_{X},\tau_{X})$ is the
finest linear topology on $X$ that coincides with $\tau_{X}$ on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets and such that
$\tau_{X}\subseteq\gamma_{X}\subseteq\tau_{\left\lVert\cdot\right\rVert_{X}}$.

The mixed topology is actually Hausdorff locally convex and the definition
given above is equivalent to the one from the literature [28, Section 2.1] by
[28, Lemmas 2.2.1, 2.2.2].

######  Definition 2.2.

We call a Saks space $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$
_sequentially complete_ if $(X,\gamma_{X})$ is sequentially complete.

Due to [28, Corollary 2.3.2] a Saks space
$(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ is sequentially complete if
and only if $(X,\tau_{X})$ is sequentially complete on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets, i.e. every
$\left\lVert\cdot\right\rVert_{X}$-bounded $\tau_{X}$-Cauchy sequence
converges in $X$. In combination with [13, Remark 2.3 (c)] this yields that a
triple $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ fulfils [17,
Assumptions 1] if and only if it is a sequentially complete Saks space.

######  Definition 2.3 ([17, Definition 3]).

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space. Let $(S_{t})_{t\geq 0}$ in $\mathcal{L}(X)$ be a semigroup on $X$.
We say that $(S_{t})_{t\geq 0}$ is _(locally) $\tau_{X}$-bi-continuous_ if

  1. (a)

it is exponentially bounded, i.e. there exist $M\geq 1$ and
$\omega\in\mathds{R}$ such that $\left\lVert
S_{t}\right\rVert_{\mathcal{L}(X)}\leq M\mathrm{e}^{\omega t}$ for all $t\geq
0$,

  2. (b)

$(S_{t})_{t\geq 0}$ is a $C_{0}$-semigroup on $(X,\tau_{X})$, i.e. for all
$x\in X$ the map $[0,\infty)\ni t\mapsto T_{t}x\in(X,\tau_{X})$ is continuous,

  3. (c)

it is (locally) bi-equicontinuous, i.e. for every $(x_{n})_{n\in\mathds{N}}$
in $X$ and $x\in X$ with $\sup\limits_{n\in\mathds{N}}\left\lVert
x_{n}\right\rVert_{X}<\infty$ and
$\tau_{X}\text{-}\lim\limits_{n\to\infty}x_{n}=x$ we have

$\tau_{X}\text{-}\lim_{n\to\infty}S_{t}(x_{n}-x)=0$

(locally) uniformly for $t\in[0,\infty)$.

As in the case of $C_{0}$-semigroups on Banach spaces we can define generators
for bi-continuous semigroups.

######  Definition 2.4 ([7, Definition 1.2.6]).

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space and $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous
semigroup on $X$. The _generator_ $(-A,D(A))$ is defined by

$\displaystyle D(A)\coloneqq$ $\displaystyle\Bigl{\\{}x\in
X\;|\;\tau_{X}\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S_{t}x-x}{t}\;\text{exists
in }X\;\text{and }\sup_{t\in(0,1]}\frac{\left\lVert
S_{t}x-x\right\rVert_{X}}{t}<\infty\Bigr{\\}},$ $\displaystyle-Ax\coloneqq$
$\displaystyle\tau_{X}\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S_{t}x-x}{t}\quad(x\in
D(A)).$

######  Remark 2.5.

There is no common agreement whether to use the here presented definition of a
generator or its negative. Throughout the entire paper we will stick to the
definition made above, i.e. $-A$ is the generator.

##  3\. Final State Observability Estimates for Bi-Continuous Semigroups

The final state observability estimate rests on the following abstract
theorem. It provides a sufficient criterion stating that an abstract
uncertainty principle (also called spectral inequality), see (UP), together
with a dissipation estimate, see (DISS), yields a final state observability
estimate, and has its roots in [18], see also [21, 23, 9, 1].

######  Theorem 3.1 ([1, Theorem A.1]).

Let $X$ and $Y$ be Banach spaces, $C\in\mathcal{L}(X;Y)$, $(S_{t})_{t\geq 0}$
a semigroup on $X$, $M\geq 1$ and $\omega\in\mathds{R}$ such that $\left\lVert
S_{t}\right\rVert_{\mathcal{L}(X)}\leq M\mathrm{e}^{\omega t}$ for all $t\geq
0$, and assume that for all $x\in X$ the map $t\mapsto\left\lVert
CS_{t}x\right\rVert_{Y}$ is measurable. Further, let $\lambda^{*}\geq 0$,
$(P_{\lambda})_{\lambda>\lambda^{*}}$ in $\mathcal{L}(X)$, $r\in[1,\infty]$,
$d_{0},d_{1},d_{3},\gamma_{1},\gamma_{2},\gamma_{3},T>0$ with
$\gamma_{1}<\gamma_{2}$, and $d_{2}\geq 1$, and assume that

$\forall x\in X\ \forall\lambda>\lambda^{*}\colon\quad\left\lVert
P_{\lambda}x\right\rVert_{X}\leq
d_{0}\mathrm{e}^{d_{1}\lambda^{\gamma_{1}}}\left\lVert
CP_{\lambda}x\right\rVert_{Y}$ (UP)

and

$\forall x\in X\ \forall\lambda>\lambda^{*}\ \forall
t\in(0,T/2]\colon\quad\left\lVert(I-P_{\lambda})S_{t}x\right\rVert_{X}\leq
d_{2}\mathrm{e}^{-d_{3}\lambda^{\gamma_{2}}t^{\gamma_{3}}}\left\lVert
x\right\rVert_{X}.$ (DISS)

Then there exists $C_{\mathrm{obs}}\geq 0$ such that for all $x\in X$ we have

$\left\lVert
S_{T}x\right\rVert_{X}\leq\begin{cases}C_{\mathrm{obs}}\left(\int_{0}^{T}\left\lVert
CS_{t}x\right\rVert_{Y}^{r}\mathrm{d}t\right)^{1/r}&\text{if
}r\in[1,\infty),\\\
C_{\mathrm{obs}}\operatorname*{ess\,sup}_{t\in[0,T]}\left\lVert
CS_{t}x\right\rVert_{Y}&\text{if }r=\infty.\end{cases}$

######  Remark 3.2.

The constant $C_{\mathrm{obs}}$ is explicit in all parameters and of the form

$\displaystyle
C_{\mathrm{obs}}=\frac{C_{1}}{T^{1/r}}\exp\left(\frac{C_{2}}{T^{\frac{\gamma_{1}\gamma_{3}}{\gamma_{2}-\gamma_{1}}}}+C_{3}T\right),$

with $T^{1/r}=1$ if $r=\infty$, and suitable constants $C_{1},C_{2},C_{3}\geq
0$ depending on the parameters; see [1, Theorem A.1] as well as [10, Theorem
2.1].

In order to obtain a version of this theorem for bi-continuous semigroups, we
need to argue on measurability of $[0,\infty)\ni t\mapsto\left\lVert
CS_{t}x\right\rVert_{Y}$ for all $x\in X$.

######  Lemma 3.3.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$, $(Y,\left\lVert\cdot\right\rVert_{Y},\tau_{Y})$ a Saks space, $C\colon
X\to Y$ linear and sequentially $\tau_{X}$-$\tau_{Y}$-continuous on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets, and $x\in X$. Then
$[0,\infty)\ni t\mapsto\left\lVert CS_{t}x\right\rVert_{Y}$ is measurable.

###### Proof.

Since $[0,\infty)\ni t\mapsto\left\lvert\left\langle
y^{\prime},CS_{t}x\right\rangle\right\rvert$ is continuous for all
$y^{\prime}\in(Y,\tau_{Y})^{\prime}$ by the assumptions on $C$ and the
exponential boundedness of $(S_{t})_{t\geq 0}$, and $\left\lVert
CS_{t}x\right\rVert_{Y}=\sup\\{\left\lvert\left\langle
y^{\prime},CS_{t}x\right\rangle\right\rvert\ |\
y^{\prime}\in(Y,\tau_{Y})^{\prime},\,\left\lVert
y^{\prime}\right\rVert_{Y^{\prime}}\leq 1\\}$ for all $t\geq 0$, we obtain
that $[0,\infty)\ni t\mapsto\left\lVert CS_{t}x\right\rVert_{Y}$ is lower
semi-continuous and hence measurable. ∎

In view of Lemma 3.3, we can apply Theorem 3.1 to obtain the following.

######  Theorem 3.4.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$, $(Y,\left\lVert\cdot\right\rVert_{Y},\tau_{Y})$ a Saks space,
$C\in\mathcal{L}(X;Y)$ such that $C$ is also sequentially
$\tau_{X}$-$\tau_{Y}$-continuous on $\left\lVert\cdot\right\rVert_{X}$-bounded
sets. Further, let $\lambda^{*}\geq 0$, $(P_{\lambda})_{\lambda>\lambda^{*}}$
a family in $\mathcal{L}(X)$, $r\in[1,\infty]$,
$d_{0},d_{1},d_{3},\gamma_{1},\gamma_{2},\gamma_{3},T>0$ with
$\gamma_{1}<\gamma_{2}$, and $d_{2}\geq 1$, and assume that

$\forall x\in X\ \forall\lambda>\lambda^{*}\colon\quad\left\lVert
P_{\lambda}x\right\rVert_{X}\leq
d_{0}\mathrm{e}^{d_{1}\lambda^{\gamma_{1}}}\left\lVert
CP_{\lambda}x\right\rVert_{Y}$ (UP’)

and

$\forall x\in X\ \forall\lambda>\lambda^{*}\ \forall
t\in(0,T/2]\colon\quad\left\lVert(I-P_{\lambda})S_{t}x\right\rVert_{X}\leq
d_{2}\mathrm{e}^{-d_{3}\lambda^{\gamma_{2}}t^{\gamma_{3}}}\left\lVert
x\right\rVert_{X}.$ (DISS’)

Then there exists $C_{\mathrm{obs}}\geq 0$ such that for all $x\in X$ we have

$\left\lVert
S_{T}x\right\rVert_{X}\leq\begin{cases}C_{\mathrm{obs}}\left(\int_{0}^{T}\left\lVert
CS_{t}x\right\rVert_{Y}^{r}\mathrm{d}t\right)^{1/r}&\text{if
}r\in[1,\infty),\\\
C_{\mathrm{obs}}\operatorname*{ess\,sup}_{t\in[0,T]}\left\lVert
CS_{t}x\right\rVert_{Y}&\text{if }r=\infty.\end{cases}$

######  Remark 3.5.

The statements in Theorem 3.1 and Theorem 3.4 can be generalised in the sense
that one can obtain an estimate with an $L_{r}$-norm of $t\mapsto\left\lVert
CS_{t}x\right\rVert_{Y}$ on a measurable subset $E\subseteq[0,T]$ with
positive Lebesgue measure; cf. e.g. [2]. However, in this case the constant
$C_{\mathrm{obs}}$ (cf. Remark 3.2) is not explicit anymore.

##  4\. Two Examples of Bi-Continuous Semigroups

In this section we consider final state observability for two important
examples: the Gauß–Weierstraß semigroup on $C_{b}(\mathds{R}^{d})$ and the
Ornstein–Uhlenbeck semigroup on $C_{b}(\mathds{R}^{d})$. We begin with the
study of restriction operators on $C_{b}(\mathds{R}^{d})$, restricting
functions to suitable subsets, and relate this to an abstract uncertainty
principle.

###  4.1. Restriction Operators on $C_{b}(\mathds{R}^{d})$ and the Uncertainty
Principle.

Let $\Omega\subseteq\mathds{R}^{d}$ be non-empty, $C\colon
C_{b}(\mathds{R}^{d})\to C_{b}(\Omega)$ the restriction operator defined by
$Cf\coloneqq f|_{\Omega}$ for $f\in C_{b}(\mathds{R}^{d})$. Then
$C\in\mathcal{L}(C_{b}(\mathds{R}^{d});C_{b}(\Omega))$. Let
$\tau_{\mathrm{co}}$ be the compact-open topology on $C_{b}(\mathds{R}^{d})$
(as well as on $C_{b}(\Omega)$). Then
$(C_{b}(\Omega),\left\lVert\cdot\right\rVert_{\infty},\tau_{\mathrm{co}})$ is
a Saks space which is sequentially complete if $\Omega$ is locally compact (in
particular if $\Omega=\mathds{R}^{d}$).

