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

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.

1 Introduction

Let X be a Banach space and $(S_t)_{t\ge 0}$ a semigroup on X, i.e. $S_0 = I$ and $S_{t+s} = S_t S_s$ for all $t,s\ge 0$. Moreover, let Y be another Banach space and $C\in \mathcal {L}(X;Y)$, a so-called observation operator, and $T>0$. Then $(S_t)_{t\ge 0}$ 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 $C_{\textrm{obs}}\ge 0$ such that

$$\begin{aligned} \left\Vert S_T x\right\Vert _X \le C_{\textrm{obs}} \left\Vert CS_{(\cdot )} x\right\Vert _{\mathcal {Z}} \qquad (x\in X). \end{aligned}$$

Put differently, we can estimate the norm of the final state $S_Tx$ by just taking into account the observations $CS_tx$ for $t\in [0,T]$. Typical applications stem from evolution equations on some function space over (a subset of) $\mathbb {R}^d$, where C is a restriction operator to a suitable subset $\Omega$ of $\mathbb {R}^d$ (or of the subset of $\mathbb {R}^d$ the functions are defined on) such that we want to control the final state on all of $\mathbb {R}^d$ 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. [1, 2, 4, 9, 10, 18, 21, 23, 25, 29] 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

$$\begin{aligned} \left\Vert S_T x\right\Vert _X \le {\left\{ \begin{array}{ll} C_{\textrm{obs}} \left( \int _0^T \left\Vert CS_t x\right\Vert _Y^r \textrm{d}t\right) ^{1/r} &{} \text {if } r\in [1,\infty ),\\ C_{\textrm{obs}}{{\,\mathrm{ess\,sup}\,}}_{t\in [0,T]} \left\Vert CS_t x\right\Vert _Y &{} \text {if } r=\infty , \end{array}\right. } \qquad (x\in X).\end{aligned}$$

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 $t\mapsto \left\Vert CS_t x\right\Vert _Y$ for all $x\in X$. Of course, strong continuity of $(S_t)_{t\ge 0}$ 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 paper 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 $C_b(\mathbb {R}^d)$, the space of bounded continuous functions (on $\mathbb {R}^d$), as well as the Ornstein–Uhlenbeck semigroup on $C_b(\mathbb {R}^d)$. 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 [4, 6, 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 Sect. 2 we review bi-continuous semigroups and then turn to final-state observability estimates in Sect. 3 together with two examples in Sect. 4. Final state observability estimates are then related with cost-uniform approximate null-controllability via duality, which we will exploit in our context in Sect. 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 $\mathbb {R}$ or $\mathbb {C}$ with a Hausdorff locally convex topology $\tau _X$ we denote by $(X,\tau _X)'$ the topological linear dual space and just write $X':=(X,\tau _X)'$ if $(X,\tau _X)$ is a Banach space. We use the symbol $\mathcal {L}(X;Y):=\mathcal {L}((X,\left\Vert \cdot \right\Vert _{X});(Y,\left\Vert \cdot \right\Vert _{Y}))$ for the space of continuous linear operators from a Banach space $(X,\left\Vert \cdot \right\Vert _{X})$ to a Banach space $(Y,\left\Vert \cdot \right\Vert _{Y})$ and denote by $\left\Vert \cdot \right\Vert _{\mathcal {L}(X;Y)}$ 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 $(X,\left\Vert \cdot \right\Vert _X)$ be a normed space and $\tau _X$ a Hausdorff locally convex topology on X.

• (a) The triple $(X,\left\Vert \cdot \right\Vert _X,\tau _{X})$ is called a Saks space if $(X,\left\Vert \cdot \right\Vert _X)$ is a Banach space, $\tau _{X}\subseteq \tau _{\left\Vert \cdot \right\Vert _X}$ and $(X,\tau _{X})'$ is norming for X where $\tau _{\left\Vert \cdot \right\Vert _X}$ denotes the $\left\Vert \cdot \right\Vert _X$-topology.

• (b) The mixed topology $\gamma _X:=\gamma (\left\Vert \cdot \right\Vert _X,\tau _X)$ is the finest linear topology on X that coincides with $\tau _X$ on $\left\Vert \cdot \right\Vert _X$-bounded sets and such that $\tau _X\subseteq \gamma _X \subseteq \tau _{\left\Vert \cdot \right\Vert _X}$.

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 $(X,\left\Vert \cdot \right\Vert _X,\tau _{X})$ sequentially complete if $(X,\gamma _X)$ is sequentially complete.

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

Definition 2.3

([17, Definition 3]) Let $(X,\left\Vert \cdot \right\Vert _X,\tau _{X})$ be a sequentially complete Saks space. Let $(S_t)_{t\ge 0}$ in $\mathcal {L}(X)$ be a semigroup on X. We say that $(S_t)_{t\ge 0}$ is (locally)  $\tau _X$-bi-continuous if

• (a) it is exponentially bounded, i.e. there exist $M\ge 1$ and $\omega \in \mathbb {R}$ such that $\left\Vert S_t\right\Vert _{\mathcal {L}(X)}\le Me^{\omega t}$ for all $t\ge 0$,

• (b) $(S_t)_{t\ge 0}$ is a $C_0$-semigroup on $(X, \tau _X)$, i.e. for all $x\in X$ the map $[0,\infty )\ni t\mapsto T_tx\in (X,\tau _X)$ is continuous,

• (c) it is (locally) bi-equicontinuous, i.e. for every $(x_n)_{n\in \mathbb {N}}$ in X and $x\in X$ with $\sup \limits _{n\in \mathbb {N}}\left\Vert x_n\right\Vert _X<\infty$ and $\tau _X\text {-}\lim \limits _{n\rightarrow \infty } x_n = x$ we have

  $$\begin{aligned} \tau _X\text {-}\lim _{n\rightarrow \infty } S_t (x_n-x) = 0 \end{aligned}$$

  (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 $(X,\left\Vert \cdot \right\Vert _X,\tau _{X})$ be a sequentially complete Saks space and $(S_t)_{t\ge 0}$ a locally $\tau _{X}$-bi-continuous semigroup on X. The generator $(-A,D(A))$ is defined by

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 [1, 9, 21, 23].

Theorem 3.1

([1, Theorem A.1]) Let X and Y be Banach spaces, $C\in \mathcal {L}(X;Y)$, $(S_t)_{t\ge 0}$ a semigroup on X, $M \ge 1$ and $\omega \in \mathbb {R}$ such that $\left\Vert S_t\right\Vert _{\mathcal {L}(X)} \le M \textrm{e}^{\omega t}$ for all $t \ge 0$, and assume that for all $x\in X$ the map $t\mapsto \left\Vert C S_t x\right\Vert _Y$ is measurable. Further, let $\lambda ^*\ge 0$, $(P_\lambda )_{\lambda >\lambda ^*}$ 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 $d_2\ge 1$, and assume that

$$\begin{aligned} \forall x\in X \ \forall \lambda > \lambda ^* :\quad \left\Vert P_\lambda x\right\Vert _{ X } \le d_0 \textrm{e}^{d_1 \lambda ^{\gamma _1}} \left\Vert C P_\lambda x\right\Vert _{Y } \end{aligned}$$

(UP)

and

$$\begin{aligned} \forall x\in X \ \forall \lambda > \lambda ^* \ \forall t\in (0,T/2] :\quad \left\Vert (I-P_\lambda ) S_t x\right\Vert _{X} \le d_2 \textrm{e}^{-d_3 \lambda ^{\gamma _2} t^{\gamma _3}} \left\Vert x\right\Vert _{X}. \end{aligned}$$

(DISS)

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

$$\begin{aligned} \left\Vert S_T x\right\Vert _{X} \le {\left\{ \begin{array}{ll} C_{\textrm{obs}} \left( \int _0^T \left\Vert CS_t x\right\Vert _Y^r \textrm{d}t\right) ^{1/r} &{} \text {if } r\in [1,\infty ),\\ C_{\textrm{obs}}{{\,\mathrm{ess\,sup}\,}}_{t\in [0,T]} \left\Vert CS_t x\right\Vert _Y &{} \text {if } r=\infty . \end{array}\right. } \end{aligned}$$

Remark 3.2

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

$$\begin{aligned} C_{\textrm{obs}} = \frac{C_1}{T^{1/r}} \exp \left( \frac{C_2}{T^{\frac{\gamma _1 \gamma _3}{\gamma _2 - \gamma _1}}} + C_3 T\right) , \end{aligned}$$

with $T^{1/r} = 1$ if $r=\infty$, and suitable constants $C_1, C_2, C_3\ge 0$ 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 $[0,\infty )\ni t\mapsto \left\Vert CS_t x\right\Vert _Y$ for all $x\in X$.

Lemma 3.3

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete Saks space, $(S_t)_{t\ge 0}$ a locally $\tau _X$-bi-continuous semigroup on X, $(Y,\left\Vert \cdot \right\Vert _Y,\tau _{Y})$ a Saks space, $C:X\rightarrow Y$ linear and sequentially $\tau _X$-$\tau _Y$-continuous on $\left\Vert \cdot \right\Vert _X$-bounded sets, and $x\in X$. Then $[0,\infty )\ni t\mapsto \left\Vert CS_t x\right\Vert _Y$ is measurable.

Proof

Since $[0,\infty ) \ni t\mapsto \left|\left\langle y', CS_tx\right\rangle \right|$ is continuous for all $y'\in (Y,\tau _Y)'$ by the assumptions on C and the exponential boundedness of $(S_t)_{t\ge 0}$, and $\left\Vert CS_t x\right\Vert _Y = \sup \{ \left|\left\langle y', CS_tx\right\rangle \right|\ | \ y'\in (Y,\tau _Y)',\, \left\Vert y'\right\Vert _{Y'}\le 1\}$ for all $t\ge 0$, we obtain that $[0,\infty )\ni t\mapsto \left\Vert CS_t x\right\Vert _Y$ is lower semi-continuous and hence measurable. $\square$

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

Theorem 3.4

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete Saks space, $(S_t)_{t\ge 0}$ a locally $\tau _X$-bi-continuous semigroup on X, $(Y,\left\Vert \cdot \right\Vert _Y,\tau _Y)$ a Saks space, $C\in \mathcal {L}(X;Y)$ such that C is also sequentially $\tau _X$-$\tau _Y$-continuous on $\left\Vert \cdot \right\Vert _X$-bounded sets. Further, let $\lambda ^* \ge 0$, $(P_\lambda )_{\lambda >\lambda ^*}$ 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 $d_2\ge 1$, and assume that

$$\begin{aligned} \forall x\in X \ \forall \lambda > \lambda ^* :\quad \left\Vert P_\lambda x\right\Vert _{ X } \le d_0 \textrm{e}^{d_1 \lambda ^{\gamma _1}} \left\Vert C P_\lambda x\right\Vert _{Y } \end{aligned}$$

(UP')

and

$$\begin{aligned} \forall x\in X \ \forall \lambda > \lambda ^* \ \forall t\in (0,T/2] :\quad \left\Vert (I-P_\lambda ) S_t x\right\Vert _{X} \le d_2 \textrm{e}^{-d_3 \lambda ^{\gamma _2} t^{\gamma _3}} \left\Vert x\right\Vert _{X}. \end{aligned}$$

(DISS')

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

$$\begin{aligned} \left\Vert S_T x \right\Vert _{X} \le {\left\{ \begin{array}{ll} C_{\textrm{obs}} \left( \int _0^T \left\Vert CS_t x\right\Vert _Y^r \textrm{d}t\right) ^{1/r} &{} \text {if } r\in [1,\infty ),\\ C_{\textrm{obs}}{{\,\mathrm{ess\,sup}\,}}_{t\in [0,T]} \left\Vert CS_t x\right\Vert _Y &{} \text {if } r=\infty . \end{array}\right. } \end{aligned}$$

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 $t\mapsto \left\Vert CS_tx\right\Vert _Y$ on a measurable subset $E\subseteq [0,T]$ with positive Lebesgue measure; cf. e.g. [2]. However, in this case the constant $C_{\textrm{obs}}$ (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 $C_b(\mathbb {R}^d)$ and the Ornstein–Uhlenbeck semigroup on $C_b(\mathbb {R}^d)$. We begin with the study of restriction operators on $C_b(\mathbb {R}^d)$, restricting functions to suitable subsets, and relate this to an abstract uncertainty principle.

4.1 Restriction operators on $C_b(\mathbb {R}^d)$ and the uncertainty principle

Let $\Omega \subseteq \mathbb {R}^d$ be non-empty, $C:C_b(\mathbb {R}^d)\rightarrow C_b(\Omega )$ the restriction operator defined by $Cf:=f|_\Omega$ for $f\in C_b(\mathbb {R}^d)$. Then $C\in \mathcal {L}(C_b(\mathbb {R}^d);C_b(\Omega ))$. Let $\tau _{\textrm{co}}$ be the compact-open topology on $C_b(\mathbb {R}^d)$ (as well as on $C_b(\Omega )$). Then $(C_b(\Omega ),\left\Vert \cdot \right\Vert _\infty ,\tau _{\textrm{co}})$ is a Saks space which is sequentially complete if $\Omega$ is locally compact (in particular if $\Omega =\mathbb {R}^d$).

Lemma 4.1

$C:(C_b(\mathbb {R}^d),\tau _{\textrm{co}})\rightarrow (C_b(\Omega ),\tau _{\textrm{co}})$ is continuous.

Proof

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

$$\begin{aligned} p_{K}^{Z}(f):=\sup _{x\in K}|f(x)|\quad (f\in C_b(Z)) \end{aligned}$$

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

$$\begin{aligned} p_{K}^{\Omega }(Cf)=\sup _{x\in K}|f|_\Omega (x)|=\sup _{x\in K}|f(x)|=p_{K}^{\mathbb {R}^{d}}(f)\quad (f\in C_b(\mathbb {R}^{d})), \end{aligned}$$

which means that C is continuous. $\square$

We now use the operator C to provide an uncertainty principle based on the well-known Logvinenko–Sereda theorem. Let $\eta \in C_c[0,\infty )$, $\mathbb {1}_{[0,1/2]} \le \eta \le \mathbb {1}_{[0,1]}$. For $\lambda >0$ let $\chi _\lambda :\mathbb {R}^d\rightarrow \mathbb {R}$, $\chi _\lambda :=\eta (\left|\cdot \right|/\lambda )$, and $P_\lambda \in \mathcal {L}(C_b(\mathbb {R}^d))$ be defined by $P_\lambda f:=(\mathcal {F}^{-1}\chi _\lambda ) *f$, where $\mathcal {F}$ denotes the Fourier transformation. By Young’s inequality and scaling properties of the Fourier transformation, we have

$$\begin{aligned} \left\Vert P_\lambda \right\Vert \le \left\Vert \mathcal {F}^{-1}\chi _\lambda \right\Vert _{L_1(\mathbb {R}^d)} = \left\Vert \mathcal {F}^{-1}\chi _1\right\Vert _{L_1(\mathbb {R}^d)} \quad (\lambda >0). \end{aligned}$$

Note that for all $f\in C_b(\mathbb {R}^d)$ and $\lambda >0$ we have $\mathcal {F}P_\lambda f = \chi _\lambda \mathcal {F}f$ and therefore ${{\,\textrm{spt}\,}}\mathcal {F}P_\lambda f \subseteq B[0,\lambda ] \subseteq [-\lambda ,\lambda ]^d$, where $B[0,\lambda ]:=\{x\in \mathbb {R}^d \mid \left|x\right|\le \lambda \}$ is the closed ball around 0 with radius $\lambda$.