######  Lemma 4.1.

$C\colon(C_{b}(\mathds{R}^{d}),\tau_{\mathrm{co}})\to(C_{b}(\Omega),\tau_{\mathrm{co}})$
is continuous.

###### Proof.

The map $C$ is clearly linear. Due to [22, Theorem 46.8] the compact-open
topology $\tau_{\mathrm{co}}$ on $C_{b}(Z)$ for a Hausdorff topological space
$Z$ is given by the system of seminorms

$p_{K}^{Z}(f)\coloneqq\sup_{x\in K}|f(x)|\quad(f\in C_{b}(Z))$

for compact $K\subseteq Z$. Let $K\subseteq\Omega$ be compact in the relative
topology. Then $K$ is compact in $\mathds{R}^{d}$ as well and

$p_{K}^{\Omega}(Cf)=\sup_{x\in K}|f|_{\Omega}(x)|=\sup_{x\in
K}|f(x)|=p_{K}^{\mathds{R}^{d}}(f)\quad(f\in C_{b}(\mathds{R}^{d})),$

which means that $C$ is continuous. ∎

We now use the operator $C$ to provide an uncertainty principle based on the
well-known Logvinenko–Sereda theorem. Let $\eta\in C_{c}[0,\infty)$,
$\mathds{1}_{[0,1/2]}\leq\eta\leq\mathds{1}_{[0,1]}$. For $\lambda>0$ let
$\chi_{\lambda}\colon\mathds{R}^{d}\to\mathds{R}$,
$\chi_{\lambda}\coloneqq\eta(\left\lvert\cdot\right\rvert/\lambda)$, and
$P_{\lambda}\in\mathcal{L}(C_{b}(\mathds{R}^{d}))$ be defined by
$P_{\lambda}f\coloneqq(\mathcal{F}^{-1}\chi_{\lambda})\ast f$, where
$\mathcal{F}$ denotes the Fourier transformation. By Young’s inequality and
scaling properties of the Fourier transformation, we have

$\left\lVert
P_{\lambda}\right\rVert\leq\left\lVert\mathcal{F}^{-1}\chi_{\lambda}\right\rVert_{L_{1}(\mathds{R}^{d})}=\left\lVert\mathcal{F}^{-1}\chi_{1}\right\rVert_{L_{1}(\mathds{R}^{d})}\quad(\lambda>0).$

Note that for all $f\in C_{b}(\mathds{R}^{d})$ and $\lambda>0$ we have
$\mathcal{F}P_{\lambda}f=\chi_{\lambda}\mathcal{F}f$ and therefore
$\operatorname{spt}\mathcal{F}P_{\lambda}f\subseteq
B[0,\lambda]\subseteq[-\lambda,\lambda]^{d}$, where
$B[0,\lambda]\coloneqq\\{x\in\mathds{R}^{d}\mid\left\lvert
x\right\rvert\leq\lambda\\}$ is the closed ball around $0$ with radius
$\lambda$.

######  Definition 4.2.

Let $\Omega\subseteq\mathds{R}^{d}$. Then $\Omega$ is called _thick_ if
$\Omega$ is measurable and there exist $L\in(0,\infty)^{d}$ and $\rho\in(0,1]$
such that

$\lambda^{d}(\Omega\cap(x+(0,L)))\geq\rho\lambda^{d}((0,L))\quad(x\in\mathds{R}^{d}),$

where $\lambda^{d}$ denotes the $d$-dimensional Lebesgue measure, and
$(0,L)\coloneqq\prod_{j=1}^{d}(0,L_{j})$ is the hypercube with sidelengths
contained in $L$.

Thus, a measurable set $\Omega\subseteq\mathds{R}^{d}$ is thick (with
parameters $L$ and $\rho$) provided the portion of $\Omega$ in every hypercube
with sidelengths contained in $L$ is at least $\rho$.

By the Logvinenko–Sereda theorem (see [19, 12]), if
$\Omega\subseteq\mathds{R}^{d}$ is a thick set, then there exist
$d_{0},d_{1}>0$ such that

$\left\lVert P_{\lambda}f\right\rVert_{C_{b}(\mathds{R}^{d})}\leq
d_{0}\mathrm{e}^{d_{1}\lambda}\left\lVert
CP_{\lambda}f\right\rVert_{C_{b}(\Omega)}\quad(\lambda>0,f\in
C_{b}(\mathds{R}^{d})).$ (LS)

Thus, (LS) yields an estimate of the form (UP’).

###  4.2. The Gauß–Weierstraß Semigroup on $C_{b}(\mathds{R}^{d})$.

Let $k\colon(0,\infty)\times\mathds{R}^{d}\to\mathds{R}$ be given by

$k_{t}(x)\coloneqq k(t,x)\coloneqq\frac{1}{(4\pi
t)^{d/2}}\mathrm{e}^{-\left\lvert
x\right\rvert^{2}/(4t)}\quad(t>0,x\in\mathds{R}^{d}),$

the so-called _Gauß–Weierstraß kernel_. For $t\geq 0$ we define
$S_{t}\in\mathcal{L}(C_{b}(\mathds{R}^{d}))$ by

$S_{t}f\coloneqq\begin{cases}f&t=0,\\\ k_{t}*f&t>0.\end{cases}$

Note that by Young’s inequality and the fact that $\left\lVert
k_{t}\right\rVert_{L_{1}(\mathds{R}^{d})}=1$ for all $t>0$ we have
$\left\lVert S_{t}f\right\rVert_{C_{b}(\mathds{R}^{d})}\leq\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}$ for $f\in C_{b}(\mathds{R}^{d})$ and
$t\geq 0$. It is easy to see that $(S_{t})_{t\geq 0}$ is a semigroup, which is
called the _Gauß–Weierstraß semigroup_. Let $\tau_{\mathrm{co}}$ be the
compact-open topology on $C_{b}(\mathds{R}^{d})$. Then $(S_{t})_{t\geq 0}$ is
locally $\tau_{\mathrm{co}}$-bi-continuous; see e.g. [17, Examples 6 (a)].

For $\lambda>0$ let $P_{\lambda}\in\mathcal{L}(C_{b}(\mathds{R}^{d}))$ be
defined as in Subsection 4.1. By [1, Proposition 3.2], there exist $d_{2}\geq
1$ and $d_{3}>0$ such that

$\left\lVert(I-P_{\lambda})S_{t}f\right\rVert_{C_{b}(\mathds{R}^{d})}\leq
d_{2}\mathrm{e}^{-d_{3}\lambda^{2}t}\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}\quad(\lambda>0,t\geq 0,f\in
C_{b}(\mathds{R}^{d})),$ (DISS(GW))

i.e. a dissipation estimate (DISS’) is fulfilled.

Thus, if $\Omega\subseteq\mathds{R}^{d}$ is thick, then (LS) and (DISS(GW))
provide the estimates (UP’) and (DISS’) and so Theorem 3.4 yields a final
state observability estimate for the Gauß–Weierstraß semigroup on
$C_{b}(\mathds{R}^{d})$.

###  4.3. The Ornstein–Uhlenbeck Semigroup on $C_{b}(\mathds{R}^{d})$.

Let $M\colon(0,\infty)\times\mathds{R}^{d}\times\mathds{R}^{d}\to\mathds{R}$
be given by

$M_{t}(x,y)\coloneqq
M(t,x,y)\coloneqq\frac{1}{\pi^{d/2}(1-\mathrm{e}^{-2t})^{d/2}}\mathrm{e}^{-\left\lvert
y-\mathrm{e}^{-t}x\right\rvert^{2}/(1-\mathrm{e}^{-2t})}\quad(t>0,x,y\in\mathds{R}^{d}),$

the so-called _Mehler kernel_. For $t\geq 0$ we define
$S_{t}\in\mathcal{L}(C_{b}(\mathds{R}^{d}))$ by

$S_{t}f\coloneqq\begin{cases}f&t=0,\\\
\int_{\mathds{R}^{d}}M_{t}(\cdot,y)f(y)\,\mathrm{d}y&t>0.\end{cases}$

Since $\int_{\mathds{R}^{d}}M_{t}(\cdot,y)\,\mathrm{d}y=1$ for all $t>0$, we
have $\left\lVert S_{t}f\right\rVert_{C_{b}(\mathds{R}^{d})}\leq\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}$ for $f\in C_{b}(\mathds{R}^{d})$ and
$t\geq 0$. It is not difficult to see that $(S_{t})_{t\geq 0}$ is a semigroup,
which is called the _Ornstein–Uhlenbeck semigroup_. Let $\tau_{\mathrm{co}}$
be the compact-open topology on $C_{b}(\mathds{R}^{d})$. Then $(S_{t})_{t\geq
0}$ is locally $\tau_{\mathrm{co}}$-bi-continuous on $C_{b}(\mathds{R}^{d})$;
see e.g. [16, Proposition 3.10].

Define $k\colon(0,1)\times\mathds{R}^{d}\to\mathds{R}$ by

$k_{s}(x)\coloneqq\frac{1}{\pi^{d/2}(1-s^{2})^{d/2}}\mathrm{e}^{-\left\lvert
x\right\rvert^{2}/(1-s^{2})}.$

Let $s\in(0,1)$. Then we obtain

$M_{\ln\frac{1}{s}}(\tfrac{1}{s}x,y)=\frac{1}{\pi^{d/2}(1-s^{2})^{d/2}}\mathrm{e}^{-\left\lvert
y-x\right\rvert^{2}/(1-s^{2})}=k_{s}(x-y)\quad(x,y\in\mathds{R}^{d}).$

Hence,

$\bigl{(}S_{\ln\frac{1}{s}}f\bigr{)}(\tfrac{1}{s}\,\cdot\,)=k_{s}\ast
f\quad(f\in C_{b}(\mathds{R}^{d})).$

For $\lambda>0$ let $P_{\lambda}\in\mathcal{L}(C_{b}(\mathds{R}^{d}))$ as in
Subsection 4.2. Since

$\mathcal{F}k_{s}(\xi)=\mathrm{e}^{-(1-s^{2})\left\lvert\xi\right\rvert^{2}/4}\eqqcolon
h_{s}(\xi)\quad(\xi\in\mathds{R}^{d}),$

for $\lambda>0$ and $s\in(0,1)$ we conclude that

$\bigl{(}(I-P_{\lambda})S_{\ln\frac{1}{s}}f\bigr{)}(\tfrac{1}{s}\,\cdot\,)=(I-P_{\lambda/s})\bigl{(}S_{\ln\frac{1}{s}}f(\tfrac{1}{s}\,\cdot\,)\bigr{)}=\mathcal{F}^{-1}\bigl{(}(1-\chi_{\lambda/s})h_{s}\bigr{)}\ast
f\quad(f\in C_{b}(\mathds{R}^{d})).$

######  Lemma 4.3.

There exist $d_{2}\geq 1$ and $d_{3}>0$ such that for $\lambda>0$ and
$s\in(0,1)$ we have

$\left\lVert\mathcal{F}^{-1}\bigl{(}(1-\chi_{\lambda})h_{s}\bigr{)}\right\rVert_{L_{1}(\mathds{R}^{d})}\leq
d_{2}\mathrm{e}^{-d_{3}\lambda^{2}(1-s^{2})}.$

###### Proof.

Let $\lambda>0$, $s\in(0,1)$, and define

$\sigma_{s,\lambda}\coloneqq(1-\chi_{\sqrt{1-s^{2}}\lambda})h_{s}\Bigl{(}\tfrac{1}{\sqrt{1-s^{2}}}\,\cdot\,\Bigr{)}=(1-\chi_{\sqrt{1-s^{2}}\lambda})\mathrm{e}^{-\left\lvert\cdot\right\rvert^{2}/4}.$