Definition 4.2

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

$$\begin{aligned} \lambda ^d(\Omega \cap (x+(0,L))) \ge \rho \lambda ^d((0,L)) \quad (x\in \mathbb {R}^d), \end{aligned}$$

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

Thus, a measurable set $\Omega \subseteq \mathbb {R}^d$ 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 [12, 19]), if $\Omega \subseteq \mathbb {R}^d$ is a thick set, then there exist $d_0,d_1>0$ such that

$$\begin{aligned} \left\Vert P_\lambda f\right\Vert _{C_b(\mathbb {R}^d)} \le d_0\textrm{e}^{d_1 \lambda } \left\Vert CP_\lambda f\right\Vert _{C_b(\Omega )} \quad (\lambda >0, f\in C_b(\mathbb {R}^d)). \end{aligned}$$

(LS)

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

4.2 The Gauß–Weierstraß semigroup on $C_b(\mathbb {R}^d)$

Let $k:(0,\infty )\times \mathbb {R}^d\rightarrow \mathbb {R}$ be given by

$$\begin{aligned} k_t(x):=k(t,x) :=\frac{1}{(4\pi t)^{d/2}} \textrm{e}^{-\left|x\right|^2/(4t)} \quad (t>0, x\in \mathbb {R}^d), \end{aligned}$$

the so-called Gauß–Weierstraß kernel. For $t\ge 0$ we define $S_t \in \mathcal {L}(C_b(\mathbb {R}^d))$ by

$$\begin{aligned} S_t f :={\left\{ \begin{array}{ll} f &{} t=0,\\ k_t * f &{} t>0. \end{array}\right. } \end{aligned}$$

Note that by Young’s inequality and the fact that $\left\Vert k_t\right\Vert _{L_1(\mathbb {R}^d)} = 1$ for all $t>0$ we have $\left\Vert S_tf\right\Vert _{C_b(\mathbb {R}^d)} \le \left\Vert f\right\Vert _{C_b(\mathbb {R}^d)}$ for $f\in C_b(\mathbb {R}^d)$ and $t\ge 0$. It is easy to see that $(S_t)_{t\ge 0}$ is a semigroup, which is called the Gauß–Weierstraß semigroup. Let $\tau _{\textrm{co}}$ be the compact-open topology on $C_b(\mathbb {R}^d)$. Then $(S_t)_{t\ge 0}$ is locally $\tau _{\textrm{co}}$-bi-continuous; see e.g. [17, Examples 6 (a)].

For $\lambda >0$ let $P_\lambda \in \mathcal {L}(C_b(\mathbb {R}^d))$ be defined as in Subsect. 4.1. By [1, Proposition 3.2], there exist $d_2\ge 1$ and $d_3>0$ such that

$$\begin{aligned} \left\Vert (I-P_\lambda ) S_tf\right\Vert _{C_b(\mathbb {R}^d)} \le d_2 \textrm{e}^{-d_3 \lambda ^{2} t} \left\Vert f\right\Vert _{C_b(\mathbb {R}^d)} \quad (\lambda >0, t\ge 0, f\in C_b(\mathbb {R}^d)), \end{aligned}$$

(DISS(GW))

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

Thus, if $\Omega \subseteq \mathbb {R}^d$ 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 $C_b(\mathbb {R}^d)$.

4.3 The Ornstein–Uhlenbeck semigroup on $C_b(\mathbb {R}^d)$

Let $M:(0,\infty )\times \mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}$ be given by

$$\begin{aligned} M_t(x,y):= & {} M(t,x,y):=\frac{1}{\pi ^{d/2} (1-\textrm{e}^{-2t})^{d/2}} \textrm{e}^{-\left|y-\textrm{e}^{-t}x\right|^2/(1-\textrm{e}^{-2t})} \\{} & {} \qquad \quad (t>0, x,y\in \mathbb {R}^d), \end{aligned}$$

the so-called Mehler kernel. For $t\ge 0$ we define $S_t \in \mathcal {L}(C_b(\mathbb {R}^d))$ by

$$\begin{aligned} S_t f :={\left\{ \begin{array}{ll} f &{} t=0,\\ \int _{\mathbb {R}^d} M_t(\cdot ,y)f(y)\,\textrm{d}y &{} t>0. \end{array}\right. } \end{aligned}$$

Since $\int _{\mathbb {R}^d} M_t(\cdot ,y)\,\textrm{d}y = 1$ for all $t>0$, we have $\left\Vert S_tf\right\Vert _{C_b(\mathbb {R}^d)} \le \left\Vert f\right\Vert _{C_b(\mathbb {R}^d)}$ for $f\in C_b(\mathbb {R}^d)$ and $t\ge 0$. It is not difficult to see that $(S_t)_{t\ge 0}$ is a semigroup, which is called the Ornstein–Uhlenbeck semigroup. Let $\tau _{\textrm{co}}$ be the compact-open topology on $C_b(\mathbb {R}^d)$. Then $(S_t)_{t\ge 0}$ is locally $\tau _{\textrm{co}}$-bi-continuous on $C_b(\mathbb {R}^d)$; see e.g. [16, Proposition 3.10].

Define $k:(0,1)\times \mathbb {R}^d\rightarrow \mathbb {R}$ by

$$\begin{aligned} k_s(x):=\frac{1}{\pi ^{d/2} (1-s^2)^{d/2}} \textrm{e}^{-\left|x\right|^2/(1-s^2)}. \end{aligned}$$

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

$$\begin{aligned} M_{\ln \frac{1}{s}} (\tfrac{1}{s}x,y) = \frac{1}{\pi ^{d/2} (1-s^2)^{d/2}} \textrm{e}^{-\left|y-x\right|^2/(1-s^2)} = k_s(x-y) \quad (x,y\in \mathbb {R}^d). \end{aligned}$$

Hence,

$$\begin{aligned} \bigl (S_{\ln \frac{1}{s}} f\bigr )(\tfrac{1}{s}\,\cdot \,) = k_s *f \quad (f\in C_b(\mathbb {R}^d)). \end{aligned}$$

For $\lambda >0$ let $P_\lambda \in \mathcal {L}(C_b(\mathbb {R}^d))$ as in Subsect. 4.2. Since

$$\begin{aligned} \mathcal {F}k_s (\xi ) = \textrm{e}^{-(1-s^2)\left|\xi \right|^2/4}=:h_s(\xi ) \quad (\xi \in \mathbb {R}^d), \end{aligned}$$

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

$$\begin{aligned} \bigl ((I-P_\lambda )S_{\ln \frac{1}{s}}f\bigr )(\tfrac{1}{s}\,\cdot \,)= & {} (I-P_{\lambda /s}) \bigl (S_{\ln \frac{1}{s}}f(\tfrac{1}{s}\,\cdot \,)\bigr )\\= & {} \mathcal {F}^{-1}\bigl ((1-\chi _{\lambda /s}) h_s\bigr ) *f \quad (f\in C_b(\mathbb {R}^d)). \end{aligned}$$

Lemma 4.3

There exist $d_2\ge 1$ and $d_3>0$ such that for $\lambda >0$ and $s\in (0,1)$ we have

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}\bigl ((1-\chi _\lambda ) h_s\bigr )\right\Vert _{L_1(\mathbb {R}^d)} \le d_2 \textrm{e}^{-d_3 \lambda ^2 (1-s^2)}.\end{aligned}$$

Proof

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

$$\begin{aligned} \sigma _{s,\lambda }:=(1-\chi _{\sqrt{1-s^2} \lambda })h_s\Bigl (\tfrac{1}{\sqrt{1-s^2}}\,\cdot \,\Bigr ) = (1-\chi _{\sqrt{1-s^2} \lambda }) \textrm{e}^{-\left|\cdot \right|^2/4}. \end{aligned}$$

Then by a linear substitution we obtain

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}\bigl ((1-\chi _\lambda ) h_s\bigr )\right\Vert _{L_1(\mathbb {R}^d)} = \left\Vert \mathcal {F}^{-1}\sigma _{s,\lambda }\right\Vert _{L_1(\mathbb {R}^d)}. \end{aligned}$$

Let $\alpha \in \mathbb {N}_0^d$, $\left|\alpha \right|\le d+1$. Then

$$\begin{aligned} \left|\partial ^\alpha \sigma _{s,\lambda }\right|\le & {} \mathbb {1}_{\{\left|\cdot \right|\ge \sqrt{1-s^2}\lambda /2\}} \left|\partial ^\alpha \textrm{e}^{-\left|\cdot \right|^2/4}\right|\\{} & {} + \sum _{\beta \in \mathbb {N}_0^d, \beta <\alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \left|\partial ^{\alpha -\beta } (1-\chi _{\sqrt{1-s^2}\lambda })\right| \left|\partial ^\beta \textrm{e}^{-\left|\cdot \right|^2/4}\right|. \end{aligned}$$

There exists $K\ge 0$ such that for all $\beta \in \mathbb {N}_0^d$ with $\beta \le \alpha$ and all $\xi \in \mathbb {R}^d$ we have

$$\begin{aligned} \left|\partial ^\beta \textrm{e}^{-\left|\cdot \right|^2/4}(\xi )\right| \le K (1+\left|\xi \right|)^{\left|\beta \right|} \textrm{e}^{-\left|\xi \right|^2/4}. \end{aligned}$$

Let

$$\begin{aligned} C_1:=\sup _{\beta \in \mathbb {N}_0^d, \beta \le \alpha , \xi \in \mathbb {R}^d} K (1+ \left|\xi \right|)^{\left|\beta \right|} \textrm{e}^{-\left|\xi \right|^2/16}. \end{aligned}$$

Then, for $\beta \le \alpha$ and $\left|\xi \right|\ge \sqrt{1-s^2}\lambda /2$ we have

$$\begin{aligned} \left|\partial ^\beta \textrm{e}^{-\left|\cdot \right|^2/4}(\xi )\right| \le C_1 \textrm{e}^{-\left|\xi \right|^2/16} \textrm{e}^{-(1-s^2)\lambda ^2/32}. \end{aligned}$$

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

$$\begin{aligned}&\left|\partial ^{\alpha -\beta } (1-\chi _{\sqrt{1-s^2}\lambda })(\xi )\right| \\&\quad \le (\sqrt{1-s^2}\lambda )^{-\left|\alpha -\beta \right|} \left|\partial ^{\alpha -\beta } \chi _1\Bigl (\tfrac{\xi }{\sqrt{1-s^2}\lambda }\Bigr )\right| \mathbb {1}_{\{\sqrt{1-s^2}\lambda /2 \le \left|\cdot \right|\le \sqrt{1-s^2}\lambda \}}(\xi ) \\&\quad \le C_2 (\sqrt{1-s^2}\lambda )^{-\left|\alpha -\beta \right|} \mathbb {1}_{\{\sqrt{1-s^2}\lambda /2 \le \left|\cdot \right|\le \sqrt{1-s^2}\lambda \}}(\xi ) \end{aligned}$$

where $C_2:=\max _{\beta <\alpha } \left\Vert \partial ^{\alpha -\beta } \chi _1\right\Vert _{C_b(\mathbb {R}^d)}$. Hence, there exists $C\ge 0$ (which is independent of s and $\lambda$) such that if $\sqrt{1-s^2}\lambda \ge 1$, then for all $\xi \in \mathbb {R}^d$ we have

$$\begin{aligned} \left|\partial ^\alpha \sigma _{s,\lambda }\right| \le C \textrm{e}^{-\left|\xi \right|^2/16} \textrm{e}^{-(1-s^2)\lambda ^2/32}. \end{aligned}$$

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

$$\begin{aligned} \left|x^\alpha \mathcal {F}^{-1}\sigma _{s,\lambda }(x)\right| = \left|\mathcal {F}^{-1}(\partial ^\alpha \sigma _{s,\lambda })(x)\right| \le C\textrm{e}^{-(1-s^2)\lambda ^2/32}. \end{aligned}$$

(1)

By choosing $j\in \{1,\ldots ,d\}$ and $\alpha :=(d+1)e_j$ for the j-th canonical unit vector $e_j$, we observe $\left\Vert x\right\Vert _\infty ^{d+1} \left|\mathcal {F}^{-1}\sigma _{s,\lambda }(x)\right| \le C\textrm{e}^{-(1-s^2)\lambda ^2/32}$ and hence

$$\begin{aligned} \left|\mathcal {F}^{-1}\sigma _{s,\lambda }(x)\right| \le C\textrm{e}^{-(1-s^2)\lambda ^2/32} \left|x\right|^{-d-1} \end{aligned}$$

(2)

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

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

$$\begin{aligned} \left\Vert \mathcal {F}^{-1}\bigl ((1-\chi _\lambda ) h_s\bigr )\right\Vert _{L_1(\mathbb {R}^d)}&= \left\Vert \mathcal {F}^{-1}\sigma _{s,\lambda }\right\Vert _{L_1(\mathbb {R}^d)} \\&\le C\textrm{e}^{-(1-s^2)\lambda ^2/32} \Biggl (\;\int _{B[0,1]} 1\,\textrm{d}x + \int _{\mathbb {R}^{d}\setminus B[0,1]} \left|x\right|^{-d-1}\,\textrm{d}x\Biggr ) \\&\le C\textrm{e}^{-(1-s^2)\lambda ^2/32}, \end{aligned}$$

where we increased C again.

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

$$\begin{aligned} \left\Vert \mathcal {F}^{-1} (\chi _\lambda h_s)\right\Vert _{L_1(\mathbb {R}^d)}&= \left\Vert \mathcal {F}^{-1} \chi _\lambda *\mathcal {F}^{-1} h_s\right\Vert _{L_1(\mathbb {R}^d)} \\&\le \left\Vert \mathcal {F}^{-1} \chi _\lambda \right\Vert _{L_1(\mathbb {R}^d)} \left\Vert k_s\right\Vert _{L_1(\mathbb {R}^d)} = \left\Vert \mathcal {F}^{-1} \chi _1\right\Vert _{L_1(\mathbb {R}^d)}, \end{aligned}$$

where the last equality follows form scaling properties of the Fourier transformation and the fact that $k_s$ is normalised in $L_1(\mathbb {R}^d)$.

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

$$\begin{aligned} \left\Vert \mathcal {F}^{-1} (\chi _\lambda h_s)\right\Vert _{L_1(\mathbb {R}^d)} \le \left\Vert \mathcal {F}^{-1} \chi _1\right\Vert _{L_1(\mathbb {R}^d)} \textrm{e}^{1/32} \textrm{e}^{-(1-s^2)\lambda ^2/32}, \end{aligned}$$

which ends the proof. $\square$

In view of Lemma 4.3, we obtain the dissipation estimate (DISS’) as follows. Note that for $t\ge 0$ we have $\textrm{e}^{2t}-1\ge 2t$. Let $t>0$ and $\lambda >0$, and set $s:=\textrm{e}^{-t}\in (0,1)$. Then, for $f\in C_b(\mathbb {R}^d)$, Young’s inequality and Lemma 4.3 yield

$$\begin{aligned} \left\Vert (I-P_\lambda )S_{t}f\right\Vert _{C_b(\mathbb {R}^d)}&= \left\Vert \bigl ((I-P_\lambda )S_{\ln \frac{1}{s}}f\bigr )(\tfrac{1}{s}\,\cdot \,)\right\Vert _{C_b(\mathbb {R}^d)}\\&\le \left\Vert \mathcal {F}^{-1}\bigl ((1-\chi _{\lambda /s})h_s\bigr )\right\Vert _{L_1(\mathbb {R}^d)} \left\Vert f\right\Vert _{C_b(\mathbb {R}^d)} \\&\le d_2 \textrm{e}^{-d_3 \lambda ^2 s^{-2} (1-s^2)}\left\Vert f\right\Vert _{C_b(\mathbb {R}^d)} = d_2 \textrm{e}^{-d_3 \lambda ^2 (\textrm{e}^{2t}-1)}\left\Vert f\right\Vert _{C_b(\mathbb {R}^d)} \\&\le d_2 \textrm{e}^{-2d_3 \lambda ^2 t}\left\Vert f\right\Vert _{C_b(\mathbb {R}^d)}. \end{aligned}$$

(DISS(OU))

Thus, if $\Omega \subseteq \mathbb {R}^d$ is thick and $C:C_b(\mathbb {R}^d)\rightarrow C_b(\Omega )$ is the restriction map as in Subsect. 4.2, then (LS) and (DISS(OU)) provide the estimates (UP’) and (DISS’) and so Theorem 3.4 yields a final state observability estimate for the Ornstein–Uhlenbeck semigroup on $C_b(\mathbb {R}^d)$.

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 [4, 6, 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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a Saks space and $T>0$. We set

$$\begin{aligned} C_{\tau ,b}([0,T];X) :=\{f\in C([0,T];(X,\tau _X))\ |\ \Vert f\Vert _{\infty }:=\sup _{t\in [0,T]}\left\Vert f(t)\right\Vert _X<\infty \} \end{aligned}$$

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

Remark 5.2

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a Saks space and $T>0$.