Then by a linear substitution we obtain

$\left\lVert\mathcal{F}^{-1}\bigl{(}(1-\chi_{\lambda})h_{s}\bigr{)}\right\rVert_{L_{1}(\mathds{R}^{d})}=\left\lVert\mathcal{F}^{-1}\sigma_{s,\lambda}\right\rVert_{L_{1}(\mathds{R}^{d})}.$

Let $\alpha\in\mathds{N}_{0}^{d}$, $\left\lvert\alpha\right\rvert\leq d+1$.
Then

$\left\lvert\partial^{\alpha}\sigma_{s,\lambda}\right\rvert\leq\mathds{1}_{\\{\left\lvert\cdot\right\rvert\geq\sqrt{1-s^{2}}\lambda/2\\}}\left\lvert\partial^{\alpha}\mathrm{e}^{-\left\lvert\cdot\right\rvert^{2}/4}\right\rvert+\sum_{\beta\in\mathds{N}_{0}^{d},\beta<\alpha}\binom{\alpha}{\beta}\left\lvert\partial^{\alpha-\beta}(1-\chi_{\sqrt{1-s^{2}}\lambda})\right\rvert\left\lvert\partial^{\beta}\mathrm{e}^{-\left\lvert\cdot\right\rvert^{2}/4}\right\rvert.$

There exists $K\geq 0$ such that for all $\beta\in\mathds{N}_{0}^{d}$ with
$\beta\leq\alpha$ and all $\xi\in\mathds{R}^{d}$ we have

$\left\lvert\partial^{\beta}\mathrm{e}^{-\left\lvert\cdot\right\rvert^{2}/4}(\xi)\right\rvert\leq
K(1+\left\lvert\xi\right\rvert)^{\left\lvert\beta\right\rvert}\mathrm{e}^{-\left\lvert\xi\right\rvert^{2}/4}.$

Let

$C_{1}\coloneqq\sup_{\beta\in\mathds{N}_{0}^{d},\beta\leq\alpha,\xi\in\mathds{R}^{d}}K(1+\left\lvert\xi\right\rvert)^{\left\lvert\beta\right\rvert}\mathrm{e}^{-\left\lvert\xi\right\rvert^{2}/16}.$

Then, for $\beta\leq\alpha$ and
$\left\lvert\xi\right\rvert\geq\sqrt{1-s^{2}}\lambda/2$ we have

$\left\lvert\partial^{\beta}\mathrm{e}^{-\left\lvert\cdot\right\rvert^{2}/4}(\xi)\right\rvert\leq
C_{1}\mathrm{e}^{-\left\lvert\xi\right\rvert^{2}/16}\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}.$

Further, for $\beta<\alpha$ and $\xi\in\mathds{R}^{d}$ we have

$\displaystyle\left\lvert\partial^{\alpha-\beta}(1-\chi_{\sqrt{1-s^{2}}\lambda})(\xi)\right\rvert$
$\displaystyle\leq(\sqrt{1-s^{2}}\lambda)^{-\left\lvert\alpha-\beta\right\rvert}\left\lvert\partial^{\alpha-\beta}\chi_{1}\Bigl{(}\tfrac{\xi}{\sqrt{1-s^{2}}\lambda}\Bigr{)}\right\rvert\mathds{1}_{\\{\sqrt{1-s^{2}}\lambda/2\leq\left\lvert\cdot\right\rvert\leq\sqrt{1-s^{2}}\lambda\\}}(\xi)$
$\displaystyle\leq
C_{2}(\sqrt{1-s^{2}}\lambda)^{-\left\lvert\alpha-\beta\right\rvert}\mathds{1}_{\\{\sqrt{1-s^{2}}\lambda/2\leq\left\lvert\cdot\right\rvert\leq\sqrt{1-s^{2}}\lambda\\}}(\xi)$

where
$C_{2}\coloneqq\max_{\beta<\alpha}\left\lVert\partial^{\alpha-\beta}\chi_{1}\right\rVert_{C_{b}(\mathds{R}^{d})}$.
Hence, there exists $C\geq 0$ (which is independent of $s$ and $\lambda$) such
that if $\sqrt{1-s^{2}}\lambda\geq 1$, then for all $\xi\in\mathds{R}^{d}$ we
have

$\left\lvert\partial^{\alpha}\sigma_{s,\lambda}\right\rvert\leq
C\mathrm{e}^{-\left\lvert\xi\right\rvert^{2}/16}\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}.$

Therefore, increasing $C$, for all $x\in\mathds{R}^{d}$ we obtain

$\left\lvert
x^{\alpha}\mathcal{F}^{-1}\sigma_{s,\lambda}(x)\right\rvert=\left\lvert\mathcal{F}^{-1}(\partial^{\alpha}\sigma_{s,\lambda})(x)\right\rvert\leq
C\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}.$ (1)

By choosing $j\in\\{1,\ldots,d\\}$ and $\alpha\coloneqq(d+1)e_{j}$ for the
$j$-th canonical unit vector $e_{j}$, we observe $\left\lVert
x\right\rVert_{\infty}^{d+1}\left\lvert\mathcal{F}^{-1}\sigma_{s,\lambda}(x)\right\rvert\leq
C\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}$ and hence

$\left\lvert\mathcal{F}^{-1}\sigma_{s,\lambda}(x)\right\rvert\leq
C\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}\left\lvert x\right\rvert^{-d-1}$ (2)

for all $x\in\mathds{R}^{d}\setminus\\{0\\}$, where we increased $C$.

Therefore, if $\sqrt{1-s^{2}}\lambda\geq 1$, we can conclude by (1) for
$\alpha=0$ and (2) that

$\displaystyle\left\lVert\mathcal{F}^{-1}\bigl{(}(1-\chi_{\lambda})h_{s}\bigr{)}\right\rVert_{L_{1}(\mathds{R}^{d})}$
$\displaystyle=\left\lVert\mathcal{F}^{-1}\sigma_{s,\lambda}\right\rVert_{L_{1}(\mathds{R}^{d})}$
$\displaystyle\leq
C\mathrm{e}^{-(1-s^{2})\lambda^{2}/32}\Biggl{(}\;\int_{B[0,1]}1\,\mathrm{d}x+\int_{\mathds{R}^{d}\setminus
B[0,1]}\left\lvert x\right\rvert^{-d-1}\,\mathrm{d}x\Biggr{)}$
$\displaystyle\leq C\mathrm{e}^{-(1-s^{2})\lambda^{2}/32},$

where we increased $C$ again.

It remains to prove the estimate for the case $\sqrt{1-s^{2}}\lambda<1$. Note
that

$\displaystyle\left\lVert\mathcal{F}^{-1}(\chi_{\lambda}h_{s})\right\rVert_{L_{1}(\mathds{R}^{d})}$
$\displaystyle=\left\lVert\mathcal{F}^{-1}\chi_{\lambda}\ast\mathcal{F}^{-1}h_{s}\right\rVert_{L_{1}(\mathds{R}^{d})}$
$\displaystyle\leq\left\lVert\mathcal{F}^{-1}\chi_{\lambda}\right\rVert_{L_{1}(\mathds{R}^{d})}\left\lVert
k_{s}\right\rVert_{L_{1}(\mathds{R}^{d})}=\left\lVert\mathcal{F}^{-1}\chi_{1}\right\rVert_{L_{1}(\mathds{R}^{d})},$

where the last equality follows form scaling properties of the Fourier
transformation and the fact that $k_{s}$ is normalised in
$L_{1}(\mathds{R}^{d})$.

Thus, for $\sqrt{1-s^{2}}\lambda<1$ we obtain

$\left\lVert\mathcal{F}^{-1}(\chi_{\lambda}h_{s})\right\rVert_{L_{1}(\mathds{R}^{d})}\leq\left\lVert\mathcal{F}^{-1}\chi_{1}\right\rVert_{L_{1}(\mathds{R}^{d})}\mathrm{e}^{1/32}\mathrm{e}^{-(1-s^{2})\lambda^{2}/32},$

which ends the proof. ∎

In view of Lemma 4.3, we obtain the dissipation estimate (DISS’) as follows.
Note that for $t\geq 0$ we have $\mathrm{e}^{2t}-1\geq 2t$. Let $t>0$ and
$\lambda>0$, and set $s\coloneqq\mathrm{e}^{-t}\in(0,1)$. Then, for $f\in
C_{b}(\mathds{R}^{d})$, Young’s inequality and Lemma 4.3 yield

$\displaystyle\left\lVert(I-P_{\lambda})S_{t}f\right\rVert_{C_{b}(\mathds{R}^{d})}$
$\displaystyle=\left\lVert\bigl{(}(I-P_{\lambda})S_{\ln\frac{1}{s}}f\bigr{)}(\tfrac{1}{s}\,\cdot\,)\right\rVert_{C_{b}(\mathds{R}^{d})}$
$\displaystyle\leq\left\lVert\mathcal{F}^{-1}\bigl{(}(1-\chi_{\lambda/s})h_{s}\bigr{)}\right\rVert_{L_{1}(\mathds{R}^{d})}\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}$ $\displaystyle\leq
d_{2}\mathrm{e}^{-d_{3}\lambda^{2}s^{-2}(1-s^{2})}\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}=d_{2}\mathrm{e}^{-d_{3}\lambda^{2}(\mathrm{e}^{2t}-1)}\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}$ $\displaystyle\leq
d_{2}\mathrm{e}^{-2d_{3}\lambda^{2}t}\left\lVert
f\right\rVert_{C_{b}(\mathds{R}^{d})}.$ (DISS(OU))

Thus, if $\Omega\subseteq\mathds{R}^{d}$ is thick and $C\colon
C_{b}(\mathds{R}^{d})\to C_{b}(\Omega)$ is the restriction map as in
Subsection 4.2, then (LS) and (4.3) provide the estimates (UP’) and (DISS’)
and so Theorem 3.4 yields a final state observability estimate for the
Ornstein–Uhlenbeck semigroup on $C_{b}(\mathds{R}^{d})$.

##  5\. Cost-Uniform Approximate Null-Controllability and Duality

In this section we want to show that cost-uniform approximate null-
controllability is equivalent to final state observability of the dual system,
which is known in the setting of norm-strongly continuous semigroups; see [6,
4, 25, 29]. In the bi-continuous setting this needs a bit of preparation so
that we can formulate the corresponding definitions. Since we work in
Hausdorff locally convex spaces, the choice of the “correct” integral may be
delicate. We therefore provide the duality for continuous control functions,
and thus relate it to a final state observability estimate of the dual system
w.r.t. a space of vector measures.

######  Definition 5.1.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a Saks space and $T>0$.
We set

$C_{\tau,b}([0,T];X)\coloneqq\\{f\in C([0,T];(X,\tau_{X}))\ |\
\|f\|_{\infty}\coloneqq\sup_{t\in[0,T]}\left\lVert
f(t)\right\rVert_{X}<\infty\\}$

where $C([0,T];(X,\tau_{X}))$ is the space of continuous functions from
$[0,T]$ to $(X,\tau_{X})$.

######  Remark 5.2.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a Saks space and $T>0$.

  1. (a)

Since the mixed topology $\gamma_{X}$ coincides with $\tau_{X}$ on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets by [28, Lemma 2.2.1], and a
subset of $X$ is $\left\lVert\cdot\right\rVert_{X}$-bounded if and only if it
is $\gamma_{X}$-bounded by [28, Corollary 2.4.1] we have

$C_{\tau,b}([0,T];X)=C_{b}([0,T];(X,\gamma_{X}))=C([0,T];(X,\gamma_{X})).$

We define two topologies on this space. First, the one given by the norm

$\left\lVert f\right\rVert_{\infty}\coloneqq\sup_{t\in[0,T]}\left\lVert
f(t)\right\rVert_{X}\qquad(f\in C_{\tau,b}([0,T];X)).$

Second, the Hausdorff locally convex topology $\gamma_{\infty}$ induced by the
directed system of seminorms given by

$p_{\gamma_{\infty}}(f)\coloneqq\sup_{t\in[0,T]}p_{\gamma_{X}}(f(t))\qquad(f\in
C_{\tau,b}([0,T];X))$

for $p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}$ where
$\mathcal{P}_{\gamma_{X}}$ is a directed system of seminorms that induces the
mixed topology $\gamma_{X}$. Clearly, $\gamma_{\infty}$ is coarser than the
$\left\lVert\cdot\right\rVert_{\infty}$-topology. Further,
$(C_{\tau,b}([0,T];X),\left\lVert\cdot\right\rVert_{\infty})$ is a Banach
space.

  2. (b)

We note that a subset $B\subseteq C_{\tau,b}([0,T];X)$ is
$\left\lVert\cdot\right\rVert_{\infty}$-bounded if and only if it is
$\gamma_{\infty}$-bounded since a subset of $X$ is
$\left\lVert\cdot\right\rVert_{X}$-bounded if and only if it is
$\gamma_{X}$-bounded by [28, 2.4.1 Corollary]. So
$((C_{\tau,b}([0,T];X),\gamma_{\infty})^{\prime},\tau_{b})$ is a topological
subspace of
$C_{\tau,b}([0,T];X)^{\prime}=((C_{\tau,b}([0,T];X),\left\lVert\cdot\right\rVert_{\infty})^{\prime},\left\lVert\cdot\right\rVert_{C_{\tau,b}([0,T];X)^{\prime}})$
where $\tau_{b}$ denotes the topology of uniform convergence on
$\gamma_{\infty}$-bounded sets. In the following we use the notation
$\left\lVert
y^{\prime}\right\rVert_{(C_{\tau,b}([0,T];X),\gamma_{\infty})^{\prime}}\coloneqq\left\lVert
y^{\prime}\right\rVert_{C_{\tau,b}([0,T];X)^{\prime}}$ for all
$y^{\prime}\in(C_{\tau,b}([0,T];X),\gamma_{\infty})^{\prime}$.

######  Proposition 5.3.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$ and $T>0$. Let $v\in C_{\tau,b}([0,T];X)$ and set

$f\colon[0,T]\to X,\;f(t)\coloneqq S_{T-t}v(t).$

Then $f\in C_{\tau,b}([0,T];X)$.

###### Proof.

We denote by $\mathcal{P}_{\tau_{X}}$ a system of directed seminorms that
generates the topology $\tau_{X}$ on $X$. Let $(t_{n})_{n\in\mathds{N}}$ be a
sequence in $[0,T]$ that converges to $t\in[0,T]$ and set
$x_{n}:=v(t_{n})-v(t)$ for $n\in\mathds{N}$. The sequence
$(x_{n})_{n\in\mathds{N}}$ is $\left\lVert\cdot\right\rVert_{X}$-bounded and
$\tau_{X}\text{-}\lim\limits_{n\to\infty}x_{n}=0$ due to our assumptions on
$v$. We have for $p\in\mathcal{P}_{\tau_{X}}$ that

$\displaystyle p(f(t_{n})-f(t))$
$\displaystyle=p(S_{T-t_{n}}v(t_{n})-S_{T-t}v(t))$ $\displaystyle\leq
p(S_{T-t_{n}}v(t_{n})-S_{{T}-t_{n}}v(t))+p(S_{T-t_{n}}v(t)-S_{T-t}v(t))$
$\displaystyle\leq p(S_{T-t_{n}}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t))$
$\displaystyle\leq\sup_{s\in[0,T]}p(S_{s}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t)).$

Combining our estimate above with the local bi-equicontinuity and
$\tau_{X}$-strong continuity of the semigroup, we deduce that
$(f(t_{n}))_{n\in\mathds{N}}$ converges to $f(t)$ in $(X,\tau_{X})$. Hence,
$f\in C([0,T];(X,\tau_{X}))$. Furthermore, as the semigroup is exponentially
bounded, there are $M\geq 1$ and $\omega\in\mathds{R}$ such that for all
$t\in[0,T]$

$\displaystyle\left\lVert f(t)\right\rVert_{X}$ $\displaystyle=\left\lVert
S_{T-t}v(t)\right\rVert_{X}\leq\left\lVert
S_{T-t}\right\rVert_{\mathcal{L}(X)}\left\lVert v(t)\right\rVert_{X}\leq
M\mathrm{e}^{\omega(T-t)}\left\lVert v(t)\right\rVert_{X}$ $\displaystyle\leq
M\mathrm{e}^{|\omega|T}\left\lVert v(t)\right\rVert_{X},$

which yields that $f$ is $\left\lVert\cdot\right\rVert_{X}$-bounded on $[0,T]$
because $v([0,T])$ is $\left\lVert\cdot\right\rVert_{X}$-bounded. ∎

######  Proposition 5.4.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a Saks space, and
$B\in\mathcal{L}(U;X)$ such that $B$ is also sequentially
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets. Let $T>0$, $u\in C_{\tau,b}([0,T];U)$ and set $f\colon[0,T]\to X$,
$f(t)\coloneqq S_{T-t}Bu(t)$. Then $f$ is $\tau_{X}$-Pettis integrable and
$\gamma_{X}$-Pettis integrable and both integrals coincide.

###### Proof.

The statement follows from [14, 2.5 Proposition (a)] and Proposition 5.3
because the map $v\colon t\mapsto Bu(t)$ belongs to $C_{\tau,b}([0,T];X)$. ∎

Now, we have everything at hand to formulate the definition of cost-uniform
approximate null-controllability in the bi-continuous setting. Let
$(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a Saks space,
$(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup on $X$ with
generator $(-A,D(A))$, and $B\in\mathcal{L}(U;X)$ such that $B$ is also
sequentially $\tau_{U}$-$\tau_{X}$-continuous on
$\left\lVert\cdot\right\rVert_{U}$-bounded sets, and $T>0$. We consider the
linear control system

$\displaystyle\begin{split}\dot{x}(t)&=-Ax(t)+Bu(t)\quad(t>0),\\\
x(0)&=x_{0}\in X,\end{split}$ (ConSys)

where $u\in C_{\tau,b}([0,T];U)$. The function $x$ is called _state function_
and $u$ is called _control function_. The unique mild solution of (ConSys) is
given by Duhamel’s formula

$x(t)=S_{t}x_{0}+\int_{0}^{t}S_{t-r}Bu(r)\mathrm{d}r\qquad(t\in[0,T])$

due to [17, Proposition 11 (a)] and Proposition 5.4. Let
$\mathcal{P}_{\tau_{X}}$ be a directed system of seminorms that induces the
topology $\tau_{X}$.

######  Definition 5.5.

We say that (ConSys) is _cost-uniform approximately $\tau_{X}$-null-
controllable in time $T$ via $C_{\tau,b}([0,T];U)$_ if there exists $C\geq 0$
such that for all $x_{0}\in X$, $\varepsilon>0$ and
$p_{\tau_{X}}\in\mathcal{P}_{\tau_{X}}$ there exists $u\in
C_{\tau,b}([0,T];U)$ with $\left\lVert u\right\rVert_{\infty}\leq C\left\lVert
x_{0}\right\rVert_{X}$ such that $p_{\tau_{X}}(x(T))\leq\varepsilon$.

We note that this definition of cost-uniform approximate $\tau_{X}$-null-
controllability does not depend on the choice of $\mathcal{P}_{\tau_{X}}$.

######  Remark 5.6.

We can analogously define the notion of cost-uniform approximate
$\gamma_{X}$-null-controllability in time $T$ via $C_{\tau,b}([0,T];U)$ by
using $p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}$ instead of
$p_{\tau_{X}}\in\mathcal{P}_{\tau_{X}}$. In view of Proposition 5.13 and
Remark 5.14 these two notions are equivalent.

Next, we prepare the definition of final state observability of the dual
system where we need to clarify which kind of duality we have to use.

######  Definition 5.7.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a Saks space and
$\mathds{K}=\mathds{R}$ or $\mathds{C}$ the scalar field of $X$.

  1. (a)

We call $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ _C-sequential_ if
$(X,\gamma_{X})$ is C-sequential, i.e. every convex sequentially open subset
of $(X,\gamma_{X})$ is already open (see [24, p. 273]).

  2. (b)

We call $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ a _Mazur space_ if
$(X,\gamma_{X})$ is a Mazur space, i.e.

$X_{\gamma}^{\prime}\coloneqq(X,\gamma_{X})^{\prime}=\\{x^{\prime}\colon
X\to\mathds{K}\ |\ x^{\prime}\;\text{linear and }\gamma_{X}\text{-sequentially
continuous}\\}$

(see [27, p. 40]).

Examples of C-sequential Saks spaces can be found in [13, Example 2.4, Remarks
3.19, 3.20, Corollary 3.23].

######  Remark 5.8.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a Saks space.

  1. (a)

If $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ is C-sequential, then it is
a Mazur space by [27, Theorem 7.4] (cf. [14, 3.6 Proposition (b)]).

  2. (b)

The space

$X^{\circ}\coloneqq\\{x^{\prime}\in X^{\prime}\ |\
x^{\prime}\;\tau_{X}\text{-sequentially continuous on
}\left\lVert\cdot\right\rVert_{X}\text{-bounded sets}\\}$

is a closed linear subspace of the norm dual $X^{\prime}$ and hence a Banach
space with norm given by $\left\lVert
x^{\circ}\right\rVert_{X^{\circ}}\coloneqq\left\lVert
x^{\circ}\right\rVert_{X^{\prime}}$ for $x^{\circ}\in X^{\circ}$ due to [8,
Proposition 2.1] (note that the proof of [8, Proposition 2.1] does not use [8,
Hypothesis A (ii)] which is the sequential completeness of
$(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$). We have
$X^{\circ}=X_{\gamma}^{\prime}$ if and only if $(X,\gamma_{X})$ is a Mazur
space by [14, 3.5 Remark].

Let $(X,\left\lVert\cdot\right\rVert_{X})$ and
$(U,\left\lVert\cdot\right\rVert_{U})$ be Banach spaces. We recall that the
dual operator $B^{\prime}$ of an element $B\in\mathcal{L}(U;X)$ is defined by
$\langle B^{\prime}x^{\prime},u\rangle\coloneqq\langle x^{\prime},Bu\rangle$
for $x^{\prime}\in X^{\prime}$ and $u\in U$.

######  Proposition 5.9.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
C-sequential Saks space and $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-
continuous semigroup on $X$. Then the operators given by
$S^{\circ}_{t}x^{\circ}\coloneqq S_{t}^{\prime}x^{\circ}$ for $t\geq 0$ and
$x^{\circ}\in X^{\circ}$ belong to $\mathcal{L}(X^{\circ})$ and form a
$\tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))$-strongly
continuous, exponentially bounded semigroup on $X^{\circ}$ where
$\tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))$ denotes the
topology of uniform convergence on $\left\lVert\cdot\right\rVert_{X}$-compact
sets.