• (a) Since the mixed topology $\gamma _X$ coincides with $\tau _X$ on $\left\Vert \cdot \right\Vert _X$-bounded sets by [28, Lemma 2.2.1], and a subset of X is $\left\Vert \cdot \right\Vert _X$-bounded if and only if it is $\gamma _X$-bounded by [28, Corollary 2.4.1] we have

  $$\begin{aligned} C_{\tau ,b}([0,T];X)=C_{b}([0,T];(X,\gamma _X))=C([0,T];(X,\gamma _X)). \end{aligned}$$

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

  $$\begin{aligned} \left\Vert f\right\Vert _{\infty }:=\sup _{t\in [0,T]}\left\Vert f(t)\right\Vert _X \qquad (f\in C_{\tau ,b}([0,T];X)). \end{aligned}$$

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

  $$\begin{aligned} p_{\gamma _\infty }(f):=\sup _{t\in [0,T]}p_{\gamma _X}(f(t)) \qquad (f\in C_{\tau ,b}([0,T];X)) \end{aligned}$$

  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 $\left\Vert \cdot \right\Vert _{\infty }$-topology. Further, $(C_{\tau ,b}([0,T];X),\left\Vert \cdot \right\Vert _{\infty })$ is a Banach space.

• (b) We note that a subset $B\subseteq C_{\tau ,b}([0,T];X)$ is $\left\Vert \cdot \right\Vert _{\infty }$-bounded if and only if it is $\gamma _{\infty }$-bounded since a subset of X is $\left\Vert \cdot \right\Vert _X$-bounded if and only if it is $\gamma _X$-bounded by [28, 2.4.1 Corollary]. So $((C_{\tau ,b}([0,T];X),\gamma _{\infty })',\tau _b)$ is a topological subspace of $C_{\tau ,b}([0,T];X)'=((C_{\tau ,b}([0,T];X),\left\Vert \cdot \right\Vert _{\infty })',\left\Vert \cdot \right\Vert _{C_{\tau ,b}([0,T];X)'})$ where $\tau _{b}$ denotes the topology of uniform convergence on $\gamma _{\infty }$-bounded sets. In the following we use the notation $\left\Vert y'\right\Vert _{(C_{\tau ,b}([0,T];X),\gamma _{\infty })'}:=\left\Vert y'\right\Vert _{C_{\tau ,b}([0,T];X)'}$ for all $y'\in (C_{\tau ,b}([0,T];X),\gamma _{\infty })'$.

Proposition 5.3

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

$$\begin{aligned} f:[0,T]\rightarrow X,\, f(t):=S_{T-t}v(t). \end{aligned}$$

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 $(t_{n})_{n\in \mathbb {N}}$ be a sequence in [0, T] that converges to $t\in [0,T]$ and set $x_{n}:=v(t_{n})-v(t)$ for $n\in \mathbb {N}$. The sequence $(x_{n})_{n\in \mathbb {N}}$ is $\left\Vert \cdot \right\Vert _X$-bounded and $\tau _{X}\text {-}\lim \limits _{n\rightarrow \infty } x_n = 0$ due to our assumptions on v. We have for $p\in \mathcal {P}_{\tau _X}$ that

$$\begin{aligned} p(f(t_{n})-f(t))&=p(S_{T-t_{n}}v(t_{n})-S_{T-t}v(t))\\&\le p(S_{T-t_{n}}v(t_{n})-S_{{T}-t_{n}}v(t))+p(S_{T-t_{n}}v(t)-S_{T-t}v(t))\\&\le p(S_{T-t_{n}}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t))\\&\le \sup _{s\in [0,T]} p(S_{s}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t)). \end{aligned}$$

Combining our estimate above with the local bi-equicontinuity and $\tau _X$-strong continuity of the semigroup, we deduce that $(f(t_{n}))_{n\in \mathbb {N}}$ 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 $M\ge 1$ and $\omega \in \mathbb {R}$ such that for all $t\in [0,T]$

$$\begin{aligned} \left\Vert f(t)\right\Vert _{X}&=\left\Vert S_{T-t}v(t)\right\Vert _{X} \le \left\Vert S_{T-t}\right\Vert _{\mathcal {L}(X)}\left\Vert v(t)\right\Vert _{X} \le Me^{\omega (T-t)}\left\Vert v(t)\right\Vert _{X}\\&\le Me^{|\omega | T}\left\Vert v(t)\right\Vert _{X}, \end{aligned}$$

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

Proposition 5.4

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete Saks space, $(S_t)_{t\ge 0}$ a locally $\tau _X$-bi-continuous semigroup on X, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, and $B\in \mathcal {L}(U;X)$ such that B is also sequentially $\tau _U$-$\tau _X$-continuous on $\left\Vert \cdot \right\Vert _{U}$-bounded sets. Let $T>0$, $u\in C_{\tau ,b}([0,T];U)$ and set $f:[0,T]\rightarrow X$, $f(t):=S_{T-t}Bu(t)$. 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 $v:t\mapsto Bu(t)$ belongs to $C_{\tau ,b}([0,T];X)$. $\square$

Now, we have everything at hand to formulate the definition of cost-uniform approximate null-controllability in the bi-continuous setting. Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete Saks space, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, $(S_t)_{t\ge 0}$ 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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets, and $T>0$. We consider the linear control system

$$\begin{aligned} \begin{aligned} \dot{x}(t)&= -Ax(t) + Bu(t) \quad (t>0),\\ x(0)&= x_0 \in X, \end{aligned} \end{aligned}$$

(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

$$\begin{aligned} x(t)=S_{t}x_{0}+\int _{0}^{t} S_{t-r}Bu(r)\textrm{d}r\qquad (t\in [0,T]) \end{aligned}$$

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 $C \ge 0$ 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 $\left\Vert u\right\Vert _{\infty }\le C \left\Vert x_0\right\Vert _X$ such that $p_{\tau _X}(x(T)) \le \varepsilon$.

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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a Saks space and $\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$ the scalar field of X.

• (a) We call $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ 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]).

• (b) We call $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ a Mazur space if $(X,\gamma _X)$ is a Mazur space, i.e.

  $$\begin{aligned} X_{\gamma }':=(X,\gamma _X)'=\{x':X\rightarrow \mathbb {K}\ |\ x'\;\text {linear and } \gamma _{X}\text {-sequentially continuous}\} \end{aligned}$$

  (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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a Saks space.

• (a) If $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ is C-sequential, then it is a Mazur space by [27, Theorem 7.4] (cf. [14, 3.6 Proposition (b)]).

• (b) The space

  $$\begin{aligned} X^{\circ }:=\{x'\in X'\ |\ x'\;\tau _X\text {-sequentially continuous on } \left\Vert \cdot \right\Vert _X\text {-bounded sets}\} \end{aligned}$$

  is a closed linear subspace of the norm dual $X'$ and hence a Banach space with norm given by $\left\Vert x^{\circ }\right\Vert _{X^{\circ }}:=\left\Vert x^{\circ }\right\Vert _{X'}$ 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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$). We have $X^{\circ }=X_{\gamma }'$ if and only if $(X,\gamma _X)$ is a Mazur space by [14, 3.5 Remark].

Let $(X,\left\Vert \cdot \right\Vert _X)$ and $(U,\left\Vert \cdot \right\Vert _U)$ be Banach spaces. We recall that the dual operator $B'$ of an element $B\in \mathcal {L}(U;X)$ is defined by $\langle B'x',u\rangle :=\langle x',Bu\rangle$ for $x'\in X'$ and $u\in U$.

Proposition 5.9

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete C-sequential Saks space and $(S_t)_{t\ge 0}$ a locally $\tau _X$-bi-continuous semigroup on X. Then the operators given by $S^{\circ }_{t}x^{\circ }:=S_{t}'x^{\circ }$ for $t\ge 0$ and $x^{\circ }\in X^{\circ }$ belong to $\mathcal {L}(X^{\circ })$ and form a $\tau _{c}(X^{\circ },(X,\left\Vert \cdot \right\Vert _{X}))$-strongly continuous, exponentially bounded semigroup on $X^{\circ }$ where $\tau _{c}(X^{\circ },(X,\left\Vert \cdot \right\Vert _{X}))$ denotes the topology of uniform convergence on $\left\Vert \cdot \right\Vert _{X}$-compact sets.

Proof

Since $(X,\gamma _{X})$ is C-sequential, in particular Mazur by Remark 5.8 (a), $(S_t)_{t\ge 0}$ is quasi-$\gamma _{X}$-equicontinuous and $X^{\circ }=X_{\gamma }'$ 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 $t\ge 0$. Furthermore, $(S^{\circ }_{t})_{t\ge 0}$ is exponentially bounded (w.r.t. $\left\Vert \cdot \right\Vert _{\mathcal {L}(X^{\circ })}$) because $(S_t)_{t\ge 0}$ is exponentially bounded. It follows that $S^{\circ }_{t}\in \mathcal {L}(X^{\circ })$ for all $t\ge 0$ and $(S^{\circ }_{t})_{t\ge 0}$ is a $\sigma (X^{\circ },X)$-strongly continuous semigroup. As $\sigma (X^{\circ },X)$ and the mixed topology $\gamma ^{\circ }:=\gamma (\Vert \cdot \Vert _{X^{\circ }},\sigma (X^{\circ },X))$ coincide on $\Vert \cdot \Vert _{X^{\circ }}$-bounded sets by Definition 2.1 (b), $(S^{\circ }_{t})_{t\ge 0}$ is also $\gamma ^{\circ }$-strongly continuous. Due to [13, Proposition 3.22 (a)] we have $\gamma ^{\circ }=\tau _{c}(X^{\circ },(X,\left\Vert \cdot \right\Vert _{X}))$. $\square$

The semigroup $(S^{\circ }_{t})_{t\ge 0}$ 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 $(S^{\circ }_{t})_{t\ge 0}$ from Proposition 5.9 is given by

and fulfils

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

Remark 5.10

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete Saks space and $(S_t)_{t\ge 0}$ a $\tau _X$-bi-continuous semigroup on X with generator $(-A,D(A))$. If

• (i) $X^{\circ }\cap \{x'\in X'\ |\ \Vert x'\Vert _{X'}\le 1\}$ is sequentially complete w.r.t. $\sigma (X^{\circ },X)$,

• (ii) every $\Vert \cdot \Vert _{X'}$-bounded $\sigma (X^{\circ },X)$-null sequence in $X^{\circ }$ is $\tau _X$-equicontinuous on $\left\Vert \cdot \right\Vert _X$-bounded sets,

(see [8, Hypothesis B and C]), then $(X^{\circ },\Vert \cdot \Vert _{X^{\circ }},\sigma (X^{\circ },X))$ is a sequentially complete Saks space and $(S^{\circ }_{t})_{t\ge 0}$ 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

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

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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ and $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ be Saks spaces, and $B\in \mathcal {L}(U;X)$ such that B is also sequentially $\tau _U$-$\tau _X$-continuous on $\left\Vert \cdot \right\Vert _{U}$-bounded sets. Then $B^{\circ }:=B'|_{X^{\circ }}\in \mathcal {L}(X^{\circ };U^{\circ })$ and is also $\sigma (X^{\circ },X)$-$\sigma (U^{\circ },U)$-continuous.

Proof

Let $x^{\circ }\in X^{\circ }$, $u\in U$ and $(u_{n})_{n\in \mathbb {N}}$ a $\left\Vert \cdot \right\Vert _{U}$-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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets, we have that $(Bu_{n})_{n\in \mathbb {N}}$ is $\left\Vert \cdot \right\Vert _{X}$-bounded and $\tau _X$-convergent to Bu. This implies

$$\begin{aligned} \langle B^{\circ }x^{\circ }, u\rangle =\langle x^{\circ }, Bu\rangle =\lim _{n\rightarrow \infty }\langle x^{\circ }, Bu_{n}\rangle , \end{aligned}$$

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

$$\begin{aligned} \left\Vert B^{\circ }x^{\circ }\right\Vert _{U^{\circ }}&=\left\Vert B^{\circ }x^{\circ }\right\Vert _{U'} =\sup _{\left\Vert u\right\Vert _{U}\le 1}|\langle B^{\circ }x^{\circ }, u\rangle | \\&=\sup _{\left\Vert u\right\Vert _{U}\le 1}|\langle x^{\circ }, Bu\rangle | \le \sup _{\left\Vert u\right\Vert _{U}\le 1}\left\Vert Bu\right\Vert _{X}\left\Vert x^{\circ }\right\Vert _{X^{\circ }}\\&=\left\Vert B\right\Vert _{\mathcal {L}(U;X)}\left\Vert x^{\circ }\right\Vert _{X^{\circ }} \end{aligned}$$

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

Remark 5.12

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

• (a) B is $\tau _U$-$\tau _X$-continuous on $\left\Vert \cdot \right\Vert _{U}$-bounded sets.

• (b) B is sequentially $\tau _U$-$\tau _X$-continuous on $\left\Vert \cdot \right\Vert _{U}$-bounded sets.

Then (a)$\Rightarrow$(b) holds. Moreover, if $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ 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. $\square$

Proposition 5.13

Let $(V,\left\Vert \cdot \right\Vert _V)$, $(W,\left\Vert \cdot \right\Vert _W)$ be Banach spaces, $(Z,\left\Vert \cdot \right\Vert _Z,\tau _Z)$ 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:

• (a) There exists $c_1\ge 0$ such that $\left\Vert F'z'\right\Vert _{V'}\le c_{1}\left\Vert G'z'\right\Vert _{W'}$ for all $z'\in Z_{\gamma }'$.

• (b) There exists $c_2\ge 0$ such that

  $$\begin{aligned} \{Fv\ |\ v\in V,\,\left\Vert v\right\Vert _V \le 1\}\subseteq \overline{\{Gw\ |\ w\in W,\,\left\Vert w\right\Vert _W \le c_2\}}^{\gamma _Z}. \end{aligned}$$

• (c) There exists $c_3\ge 0$ such that for all $v\in V$, $\varepsilon >0$ and $p_{\gamma _Z}\in \mathcal {P}_{\gamma _Z}$ there is $w\in W$ with $\left\Vert w\right\Vert _W \le c_3\left\Vert v\right\Vert _V$ such that $p_{\gamma _Z}(Fv+Gw)\le \varepsilon$.

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

$$\begin{aligned} \sup _{z\in M} {{\,\textrm{Re}\,}}\langle z',z \rangle \le \sup _{z\in N} {{\,\textrm{Re}\,}}\langle z',z \rangle \end{aligned}$$

(3)

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

$$\begin{aligned} \sup _{z\in M} |\langle z',z \rangle | \le \sup _{z\in N} |\langle z',z \rangle | \end{aligned}$$

for all $z'\in Z_{\gamma }'$. Hence, setting $N:=\{Fv\ |\ v\in V,\,\left\Vert v\right\Vert _V \le 1\}$, $M:=\{Gw\ |\ w\in W,\,\left\Vert w\right\Vert _W \le c_2\}$ 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 $U_{\varepsilon , p_{\gamma _Z}}:=\{z\in Z\;|\;p_{\gamma _Z}(z)\le \varepsilon \}$. For any $M\subseteq Z$ we have

$$\begin{aligned} \overline{M}^{\gamma _Z}=\bigcap _{\varepsilon >0,\,p_{\gamma _Z}\in \mathcal {P}_{\gamma _Z}}M+U_{\varepsilon , p_{\gamma _Z}} \end{aligned}$$

(4)

(see e.g. [11, 2.1.4 Proposition]). Let assertion (b) hold, $v\in V$ with $v\ne 0$, $\varepsilon >0$ and $p_{\gamma _Z}\in \mathcal {P}_{\gamma _Z}$. Then there are $\widetilde{w}\in W$ with $\left\Vert \widetilde{w}\right\Vert _W \le c_2$ and $z\in Z$ with $p_{\gamma _Z}(z)\le \tfrac{\varepsilon }{\left\Vert v\right\Vert _{V}}$ such that $F\left( -\tfrac{v}{\left\Vert v\right\Vert _{V}}\right) =G\widetilde{w}+z$ by (b) and (4). From writing

$$\begin{aligned} -Fv=\left\Vert v\right\Vert _{V}F\left( \frac{-v}{\left\Vert v\right\Vert _{V}}\right) =\left\Vert v\right\Vert _{V}(G\widetilde{w}+z)=G(\left\Vert v\right\Vert _{V}\widetilde{w})+\left\Vert v\right\Vert _{V}z, \end{aligned}$$

setting $w:=\left\Vert v\right\Vert _{V}\widetilde{w}$, and using $\left\Vert w\right\Vert _W=\left\Vert v\right\Vert _{V}\left\Vert \widetilde{w}\right\Vert _{W}\le c_{2}\left\Vert v\right\Vert _{V}$ and

$$\begin{aligned} p_{\gamma _Z}(Fv+Gw) =p_{\gamma _Z}(-\left\Vert v\right\Vert _{V}z) =\left\Vert v\right\Vert _{V}p_{\gamma _Z}(z) \le \left\Vert v\right\Vert _{V}\frac{\varepsilon }{\left\Vert v\right\Vert _{V}} =\varepsilon , \end{aligned}$$