###### Proof.

Since $(X,\gamma_{X})$ is C-sequential, in particular Mazur by Remark 5.8 (a),
$(S_{t})_{t\geq 0}$ is quasi-$\gamma_{X}$-equicontinuous and
$X^{\circ}=X_{\gamma}^{\prime}$ by [13, Theorem 3.17 (a)] and Remark 5.8 (b).
In particular, $S^{\circ}_{t}$ is the $\gamma_{X}$-dual map of $S_{t}$ for all
$t\geq 0$. Furthermore, $(S^{\circ}_{t})_{t\geq 0}$ is exponentially bounded
(w.r.t. $\left\lVert\cdot\right\rVert_{\mathcal{L}(X^{\circ})}$) because
$(S_{t})_{t\geq 0}$ is exponentially bounded. It follows that
$S^{\circ}_{t}\in\mathcal{L}(X^{\circ})$ for all $t\geq 0$ and
$(S^{\circ}_{t})_{t\geq 0}$ is a $\sigma(X^{\circ},X)$-strongly continuous
semigroup. As $\sigma(X^{\circ},X)$ and the mixed topology
$\gamma^{\circ}\coloneqq\gamma(\|\cdot\|_{X^{\circ}},\sigma(X^{\circ},X))$
coincide on $\|\cdot\|_{X^{\circ}}$-bounded sets by Definition 2.1 (b),
$(S^{\circ}_{t})_{t\geq 0}$ is also $\gamma^{\circ}$-strongly continuous. Due
to [13, Proposition 3.22 (a)] we have
$\gamma^{\circ}=\tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))$. ∎

The semigroup $(S^{\circ}_{t})_{t\geq 0}$ in the setting of Proposition 5.9
resembles a bi-continuous semigroup. For instance, we note that the generator
$(-A^{\circ},D(A^{\circ}))$ of $(S^{\circ}_{t})_{t\geq 0}$ from Proposition
5.9 is given by

$\displaystyle D(A^{\circ})$ $\displaystyle=\Bigl{\\{}x^{\circ}\in X^{\circ}\
|\ \tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S^{\circ}_{t}x^{\circ}-x^{\circ}}{t}\;\text{exists
in }X^{\circ}\Bigr{\\}},$ $\displaystyle-A^{\circ}x$
$\displaystyle=\tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S^{\circ}_{t}x^{\circ}-x^{\circ}}{t}\qquad(x^{\circ}\in
D(A^{\circ}))$

and fulfils

$\displaystyle D(A^{\circ})$ $\displaystyle=\Bigl{\\{}x^{\circ}\in X^{\circ}\
|\ \sigma(X^{\circ},X)\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S^{\circ}_{t}x^{\circ}-x^{\circ}}{t}\;\text{ex.\leavevmode\nobreak\
in }X^{\circ},\;\sup_{t\in(0,1]}\frac{\left\lVert
S^{\circ}_{t}x^{\circ}-x^{\circ}\right\rVert_{X^{\circ}}}{t}<\infty\Bigr{\\}},$
$\displaystyle-A^{\circ}x$
$\displaystyle=\sigma(X^{\circ},X)\text{-}\lim_{t\to
0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}}\frac{S^{\circ}_{t}x^{\circ}-x^{\circ}}{t}\qquad(x^{\circ}\in
D(A^{\circ}))$

by [5, I.1.10 Proposition] for the mixed topology
$\gamma^{\circ}=\tau_{c}(X^{\circ},(X,\left\lVert\cdot\right\rVert_{X}))$ and
the exponential boundedness of $(S^{\circ}_{t})_{t\geq 0}$ (cf. [14, p. 6] in
the bi-continuous setting). What is missing for bi-continuity are sequential
completeness of the Saks space
$(X^{\circ},\|\cdot\|_{X^{\circ}},\sigma(X^{\circ},X))$ and (local) bi-
equicontinuity.

######  Remark 5.10.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
Saks space and $(S_{t})_{t\geq 0}$ a $\tau_{X}$-bi-continuous semigroup on $X$
with generator $(-A,D(A))$. If

  1. (i)

$X^{\circ}\cap\\{x^{\prime}\in X^{\prime}\ |\ \|x^{\prime}\|_{X^{\prime}}\leq
1\\}$ is sequentially complete w.r.t. $\sigma(X^{\circ},X)$,

  2. (ii)

every $\|\cdot\|_{X^{\prime}}$-bounded $\sigma(X^{\circ},X)$-null sequence in
$X^{\circ}$ is $\tau_{X}$-equicontinuous on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets,

(see [8, Hypothesis B and C]), then
$(X^{\circ},\|\cdot\|_{X^{\circ}},\sigma(X^{\circ},X))$ is a sequentially
complete Saks space and $(S^{\circ}_{t})_{t\geq 0}$ is a locally
$\sigma(X^{\circ},X)$-bi-continuous semigroup on $X^{\circ}$ by [8,
Proposition 2.4] with generator $(-A^{\circ},D(A^{\circ}))$ fulfilling

$\displaystyle D(A^{\circ})$ $\displaystyle=\\{x^{\circ}\in X^{\circ}\ |\
\exists y^{\circ}\in X^{\circ}\ \forall x\in D(A):\langle-
Ax,x^{\circ}\rangle=\langle x,y^{\circ}\rangle\\},$
$\displaystyle-A^{\circ}x^{\circ}$ $\displaystyle=y^{\circ}\quad(x^{\circ}\in
D(A^{\circ})),$

by [3, Lemma 1].

We refer to [14, 3.9 Example] for examples of sequentially complete Saks
spaces satisfying (i) and (ii) of Remark 5.10.

######  Proposition 5.11.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ and
$(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ be Saks spaces, and
$B\in\mathcal{L}(U;X)$ such that $B$ is also sequentially
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets. Then $B^{\circ}\coloneqq
B^{\prime}|_{X^{\circ}}\in\mathcal{L}(X^{\circ};U^{\circ})$ and is also
$\sigma(X^{\circ},X)$-$\sigma(U^{\circ},U)$-continuous.

###### Proof.

Let $x^{\circ}\in X^{\circ}$, $u\in U$ and $(u_{n})_{n\in\mathds{N}}$ a
$\left\lVert\cdot\right\rVert_{U}$-bounded sequence in $U$ that
$\tau_{U}$-converges to $u$. Since $B\in\mathcal{L}(U;X)$ and $B$ is also
sequentially $\tau_{U}$-$\tau_{X}$-continuous on
$\left\lVert\cdot\right\rVert_{U}$-bounded sets, we have that
$(Bu_{n})_{n\in\mathds{N}}$ is $\left\lVert\cdot\right\rVert_{X}$-bounded and
$\tau_{X}$-convergent to $Bu$. This implies

$\langle B^{\circ}x^{\circ},u\rangle=\langle
x^{\circ},Bu\rangle=\lim_{n\to\infty}\langle x^{\circ},Bu_{n}\rangle,$

yielding $B^{\circ}x^{\circ}\in U^{\circ}$ and the
$\sigma(X^{\circ},X)$-$\sigma(U^{\circ},U)$-continuity of $B^{\circ}$.
Furthermore, we note that

$\displaystyle\left\lVert B^{\circ}x^{\circ}\right\rVert_{U^{\circ}}$
$\displaystyle=\left\lVert
B^{\circ}x^{\circ}\right\rVert_{U^{\prime}}=\sup_{\left\lVert
u\right\rVert_{U}\leq 1}|\langle
B^{\circ}x^{\circ},u\rangle|=\sup_{\left\lVert u\right\rVert_{U}\leq
1}|\langle x^{\circ},Bu\rangle|\leq\sup_{\left\lVert u\right\rVert_{U}\leq
1}\left\lVert Bu\right\rVert_{X}\left\lVert x^{\circ}\right\rVert_{X^{\circ}}$
$\displaystyle=\left\lVert B\right\rVert_{\mathcal{L}(U;X)}\left\lVert
x^{\circ}\right\rVert_{X^{\circ}}$

and thus $B^{\circ}\in\mathcal{L}(X^{\circ};U^{\circ})$. ∎

######  Remark 5.12.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ and
$(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ be Saks spaces,
$B\in\mathcal{L}(U;X)$. Consider the following assertions:

  1. (a)

$B$ is $\tau_{U}$-$\tau_{X}$-continuous on
$\left\lVert\cdot\right\rVert_{U}$-bounded sets.

  2. (b)

$B$ is sequentially $\tau_{U}$-$\tau_{X}$-continuous on
$\left\lVert\cdot\right\rVert_{U}$-bounded sets.

Then (a)$\Rightarrow$(b) holds. Moreover, if
$(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ is C-sequential, then
(b)$\Rightarrow$(a) holds.

###### Proof.

The implication (a)$\Rightarrow$(b) is obviously true. Let assertion (b) hold.
It follows from [28, Theorem 2.3.1] that $B$ is sequentially
$\gamma_{U}$-$\tau_{X}$-continuous. Hence, (a) holds by [27, Theorem 7.4] if
$(U,\gamma_{U})$ is C-sequential. ∎

######  Proposition 5.13.

Let $(V,\left\lVert\cdot\right\rVert_{V})$,
$(W,\left\lVert\cdot\right\rVert_{W})$ be Banach spaces,
$(Z,\left\lVert\cdot\right\rVert_{Z},\tau_{Z})$ a Saks space,
$\mathcal{P}_{\gamma_{Z}}$ a directed system of seminorms that induces the
topology $\gamma_{Z}$, $F\in\mathcal{L}(V;Z)$ and $G\in\mathcal{L}(W;Z)$. Then
the following assertions are equivalent:

  1. (a)

There exists $c_{1}\geq 0$ such that $\left\lVert
F^{\prime}z^{\prime}\right\rVert_{V^{\prime}}\leq c_{1}\left\lVert
G^{\prime}z^{\prime}\right\rVert_{W^{\prime}}$ for all $z^{\prime}\in
Z_{\gamma}^{\prime}$.

  2. (b)

There exists $c_{2}\geq 0$ such that

$\\{Fv\ |\ v\in V,\,\left\lVert v\right\rVert_{V}\leq
1\\}\subseteq\overline{\\{Gw\ |\ w\in W,\,\left\lVert w\right\rVert_{W}\leq
c_{2}\\}}^{\gamma_{Z}}.$

  3. (c)

There exists $c_{3}\geq 0$ such that for all $v\in V$, $\varepsilon>0$ and
$p_{\gamma_{Z}}\in\mathcal{P}_{\gamma_{Z}}$ there is $w\in W$ with
$\left\lVert w\right\rVert_{W}\leq c_{3}\left\lVert v\right\rVert_{V}$ such
that $p_{\gamma_{Z}}(Fv+Gw)\leq\varepsilon$.

Moreover, we can choose $c_{1}=c_{2}=c_{3}$.

###### Proof.

First, we note that if $M$ and $N$ are convex sets in $Z$, then we have
$N\subseteq\overline{M}^{\gamma_{Z}}$ if and only if

$\sup_{z\in M}\operatorname{Re}\langle z^{\prime},z\rangle\leq\sup_{z\in
N}\operatorname{Re}\langle z^{\prime},z\rangle$ (3)

for all $z^{\prime}\in Z_{\gamma}^{\prime}$ by [4, p. 220]. Second, let
$z^{\prime}\in Z_{\gamma}^{\prime}$. For every $z\in Z$ there is
$\lambda_{z}\in\mathds{C}$ with $|\lambda_{z}|\leq 1$ such that $|\langle
z^{\prime},z\rangle|=\operatorname{Re}\langle z^{\prime},\lambda_{z}z\rangle$.
Thus, if $M$ and $N$ are additionally circled sets, then (3) is equivalent to

$\sup_{z\in M}|\langle z^{\prime},z\rangle|\leq\sup_{z\in N}|\langle
z^{\prime},z\rangle|$

for all $z^{\prime}\in Z_{\gamma}^{\prime}$. Hence, setting $N\coloneqq\\{Fv\
|\ v\in V,\,\left\lVert v\right\rVert_{V}\leq 1\\}$, $M\coloneqq\\{Gw\ |\ w\in
W,\,\left\lVert w\right\rVert_{W}\leq c_{2}\\}$ and observing that
$N,M\subseteq Z$ are convex, circled sets, we obtain the equivalence of (a)
and (b), and that we can choose $c_{1}=c_{2}$.