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

Now, let assertion (c) hold. Let $v\in V$ with $\left\Vert v\right\Vert _{V}\le 1$, $\varepsilon >0$ and $p_{\gamma _Z}\in \mathcal {P}_{\gamma _Z}$. Then there is $\widetilde{w}\in W$ with $\left\Vert \widetilde{w}\right\Vert _{W}\le c_3$ such that $p_{\gamma _Z}(Fv+G\widetilde{w})\le \varepsilon$. Setting $w:=-\widetilde{w}$, using $\left\Vert w\right\Vert =\left\Vert \widetilde{w}\right\Vert _{W}\le c_3$ 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 $c_2=c_3$. $\square$

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 $(Z,\left\Vert \cdot \right\Vert _Z,\tau _Z)$ be a Saks space. Since the mixed topology $\gamma _Z$ coincides with $\tau _Z$ on $\left\Vert \cdot \right\Vert _Z$-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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete C-sequential Saks space, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, $(S_t)_{t\ge 0}$ 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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets.

• (a) $(S_{t}B)_{t\ge 0}$ is quasi-$\gamma _U$-$\gamma _X$-equicontinuous.

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

  $$\begin{aligned} \mathcal {B}^{T}u:=\int _{0}^{T}S_{T-t}Bu(t)\textrm{d}t. \end{aligned}$$

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

• (c) Let ${\mathcal {B}^{T}}^{\circ }x^{\circ }:={\mathcal {B}^{T}}'x^{\circ }$ for $x^{\circ }\in X^{\circ }$. Then ${\mathcal {B}^{T}}^{\circ }x^{\circ }\in (C_{\tau ,b}([0,T];U),\gamma _{\infty })'$ for all $x^{\circ }\in X^{\circ }$.

Proof

(a) Let $M\subseteq U$ be a $\left\Vert \cdot \right\Vert _{U}$-bounded set. Then the restriction $B|_M:M \rightarrow X$ of B to M is $\tau _{U}|_M$-$\tau _X$-continuous. Since $B\in \mathcal {L}(U;X)$, the set B(M) is $\left\Vert \cdot \right\Vert _{X}$-bounded. As the mixed topology $\gamma _X$ coincides with $\tau _X$ on $\left\Vert \cdot \right\Vert _{X}$-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)] $(S_{t}B)_{t\ge 0}$ 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 $V_{p_{\gamma _X}}:=\{x\in X\;|\;p_{\gamma _X}(x)<1\}$ and denote its polar set by $V_{p_{\gamma _X}}^{\circ }:=\{x'\in X_{\gamma }'\;|\;\forall \;x\in V_{p_{\gamma _X}}:\;|x'(x)|\le 1\}$. It follows from part (a) that there are $C\ge 0$ and $p_{\gamma _U}\in \mathcal {P}_{\gamma _U}$ such that for all $u\in C_{\tau ,b}([0,T];U)$ we have

$$\begin{aligned} p_{\gamma _X}(\mathcal {B}^{T}u)&=\sup _{x'\in V_{p_{\gamma _X}}^{\circ }}\Bigl |\langle x',\int _{0}^{T}S_{T-t}Bu(t)\textrm{d}t\rangle \Bigr | \le \sup _{x'\in V_{p_{\gamma _X}}^{\circ }}\int _{0}^{T}|\langle x',S_{T-t}Bu(t)\rangle |\textrm{d}t \nonumber \\&\le T\sup _{x'\in V_{p_{\gamma _X}}^{\circ }}\sup _{t\in [0,T]}|\langle x',S_{T-t}Bu(t)\rangle | =T\sup _{t\in [0,T]}p_{\gamma _X}(S_{T-t}Bu(t))\nonumber \\&\underset{{(a)}}{\le } C T\sup _{t\in [0,T]}p_{\gamma _U}(u(t)) \end{aligned}$$

(5)

where we used [20, Proposition 22.14] in the first and second to last equation to get from $p_{\gamma _X}$ to $\sup _{x'\in V_{p_{\gamma _X}}^{\circ }}$ and back. Thus, $\mathcal {B}^{T}$ is $\gamma _\infty$-$\gamma _X$-continuous. Furthermore, since $X_{\gamma }'$ is norming for X by [15, Lemma 5.5 (a)], we may choose $\mathcal {P}_{\gamma _X}$ such that $\left\Vert x\right\Vert _X=\sup _{p_{\gamma _X}\in \mathcal {P}_{\gamma _X}}p_{\gamma _X}(x)$ for all $x\in X$ by [13, Remark 2.2 (c)]. Due to our previous estimates and the exponential boundedness of $(S_{t})_{t\ge 0}$ we obtain

$$\begin{aligned} \left\Vert \mathcal {B}^{T}u\right\Vert _{X}&=\sup _{p_{\gamma _X}\in \mathcal {P}_{\gamma _X}}p_{\gamma _X}(\mathcal {B}^{T}u) \underset{(5)}{\le } T\sup _{p_{\gamma _X}\in \mathcal {P}_{\gamma _X}}\sup _{t\in [0,T]}p_{\gamma _X}(S_{T-t}Bu(t))\\&=T\sup _{t\in [0,T]}\left\Vert S_{T-t}Bu(t)\right\Vert _{X} \le T\sup _{t\in [0,T]}\left\Vert S_{T-t}\right\Vert _{\mathcal {L}(X)}\left\Vert Bu(t)\right\Vert _{X}\\&\le TMe^{|\omega |T}\left\Vert B\right\Vert _{\mathcal {L}(U;X)}\sup _{t\in [0,T]}\left\Vert u(t)\right\Vert _{U}, \end{aligned}$$

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

(c) It follows from $X^{\circ }=X_{\gamma }'$ 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 ${\mathcal {B}^{T}}^{\circ }x^{\circ }\in (C_{\tau ,b}([0,T];U),\gamma _{\infty })'$ for all $x^{\circ }\in X^{\circ }$. $\square$

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 $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete C-sequential Saks space, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, $(S_t)_{t\ge 0}$ 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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets, and $T>0$. Using Proposition 5.9 and Proposition 5.11, the dual system of (ConSys) is given by

$$\begin{aligned} \begin{aligned} \dot{x}(t)&= -A^{\circ }x(t) \quad (t>0),\\ y(t)&= B^{\circ }x(t) \quad (t\ge 0),\\ x(0)&= x_0 \in X^\circ . \end{aligned} \end{aligned}$$

(ObsSys)

Definition 5.16

We say that (ObsSys) satisfies a final state observability estimate in $(C_{\tau ,b}([0,T];U),\gamma _{\infty })'$ if there exists $C_{\textrm{obs}}\ge 0$ such that

$$\begin{aligned} \left\Vert S_{T}^{\circ }x^{\circ }\right\Vert _{X^{\circ }} \le C_{\textrm{obs}}\left\Vert {\mathcal {B}^{T}}^{\circ }x^{\circ }\right\Vert _{(C_{\tau ,b}([0,T];U),\gamma _{\infty })'} \end{aligned}$$

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 $(C_{\tau ,b}([0,T];U),\gamma _{\infty })'$, 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 $M(\Omega ;(U,\vartheta _U)')$ the space of all finitely additive vector measures $\nu :\mathscr {B}(\Omega )\rightarrow (U,\vartheta _U)'$, 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

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

• (ii) there exist $p\in \mathcal {P}_{\vartheta _U}$ and $C\ge 0$ such that

  $$\begin{aligned} \sup _{(\mathcal {N},\mathcal {U}_p)}\bigl |\sum _{(N,u)\in (\mathcal {N},\mathcal {U}_p)}\nu (N)u\bigr |\le C \end{aligned}$$

  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 $p(u)\le 1$ for all $u\in \mathcal {U}_p$.

Let $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ be a Saks space and $T>0$. By [26, Theorem 1], the compactness of [0, T] and Remark 5.2 (a) the map

$$\begin{aligned} \Theta _\gamma :M([0,T];U_{\gamma }')\rightarrow (C_{\tau ,b}([0,T];U),\gamma _\infty )',\; \Theta _\gamma (\nu )(u):=\int _{0}^{T}u(t)\textrm{d}\nu , \end{aligned}$$

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

$$\begin{aligned} \Theta _{\left\Vert \cdot \right\Vert }:M([0,T];U')\rightarrow (C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )',\; \Theta _{\left\Vert \cdot \right\Vert }(\nu )(u):=\int _{0}^{T}u(t)\textrm{d}\nu , \end{aligned}$$

is a topological isomorphism w.r.t. the semivariation norm on $M([0,T];U')$ and the dual norm on $(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'$ and

$$\begin{aligned} \left\Vert \Theta _{\left\Vert \cdot \right\Vert }(\nu )\right\Vert _{(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'} =\left\Vert \nu \right\Vert _{var}\qquad (\nu \in M([0,T];U')) \end{aligned}$$

where the semivariation norm is given by

$$\begin{aligned} \left\Vert \nu \right\Vert _{var}:=\sup _{(\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})}\bigl |\sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})}\nu (N)u\bigr | \qquad (\nu \in M([0,T];U')) \end{aligned}$$

and the supremum is taken over all $(\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})$ as in (ii) above with p replaced by $\left\Vert \cdot \right\Vert _U$. We note that it follows from $U_{\gamma }'$ being a topological subspace of $U'$ (see [5, I.1.18 Proposition]) and $\gamma _{U}$ being coarser than the $\left\Vert \cdot \right\Vert _{U}$-topology, that $M([0,T];U_{\gamma }')$ is a topological subspace of $M([0,T];U')$ (if equipped with the relative topology).

Theorem 5.17

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete C-sequential Saks space, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, $(S_t)_{t\ge 0}$ 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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets. For $T>0$ we have $B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda \in M([0,T];U_{\gamma }')$ 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

$$\begin{aligned} \left\Vert {\mathcal {B}^{T}}^{\circ }x^{\circ }\right\Vert _{(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'}=\left\Vert B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda \right\Vert _{var} \qquad (x^{\circ }\in X^{\circ }) \end{aligned}$$

where

$$\begin{aligned} (B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda )(N)u :=\int _{N}\langle B^{\circ }S_{t}^{\circ }x^{\circ },u\rangle \textrm{d}t \qquad (N\in \mathscr {B}([0,T]),\,u\in U) \end{aligned}$$

for the Lebesgue measure $\lambda$.

Proof

First, we recall that $X^{\circ }=X_\gamma '$ 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 the map $\mathcal {B}^{T}$ is $\gamma _\infty$-$\gamma _X$-continuous and

$$\begin{aligned} \langle {\mathcal {B}^{T}}^{\circ }x^{\circ },u\rangle =\int _{0}^{T}\langle x^{\circ },S_{T-t}Bu(t)\rangle \textrm{d}t =\int _{0}^{T}\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u(t)\rangle \textrm{d}t \end{aligned}$$

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

For $N\in \mathscr {B}([0,T])$ and $u\in U$ we define $(B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda )(N)u :=\int _{N}\langle B^{\circ }S_{t}^{\circ }x^{\circ },u\rangle \textrm{d}t$ and show that $B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda \in M([0,T];U_{\gamma }')$. By the proof of Proposition 5.15 (b) there are $C\ge 0$ and $p_{\gamma _U}\in \mathcal {P}_{\gamma _U}$ such that

$$\begin{aligned} |\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u\rangle | =|\langle x^{\circ },S_{T-t}Bu\rangle | \le Cp_{\gamma _U}(u) \end{aligned}$$

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 $B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda :\mathscr {B}([0,T])\rightarrow U_{\gamma }'$ is a well-defined finitely additive vector measure and that

$$\begin{aligned} (B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(\varvec{\cdot })u=\int _{(\varvec{\cdot })}\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u\rangle \textrm{d}t \qquad (u\in U) \end{aligned}$$

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 $p_{\gamma _U}(u)\le 1$ for all $u\in \mathcal {U}_{p_{\gamma _U}}$. Then we have

$$\begin{aligned} \bigl |\sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{p_{\gamma _U}})}(B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(N)u\bigr |&=\bigl |\sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{p_{\gamma _U}})} \int _{N}\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u\rangle \textrm{d}t\bigr |\\&\le \sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{p_{\gamma _U}})}\lambda (N)Cp_{\gamma _U}(u)\\&\le C\sum _{N\in \mathcal {N}}\lambda (N) = C\lambda ([0,T]) = CT, \end{aligned}$$

yielding $B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda \in M([0,T];U_{\gamma }')$. Analogously, $B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda \in M([0,T];U_{\gamma }')$.

Finally, since

$$\begin{aligned} \langle {\mathcal {B}^{T}}^{\circ }x^{\circ },u\rangle&=\int _{0}^{T}\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u(t)\rangle \textrm{d}t =\int _{0}^{T}u(t)\textrm{d}(B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )\\&=\Theta _\gamma (B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(u) \end{aligned}$$

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

$$\begin{aligned} \langle {\mathcal {B}^{T}}^{\circ }x^{\circ },u\rangle =\Theta _\gamma (B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(u) =\Theta _{\left\Vert \cdot \right\Vert }(B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(u) \end{aligned}$$

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

$$\begin{aligned} \left\Vert {\mathcal {B}^{T}}^{\circ }x^{\circ }\right\Vert _{(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'}&=\left\Vert \Theta _{\left\Vert \cdot \right\Vert }(B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )\right\Vert _{(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'}\\&=\left\Vert B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda \right\Vert _{var}. \end{aligned}$$

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

$$\begin{aligned} \sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})}(B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda )(N)u&= \sum _{(N,u)\in (\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})} \int _{N}\langle B^{\circ }S_{T-t}^{\circ }x^{\circ },u\rangle \textrm{d}t \\&= -\sum _{(T-N,u)\in (T-\mathcal {N},\mathcal {U}_{\left\Vert \cdot \right\Vert _U})}\; \int _{T-N}\langle B^{\circ }S_{t}^{\circ }x^{\circ },u\rangle \textrm{d}t. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \left\Vert {\mathcal {B}^{T}}^{\circ }x^{\circ }\right\Vert _{(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _\infty )'} = \left\Vert B^{\circ }S_{T-(\cdot )}^{\circ }x^{\circ }\odot \lambda \right\Vert _{var} = \left\Vert B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda \right\Vert _{var}. \end{aligned}$$

$\square$

Theorem 5.18

Let $(X,\left\Vert \cdot \right\Vert _X,\tau _X)$ be a sequentially complete C-sequential Saks space, $(U,\left\Vert \cdot \right\Vert _U,\tau _U)$ a Saks space, $(S_t)_{t\ge 0}$ 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 $\left\Vert \cdot \right\Vert _{U}$-bounded sets, and $T>0$. Then the following assertions are equivalent:

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

• (b) The system in (ObsSys) satisfies a final state observability estimate in $(C_{\tau ,b}([0,T];U),\gamma _{\infty })'$.

If additionally $\tau _U=\tau _{\left\Vert \cdot \right\Vert _U}$, then each of the preceding assertions is equivalent to:

• (c) There exists $C_{\textrm{obs}}\ge 0$ such that

  $$\begin{aligned} \forall \;x^{\circ }\in X^{\circ }:\quad \left\Vert S_{T}^{\circ }x^{\circ }\right\Vert _{X^{\circ }}\le C_{\textrm{obs}}\left\Vert B^{\circ }S_{(\cdot )}^{\circ }x^{\circ }\odot \lambda \right\Vert _{var}. \end{aligned}$$

Proof

This statement follows from the equivalence of (a) and (c) in Proposition 5.13 with $V:=Z:=X$, $W:=C_{\tau ,b}([0,T];U)$ equipped with $\left\Vert \cdot \right\Vert _{\infty }$, $F:=S_{T}$ and $G:=\mathcal {B}^{T}$ in combination with Theorem 5.17, Remark 5.14 and $X^{\circ }=X_\gamma '$. $\square$

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

$$\begin{aligned} (C_{\tau ,b}([0,T];U),\gamma _{\infty })'=(C([0,T];(U,\left\Vert \cdot \right\Vert _{U})),\left\Vert \cdot \right\Vert _{\infty })'=M([0,T];U') \end{aligned}$$

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