Let us turn to the equivalence of (b) and (c). For $\varepsilon>0$ and
$p_{\gamma_{Z}}\in\mathcal{P}_{\gamma_{Z}}$ we set
$U_{\varepsilon,p_{\gamma_{Z}}}\coloneqq\\{z\in
Z\;|\;p_{\gamma_{Z}}(z)\leq\varepsilon\\}$. For any $M\subseteq Z$ we have

$\overline{M}^{\gamma_{Z}}=\bigcap_{\varepsilon>0,\,p_{\gamma_{Z}}\in\mathcal{P}_{\gamma_{Z}}}M+U_{\varepsilon,p_{\gamma_{Z}}}$
(4)

(see e.g. [11, 2.1.4 Proposition]). Let assertion (b) hold, $v\in V$ with
$v\neq 0$, $\varepsilon>0$ and $p_{\gamma_{Z}}\in\mathcal{P}_{\gamma_{Z}}$.
Then there are $\widetilde{w}\in W$ with
$\left\lVert\widetilde{w}\right\rVert_{W}\leq c_{2}$ and $z\in Z$ with
$p_{\gamma_{Z}}(z)\leq\tfrac{\varepsilon}{\left\lVert v\right\rVert_{V}}$ such
that $F\left(-\tfrac{v}{\left\lVert
v\right\rVert_{V}}\right)=G\widetilde{w}+z$ by (b) and (4). From writing

$-Fv=\left\lVert v\right\rVert_{V}F\left(\frac{-v}{\left\lVert
v\right\rVert_{V}}\right)=\left\lVert
v\right\rVert_{V}(G\widetilde{w}+z)=G(\left\lVert
v\right\rVert_{V}\widetilde{w})+\left\lVert v\right\rVert_{V}z,$

setting $w\coloneqq\left\lVert v\right\rVert_{V}\widetilde{w}$, and using
$\left\lVert w\right\rVert_{W}=\left\lVert
v\right\rVert_{V}\left\lVert\widetilde{w}\right\rVert_{W}\leq c_{2}\left\lVert
v\right\rVert_{V}$ and

$p_{\gamma_{Z}}(Fv+Gw)=p_{\gamma_{Z}}(-\left\lVert
v\right\rVert_{V}z)=\left\lVert
v\right\rVert_{V}p_{\gamma_{Z}}(z)\leq\left\lVert
v\right\rVert_{V}\frac{\varepsilon}{\left\lVert
v\right\rVert_{V}}=\varepsilon,$

we conclude that (c) holds (the case $v=0$ is obvious).

Now, let assertion (c) hold. Let $v\in V$ with $\left\lVert
v\right\rVert_{V}\leq 1$, $\varepsilon>0$ and
$p_{\gamma_{Z}}\in\mathcal{P}_{\gamma_{Z}}$. Then there is $\widetilde{w}\in
W$ with $\left\lVert\widetilde{w}\right\rVert_{W}\leq c_{3}$ such that
$p_{\gamma_{Z}}(Fv+G\widetilde{w})\leq\varepsilon$. Setting
$w\coloneqq-\widetilde{w}$, using $\left\lVert
w\right\rVert=\left\lVert\widetilde{w}\right\rVert_{W}\leq c_{3}$ and
$Fv=Gw+Fv+G\widetilde{w}$, we see that (b) holds due to (4). The proof of the
equivalence of (b) and (c) also shows that we can choose $c_{2}=c_{3}$. ∎

The proof of the equivalence of (a) and (b) is just an adaptation of the proof
of [4, Theorem 2.2, (iii)$\Leftrightarrow$(iv)].

######  Remark 5.14.

Let $(Z,\left\lVert\cdot\right\rVert_{Z},\tau_{Z})$ be a Saks space. Since the
mixed topology $\gamma_{Z}$ coincides with $\tau_{Z}$ on
$\left\lVert\cdot\right\rVert_{Z}$-bounded sets, we may equivalently replace
the $\gamma_{Z}$-closure by the $\tau_{Z}$-closure in Proposition 5.13 (b) and
thus $\mathcal{P}_{\gamma_{Z}}$ in (c) by a directed system of seminorms
$\mathcal{P}_{\tau_{Z}}$ that induces the topology $\tau_{Z}$, too.

######  Proposition 5.15.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
C-sequential Saks space, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$ and $B\in\mathcal{L}(U;X)$ such that $B$ is also
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets.

  1. (a)

$(S_{t}B)_{t\geq 0}$ is quasi-$\gamma_{U}$-$\gamma_{X}$-equicontinuous.

  2. (b)

Let $T>0$ and $\mathcal{B}^{T}\colon C_{\tau,b}([0,T];U)\to X$ be given by

$\mathcal{B}^{T}u\coloneqq\int_{0}^{T}S_{T-t}Bu(t)\mathrm{d}t.$

Then
$\mathcal{B}^{T}\in\mathcal{L}(C_{\tau,b}([0,T];U);X)=\mathcal{L}((C_{\tau,b}([0,T];U),\left\lVert\cdot\right\rVert_{\infty});(X,\left\lVert\cdot\right\rVert_{X}))$
and $\mathcal{B}^{T}$ is also $\gamma_{\infty}$-$\gamma_{X}$-continuous.

  3. (c)

Let
${\mathcal{B}^{T}}^{\circ}x^{\circ}\coloneqq{\mathcal{B}^{T}}^{\prime}x^{\circ}$
for $x^{\circ}\in X^{\circ}$. Then
${\mathcal{B}^{T}}^{\circ}x^{\circ}\in(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}$
for all $x^{\circ}\in X^{\circ}$.

###### Proof.

(a) Let $M\subseteq U$ be a $\left\lVert\cdot\right\rVert_{U}$-bounded set.
Then the restriction $B|_{M}\colon M\to X$ of $B$ to $M$ is
$\tau_{U}|_{M}$-$\tau_{X}$-continuous. Since $B\in\mathcal{L}(U;X)$, the set
$B(M)$ is $\left\lVert\cdot\right\rVert_{X}$-bounded. As the mixed topology
$\gamma_{X}$ coincides with $\tau_{X}$ on
$\left\lVert\cdot\right\rVert_{X}$-bounded sets by Definition 2.1 (b), it
follows that $B|_{M}$ is $\tau_{U}|_{M}$-$\gamma_{X}$-continuous, yielding
that $B$ is $\gamma_{U}$-$\gamma_{X}$-continuous by [5, I.1.7 Corollary]. Due
to [13, Theorem 3.17 (a)] $(S_{t}B)_{t\geq 0}$ is
quasi-$\gamma_{U}$-$\gamma_{X}$-equicontinuous, proving part (a).

(b) Let $\mathcal{P}_{\gamma_{X}}$ and $\mathcal{P}_{\gamma_{U}}$ be directed
systems of seminorms that induce the mixed topologies $\gamma_{X}$ and
$\gamma_{U}$, respectively. For $p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}$ we
set $V_{p_{\gamma_{X}}}\coloneqq\\{x\in X\;|\;p_{\gamma_{X}}(x)<1\\}$ and
denote its polar set by $V_{p_{\gamma_{X}}}^{\circ}\coloneqq\\{x^{\prime}\in
X_{\gamma}^{\prime}\;|\;\forall\;x\in V_{p_{\gamma_{X}}}:\;|x^{\prime}(x)|\leq
1\\}$. It follows from part (a) that there are $C\geq 0$ and
$p_{\gamma_{U}}\in\mathcal{P}_{\gamma_{U}}$ such that for all $u\in
C_{\tau,b}([0,T];U)$ we have

$\displaystyle p_{\gamma_{X}}(\mathcal{B}^{T}u)$
$\displaystyle=\sup_{x^{\prime}\in V_{p_{\gamma_{X}}}^{\circ}}\Bigl{|}\langle
x^{\prime},\int_{0}^{T}S_{T-t}Bu(t)\mathrm{d}t\rangle\Bigr{|}\leq\sup_{x^{\prime}\in
V_{p_{\gamma_{X}}}^{\circ}}\int_{0}^{T}|\langle
x^{\prime},S_{T-t}Bu(t)\rangle|\mathrm{d}t$ $\displaystyle\leq
T\sup_{x^{\prime}\in V_{p_{\gamma_{X}}}^{\circ}}\sup_{t\in[0,T]}|\langle
x^{\prime},S_{T-t}Bu(t)\rangle|=T\sup_{t\in[0,T]}p_{\gamma_{X}}(S_{T-t}Bu(t))$
(5)
$\displaystyle\underset{\mathclap{(a)}}{\leq}CT\sup_{t\in[0,T]}p_{\gamma_{U}}(u(t))$

where we used [20, Proposition 22.14] in the first and second to last equation
to get from $p_{\gamma_{X}}$ to $\sup_{x^{\prime}\in
V_{p_{\gamma_{X}}}^{\circ}}$ and back. Thus, $\mathcal{B}^{T}$ is
$\gamma_{\infty}$-$\gamma_{X}$-continuous. Furthermore, since
$X_{\gamma}^{\prime}$ is norming for $X$ by [15, Lemma 5.5 (a)], we may choose
$\mathcal{P}_{\gamma_{X}}$ such that $\left\lVert
x\right\rVert_{X}=\sup_{p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}}p_{\gamma_{X}}(x)$
for all $x\in X$ by [13, Remark 2.2 (c)]. Due to our previous estimates and
the exponential boundedness of $(S_{t})_{t\geq 0}$ we obtain

$\displaystyle\left\lVert\mathcal{B}^{T}u\right\rVert_{X}$
$\displaystyle=\sup_{p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}}p_{\gamma_{X}}(\mathcal{B}^{T}u)\underset{\eqref{eq:gamma_cont_conv}}{\leq}T\sup_{p_{\gamma_{X}}\in\mathcal{P}_{\gamma_{X}}}\sup_{t\in[0,T]}p_{\gamma_{X}}(S_{T-t}Bu(t))$
$\displaystyle=T\sup_{t\in[0,T]}\left\lVert S_{T-t}Bu(t)\right\rVert_{X}\leq
T\sup_{t\in[0,T]}\left\lVert S_{T-t}\right\rVert_{\mathcal{L}(X)}\left\lVert
Bu(t)\right\rVert_{X}$ $\displaystyle\leq TM\mathrm{e}^{|\omega|T}\left\lVert
B\right\rVert_{\mathcal{L}(U;X)}\sup_{t\in[0,T]}\left\lVert
u(t)\right\rVert_{U},$

yielding $\mathcal{B}^{T}\in\mathcal{L}(C_{\tau,b}([0,T];U);X)$.

(c) It follows from $X^{\circ}=X_{\gamma}^{\prime}$ by Remark 5.8 (a) and part
(b) that ${\mathcal{B}^{T}}^{\circ}$ is the dual map of the
$\gamma_{\infty}$-$\gamma_{X}$-continuous map $\mathcal{B}^{T}$ and hence
${\mathcal{B}^{T}}^{\circ}x^{\circ}\in(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}$
for all $x^{\circ}\in X^{\circ}$. ∎

Now, we are ready to write down the dual system of (ConSys) and to phrase the
kind of final state observability of this dual system we are seeking for. Let
$(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
C-sequential Saks space, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$ with generator $(-A,D(A))$, and $B\in\mathcal{L}(U;X)$ such that $B$ is
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets, and $T>0$. Using Proposition 5.9 and Proposition 5.11, the dual system
of (ConSys) is given by

$\displaystyle\begin{split}\dot{x}(t)&=-A^{\circ}x(t)\quad(t>0),\\\
y(t)&=B^{\circ}x(t)\quad(t\geq 0),\\\ x(0)&=x_{0}\in X^{\circ}.\end{split}$
(ObsSys)

######  Definition 5.16.

We say that (ObsSys) satisfies a _final state observability estimate in
$(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}$_ if there exists
$C_{\mathrm{obs}}\geq 0$ such that

$\left\lVert S_{T}^{\circ}x^{\circ}\right\rVert_{X^{\circ}}\leq
C_{\mathrm{obs}}\left\lVert{\mathcal{B}^{T}}^{\circ}x^{\circ}\right\rVert_{(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}}$

for all $x^{\circ}\in X^{\circ}$.

We spend the remaining part of this section with proving that cost-uniform
approximate $\tau_{X}$-null-controllability in time $T$ via
$C_{\tau,b}([0,T];U)$ of (ConSys) is equivalent to a final state observability
estimate of (ObsSys) in $(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}$, and
that the latter space is actually a certain space of vector measures.

Let $\Omega$ be a Hausdorff locally compact space, $(U,\vartheta_{U})$ a
Hausdorff locally convex space and $\mathcal{P}_{\vartheta_{U}}$ a directed
system of seminorms that induces $\vartheta_{U}$. We denote by
$\mathscr{B}(\Omega)$ the Borel $\sigma$-algebra on $\Omega$, by $M(\Omega)$
the space of all bounded complex (or real) Borel measures on $\Omega$, and by
$M(\Omega;(U,\vartheta_{U})^{\prime})$ the space of all finitely additive
vector measures $\nu\colon\mathscr{B}(\Omega)\to(U,\vartheta_{U})^{\prime}$,
i.e. $\nu(N_{1}\cup N_{2})=\nu(N_{1})+\nu(N_{2})$ for all disjoint
$N_{1},N_{2}\in\mathscr{B}(\Omega)$, such that

  1. (i)

$\nu(\cdot)u\in M(\Omega)$ for all $u\in U$, and

  2. (ii)

there exist $p\in\mathcal{P}_{\vartheta_{U}}$ and $C\geq 0$ such that

$\sup_{(\mathcal{N},\mathcal{U}_{p})}\bigl{|}\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{p})}\nu(N)u\bigr{|}\leq
C$

where the supremum is taken over all finite partitions $\mathcal{N}$ of
$\Omega$ into disjoint Borel sets and all finite sets $\mathcal{U}_{p}$ in $U$
such that $p(u)\leq 1$ for all $u\in\mathcal{U}_{p}$.

Let $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ be a Saks space and $T>0$.
By [26, Theorem 1], the compactness of $[0,T]$ and Remark 5.2 (a) the map

$\Theta_{\gamma}\colon
M([0,T];U_{\gamma}^{\prime})\to(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime},\;\Theta_{\gamma}(\nu)(u)\coloneqq\int_{0}^{T}u(t)\mathrm{d}\nu,$

is a linear isomorphism. By the same theorem in combination with [26, Lemma 4]
the map

$\Theta_{\left\lVert\cdot\right\rVert}\colon
M([0,T];U^{\prime})\to(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime},\;\Theta_{\left\lVert\cdot\right\rVert}(\nu)(u)\coloneqq\int_{0}^{T}u(t)\mathrm{d}\nu,$

is a topological isomorphism w.r.t. the semivariation norm on
$M([0,T];U^{\prime})$ and the dual norm on
$(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}$
and

$\left\lVert\Theta_{\left\lVert\cdot\right\rVert}(\nu)\right\rVert_{(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}}=\left\lVert\nu\right\rVert_{var}\qquad(\nu\in
M([0,T];U^{\prime}))$

where the semivariation norm is given by

$\left\lVert\nu\right\rVert_{var}\coloneqq\sup_{(\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})}\bigl{|}\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})}\nu(N)u\bigr{|}\qquad(\nu\in
M([0,T];U^{\prime}))$

and the supremum is taken over all
$(\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})$ as in (ii)
above with $p$ replaced by $\left\lVert\cdot\right\rVert_{U}$. We note that it
follows from $U_{\gamma}^{\prime}$ being a topological subspace of
$U^{\prime}$ (see [5, I.1.18 Proposition]) and $\gamma_{U}$ being coarser than
the $\left\lVert\cdot\right\rVert_{U}$-topology, that
$M([0,T];U_{\gamma}^{\prime})$ is a topological subspace of
$M([0,T];U^{\prime})$ (if equipped with the relative topology).

######  Theorem 5.17.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
C-sequential Saks space, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$ and $B\in\mathcal{L}(U;X)$ such that $B$ is also
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets. For $T>0$ we have $B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda\in
M([0,T];U_{\gamma}^{\prime})$ and
${\mathcal{B}^{T}}^{\circ}x^{\circ}=\Theta_{\gamma}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)$
for all $x^{\circ}\in X^{\circ}$ as well as

$\left\lVert{\mathcal{B}^{T}}^{\circ}x^{\circ}\right\rVert_{(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}}=\left\lVert
B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda\right\rVert_{var}\qquad(x^{\circ}\in
X^{\circ})$

where

$(B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda)(N)u\coloneqq\int_{N}\langle
B^{\circ}S_{t}^{\circ}x^{\circ},u\rangle\mathrm{d}t\qquad(N\in\mathscr{B}([0,T]),\,u\in
U)$

for the Lebesgue measure $\lambda$.

###### Proof.

First, we recall that $X^{\circ}=X_{\gamma}^{\prime}$ by Remark 5.8 (a) as the
C-sequential space $(X,\gamma_{X})$ is Mazur. Let $x^{\circ}\in X^{\circ}$.
Due to Proposition 5.15 (b) and Proposition 5.9 $\mathcal{B}^{T}$ is
$\gamma_{\infty}$-$\gamma_{X}$-continuous and

$\langle{\mathcal{B}^{T}}^{\circ}x^{\circ},u\rangle=\int_{0}^{T}\langle
x^{\circ},S_{T-t}Bu(t)\rangle\mathrm{d}t=\int_{0}^{T}\langle
B^{\circ}S_{T-t}^{\circ}x^{\circ},u(t)\rangle\mathrm{d}t$

for all $u\in C_{\tau,b}([0,T];U)$.

For $N\in\mathscr{B}([0,T])$ and $u\in U$ we define
$(B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda)(N)u\coloneqq\int_{N}\langle
B^{\circ}S_{t}^{\circ}x^{\circ},u\rangle\mathrm{d}t$ and show that
$B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda\in
M([0,T];U_{\gamma}^{\prime})$. By the proof of Proposition 5.15 (b) there are
$C\geq 0$ and $p_{\gamma_{U}}\in\mathcal{P}_{\gamma_{U}}$ such that

$|\langle B^{\circ}S_{T-t}^{\circ}x^{\circ},u\rangle|=|\langle
x^{\circ},S_{T-t}Bu\rangle|\leq Cp_{\gamma_{U}}(u)$

for all $t\in[0,T]$ and all $u\in U$. In combination with the continuity of
the map $t\mapsto\langle B^{\circ}S_{T-t}^{\circ}x^{\circ},u\rangle$ on
$[0,T]$ by Proposition 5.9 and Proposition 5.11 this implies that
$B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda\colon\mathscr{B}([0,T])\to
U_{\gamma}^{\prime}$ is a well-defined finitely additive vector measure and
that

$(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(\boldsymbol{\cdot})u=\int_{(\boldsymbol{\cdot})}\langle
B^{\circ}S_{T-t}^{\circ}x^{\circ},u\rangle\mathrm{d}t\qquad(u\in U)$

belongs to $M(\Omega)$. Let $\mathcal{N}$ be a finite partition of $[0,T]$
into disjoint Borel sets and $\mathcal{U}_{p_{\gamma_{U}}}$ a finite subset of
$U$ such that $p_{\gamma_{U}}(u)\leq 1$ for all
$u\in\mathcal{U}_{p_{\gamma_{U}}}$. Then we have

$\displaystyle\bigl{|}\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{p_{\gamma_{U}}})}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(N)u\bigr{|}$
$\displaystyle=\bigl{|}\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{p_{\gamma_{U}}})}\int_{N}\langle
B^{\circ}S_{T-t}^{\circ}x^{\circ},u\rangle\mathrm{d}t\bigr{|}$
$\displaystyle\leq\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{p_{\gamma_{U}}})}\lambda(N)Cp_{\gamma_{U}}(u)$
$\displaystyle\leq C\sum_{N\in\mathcal{N}}\lambda(N)=C\lambda([0,T])=CT,$

yielding $B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda\in
M([0,T];U_{\gamma}^{\prime})$. Analogously,
$B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda\in
M([0,T];U_{\gamma}^{\prime})$.

Finally, since

$\displaystyle\langle{\mathcal{B}^{T}}^{\circ}x^{\circ},u\rangle$
$\displaystyle=\int_{0}^{T}\langle
B^{\circ}S_{T-t}^{\circ}x^{\circ},u(t)\rangle\mathrm{d}t=\int_{0}^{T}u(t)\mathrm{d}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)$
$\displaystyle=\Theta_{\gamma}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(u)$

for all $u\in C_{\tau,b}([0,T];U)$ and

$\langle{\mathcal{B}^{T}}^{\circ}x^{\circ},u\rangle=\Theta_{\gamma}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(u)=\Theta_{\left\lVert\cdot\right\rVert}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(u)$

for all $u\in C([0,T];(U,\left\lVert\cdot\right\rVert_{U}))$, it holds that

$\displaystyle\left\lVert{\mathcal{B}^{T}}^{\circ}x^{\circ}\right\rVert_{(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}}$
$\displaystyle=\left\lVert\Theta_{\left\lVert\cdot\right\rVert}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)\right\rVert_{(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}}$
$\displaystyle=\left\lVert
B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda\right\rVert_{var}.$

Let $\mathcal{N}$ be a finite partition of $[0,T]$ and
$\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}}$ a finite set in $U$ such that
$\left\lVert u\right\rVert_{U}\leq 1$ for all
$u\in\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}}$. Then
$T-\mathcal{N}\coloneqq\\{T-N\;|\;N\in\mathcal{N}\\}$ is also a finite
partition of $[0,T]$, and

$\displaystyle\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})}(B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda)(N)u$
$\displaystyle=\sum_{(N,u)\in(\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})}\int_{N}\langle
B^{\circ}S_{T-t}^{\circ}x^{\circ},u\rangle\mathrm{d}t$
$\displaystyle=-\sum_{(T-N,u)\in(T-\mathcal{N},\mathcal{U}_{\left\lVert\cdot\right\rVert_{U}})}\;\int_{T-N}\langle
B^{\circ}S_{t}^{\circ}x^{\circ},u\rangle\mathrm{d}t.$

Thus, we obtain

$\left\lVert{\mathcal{B}^{T}}^{\circ}x^{\circ}\right\rVert_{(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}}=\left\lVert
B^{\circ}S_{T-(\cdot)}^{\circ}x^{\circ}\odot\lambda\right\rVert_{var}=\left\lVert
B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda\right\rVert_{var}.\qed$

######  Theorem 5.18.

Let $(X,\left\lVert\cdot\right\rVert_{X},\tau_{X})$ be a sequentially complete
C-sequential Saks space, $(U,\left\lVert\cdot\right\rVert_{U},\tau_{U})$ a
Saks space, $(S_{t})_{t\geq 0}$ a locally $\tau_{X}$-bi-continuous semigroup
on $X$ and $B\in\mathcal{L}(U;X)$ such that $B$ is also
$\tau_{U}$-$\tau_{X}$-continuous on $\left\lVert\cdot\right\rVert_{U}$-bounded
sets, and $T>0$. Then the following assertions are equivalent:

  1. (a)

The system in (ConSys) is cost-uniform approximately $\tau_{X}$-null-
controllable in time $T$ via $C_{\tau,b}([0,T];U)$.

  2. (b)

The system in (ObsSys) satisfies a final state observability estimate in
$(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}$.

If additionally $\tau_{U}=\tau_{\left\lVert\cdot\right\rVert_{U}}$, then each
of the preceding assertions is equivalent to:

  1. (c)

There exists $C_{\mathrm{obs}}\geq 0$ such that

$\forall\;x^{\circ}\in X^{\circ}:\quad\left\lVert
S_{T}^{\circ}x^{\circ}\right\rVert_{X^{\circ}}\leq C_{\mathrm{obs}}\left\lVert
B^{\circ}S_{(\cdot)}^{\circ}x^{\circ}\odot\lambda\right\rVert_{var}.$

###### Proof.

This statement follows from the equivalence of (a) and (c) in Proposition 5.13
with $V\coloneqq Z\coloneqq X$, $W\coloneqq C_{\tau,b}([0,T];U)$ equipped with
$\left\lVert\cdot\right\rVert_{\infty}$, $F\coloneqq S_{T}$ and
$G\coloneqq\mathcal{B}^{T}$ in combination with Theorem 5.17, Remark 5.14 and
$X^{\circ}=X_{\gamma}^{\prime}$. ∎

Even in the setting of Banach spaces, i.e.
$\tau_{X}=\tau_{\left\lVert\cdot\right\rVert_{X}}$,
$\tau_{U}=\tau_{\left\lVert\cdot\right\rVert_{U}}$, where we have
$C_{\tau,b}([0,T];U)=C([0,T];(U,\left\lVert\cdot\right\rVert_{U}))$ and

$(C_{\tau,b}([0,T];U),\gamma_{\infty})^{\prime}=(C([0,T];(U,\left\lVert\cdot\right\rVert_{U})),\left\lVert\cdot\right\rVert_{\infty})^{\prime}=M([0,T];U^{\prime})$

as well as $X^{\circ}=X^{\prime}$, the results of Theorem 5.18 seem to be new.

## Statements and Declarations

We thank F. Gabel for valuable comments and corrections. K. Kruse acknowledges
the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research
Training Group GRK 2583 “Modeling, Simulation and Optimization of Fluid
Dynamic Applications”.

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52 |  | outer-1 |  | 11.96 | 28.7782 | 1.28205 | no
53 |  | outer-2 |  | 14.867 | 22.4266 | 1.28966 | no
56 |  |  |  | 15.8499 | 24.8177 | 1.35955 | no
60 |  |  |  | 15.94 | 11.1737 | 1.57827 | no
64 |  |  |  | 15.7765 | 20.6513 | 1.89283 | no
67 |  | S7-228 |  | 12.0222 | 7.92802 | 1.50092 | no
75 | S93 |  |  | 15.38 | 1.09141 | 1.80819 | no
76 | S26 | S0-31 | E13 | 15.4293 | 0.721656 | 1.65122 | no
77 | S7 | S0-11 | E12 | 15.5999 | 0.497586 | 1.6382 | no
87 |  |  |  | 16.7216 | 14.4812 | 1.86302 | no
90 |  |  |  | 13.7248 | 10.1881 | 1.44307 | no
91 |  |  |  | 13.8585 | 14.4931 | 1.75828 | no
100 |  |  |  | 12.2566 | 8.03726 | 1.12067 | no
102 |  |  |  | 11.2038 | 10.6092 | 1.84185 | no
118 |  |  |  | 13.1939 | 11.152 | 0.922799 | no
123 |  | S7-30 |  | 14.05 | 7.01209 | 1.2573 | no
126 |  |  |  | 14.9965 | 6.36436 | 0.720289 | no
136 |  | S4-258 | E60 | 12.45 | 4.67586 | 0.651513 | no
149 |  | S9-1 | E78 | 12.76 | 9.46204 | 1.60932 | no
155 |  |  |  | 12.7962 | 9.49969 | 1.37407 | no
Table 8: Stars consistent with belonging to the inner warp feature. index | MPE | UCLA | Paumard | Mag | R | Delta Evidence | Orbit
---|---|---|---|---|---|---|---
6 | S60 |  |  | 16.76 | 0.113085 | 0.629584 | yes
30 | R70 | S4-36 | E54 | 12.82 | 2.08898 | 0.656347 | yes
10 |  | S7-236 |  | 12.666 | 7.94508 | 1.3941 | no
30 |  | S7-10 | E73 | 11.62 | 7.70832 | 1.05606 | no
36 |  | S9-23 | E77 | 13.57 | 9.23117 | 1.30484 | no
40 |  | S10-5 | E83 | 11.97 | 10.1532 | 1.78867 | no
65 |  |  |  | 15.3536 | 8.67658 | 1.53632 | no
84 |  | S5-237 |  | 13.25 | 5.59141 | 1.15228 | no
85 |  | S4-169 | E57 | 13.53 | 4.43282 | 1.95868 | no
86 |  | S6-82 | E72 | 13.56 | 6.73303 | 1.50384 | no
96 |  | S7-16 |  | 12.6886 | 7.42579 | 1.71009 | no
99 |  | S6-63 | E69 | 11.36 | 6.57862 | 0.921401 | no
103 |  |  | E65 | 12.13 | 6.3003 | 1.06481 | no
105 |  |  |  | 12.9989 | 6.79444 | 1.0242 | no
106 |  |  |  | 12.2293 | 7.36999 | 1.69583 | no
143 |  | S2-22 |  | 13.03 | 2.33073 | 1.99339 | no
145 |  | S5-231 | E64 | 12.01 | 5.81382 | 1.60709 | no
150 |  | S6-81 | E67 | 11.08 | 6.3722 | 1.47119 | no
153 |  |  | E63 | 9.328 | 5.297 | 0.938662 | no
Table 9: Stars consistent with beging on outer warp of the clockwise disk. MPE | UCLA | Paumard | Mag | R |  $\Delta$ Evidence | Orbit
---|---|---|---|---|---|---
S9 | S0-5 | E9 | 15.15 | 0.091365 | 1.38003 | yes
S60 |  |  | 16.76 | 0.113116 | 0.589293 | yes
| S5-235 |  | 13.21 | 5.33721 | 1.95023 | no
|  |  | 14.78 | 6.96797 | 0.958125 | no
|  |  | 13.71 | 7.06065 | 1.15084 | no
| S8-4 | E75 | 11.12 | 8.54245 | 0.321072 | no
| S9-23 | E77 | 13.57 | 9.2312 | 0.983455 | no
| S10-5 | E83 | 11.97 | 10.1532 | 0.673866 | no
| S10-4 | E84 | 11.29 | 10.2412 | 0.360631 | no
|  |  | 13.8392 | 12.0786 | 1.61299 | no
| S10-48 | E86 | 15.11 | 10.7369 | 1.76352 | no
| S11-5 | E88 | 11.70 | 11.7584 | 1.19523 | no
|  |  | 13.87 | 19.3905 | 1.95884 | no
|  |  | 14.4037 | 16.7276 | 0.872181 | no
|  |  | 15.3588 | 11.2452 | 1.91458 | no
| S11-21 | E87 | 13.51 | 11.2396 | 0.531913 | no
|  |  | 15.3722 | 9.27017 | 1.86716 | no
|  |  | 15.3536 | 8.67654 | 1.52626 | no
|  |  | 14.3489 | 15.152 | 1.82959 | no
| S1-33 |  | 15.06 | 1.2391 | 1.43069 | no
| S2-58 |  | 14.04 | 2.45128 | 0.99258 | no
| S4-169 | E57 | 13.53 | 4.4328 | 1.72397 | no
|  |  | 16.7216 | 14.4813 | 1.87428 | no
| S2-76 |  | 15.15 | 2.81263 | 1.45724 | no
|  |  | 12.6724 | 11.3924 | 1.95734 | no
|  |  | 11.4295 | 9.36258 | 0.50663 | no
| S9-20 | E76 | 13.25 | 9.11447 | 1.2888 | no
| S7-16 |  | 12.6886 | 7.42576 | 0.981402 | no
|  |  | 12.1135 | 8.81131 | 1.71294 | no
| S5-187 |  | 13.21 | 5.78784 | 1.15952 | no
|  |  | 12.2293 | 7.36998 | 1.94377 | no
| S4-71 | E55 | 12.61 | 4.13535 | 1.1851 | no
| S3-30 | E47 | 12.65 | 3.4013 | 1.13645 | no
| S3-374 | E53 | 12.73 | 3.93736 | 1.02151 | no
| S2-13 | E33 | 11.71 | 2.19857 | 1.75181 | no
| S1-24 | E26 | 11.82 | 1.7597 | 1.69777 | no
| S2-22 |  | 13.03 | 2.33074 | 1.45088 | no
| S5-231 | E64 | 12.01 | 5.81383 | 1.35316 | no
| S7-161 |  | 13.6985 | 7.36771 | 1.74088 | no
|  |  | 15.70 | 1.05406 | 1.95867 | no
| S3-190 |  | 14.06 | 3.47265 | 1.7133 | no
Table 10: Stars consistent with beging on the newly detected disk. MPE | UCLA | Paumard | Mag | R |  $\Delta$ Evidence | Orbit
---|---|---|---|---|---|---
S4 | S0-3 | E6 | 14.58 | 0.091998 | 1.90794 | yes
S14 |  | E2 | 15.63 | 0.00271191 | 0.0173749 | yes
S2 |  | E1 | 14.16 | 0.0107373 | 1.44597 | yes
|  |  | 12.4966 | 11.3631 | 1.85805 | no
| IRS10W |  | 10.9925 | 8.29025 | 1.99354 | no
|  | E90 | 11.57 | 12.7704 | 0.561536 | no
|  |  | 13.6698 | 9.79173 | 0.463197 | no
|  |  | 13.71 | 7.06064 | 1.24503 | no
| S7-10 | E73 | 11.62 | 7.70833 | 1.77437 | no
| S8-15 |  | 13.20 | 8.19548 | 1.24777 | no
|  |  | 13.3406 | 12.2262 | 0.875552 | no
|  |  | 14.1015 | 13.2585 | 1.55044 | no
|  |  | 13.7893 | 14.4404 | 1.44733 | no
|  |  | 13.8392 | 12.0786 | 1.7897 | no
|  |  | 14.7092 | 20.9025 | 1.34779 | no
|  |  | 12.7599 | 20.3779 | 1.0109 | no
|  |  | 13.87 | 19.3904 | 0.882945 | no
|  |  | 14.4037 | 16.7275 | 1.21626 | no
| outer-1 |  | 11.96 | 28.7817 | 1.22527 | no
|  |  | 15.4278 | 14.3609 | 0.223891 | no
|  |  | 15.0569 | 10.0754 | 0.611814 | no
|  |  | 15.94 | 11.1737 | 1.2928 | no
| S11-21 | E87 | 13.51 | 11.2396 | 1.53688 | no
|  |  | 15.3722 | 9.27012 | 0.551794 | no
|  |  | 16.6731 | 21.158 | 0.798925 | no
|  |  | 15.7765 | 20.651 | 1.18804 | no
|  |  | 15.41 | 5.64006 | 1.65611 | no
|  |  | 14.3489 | 15.152 | 0.281202 | no
|  |  | 16.5563 | 11.611 | 1.77148 | no
|  | E46 | 10.703 | 3.38768 | 0.919629 | no
| S2-4 | E28 | 12.31 | 2.08402 | 1.69374 | no
| S2-50 |  | 15.34 | 2.27306 | 0.983683 | no
| S3-3 |  | 15.1047 | 3.14943 | 1.12699 | no
S93 |  |  | 15.38 | 1.09142 | 1.20983 | no
S7 | S0-11 | E12 | 15.5999 | 0.496342 | 1.10106 | no
| S3-314 |  | 15.50 | 3.83191 | 1.05654 | no
| S3-2 |  | 12.25 | 3.11262 | 1.6286 | no
|  |  | 16.7216 | 14.4813 | 0.294344 | no
| S2-76 |  | 15.15 | 2.81263 | 1.73484 | no
| S9-114 | E79 | 11.00 | 9.48721 | 0.556273 | no
|  |  | 11.2038 | 10.6092 | 0.709141 | no
| S5-187 |  | 13.21 | 5.78785 | 1.98491 | no
| S4-196 |  | 14.4433 | 4.52485 | 1.68582 | no
| S8-181 | E74 | 11.88 | 8.41553 | 1.14199 | no
|  |  | 12.3832 | 9.51841 | 1.17089 | no
| S9-143 |  | 12.8182 | 9.01071 | 1.68612 | no
| S8-196 |  | 12.557 | 8.59268 | 1.07554 | no
| S5-34 |  | 13.7426 | 5.09913 | 0.966934 | no
| S3-26 | E45 | 12.73 | 3.33865 | 1.81866 | no
| S7-180 | E58 | 13.5213 | 7.52872 | 1.9775 | no
| S2-6 | E30 | 12.27 | 2.098 | 1.8105 | no
| S5-236 |  | 13.2592 | 5.70296 | 1.91062 | no
| S7-161 |  | 13.6985 | 7.3677 | 1.75061 | no