# Logarithmic stability estimates for initial data in Ornstein-Uhlenbeck
equation on $L^{2}$-space

S. E. Chorfi  and  L. Maniar  S. E. Chorfi, L. Maniar, Cadi Ayyad University,
Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), B.P. 2390, Marrakesh,
Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we continue the investigation on the connection between
observability and inverse problems for a class of parabolic equations with
unbounded first order coefficients. We prove new logarithmic stability
estimates for a class of initial data in the Ornstein-Uhlenbeck equation posed
on $L^{2}\left(\mathbb{R}^{N}\right)$ with respect to the Lebesgue measure.
The proofs combine observability and logarithmic convexity results that
include a non-analytic semigroup case. This completes the picture of the
recent results obtained for the analytic Ornstein-Uhlenbeck semigroup on
$L^{2}$-space with invariant measure.

###### Key words and phrases:

Observability, controllability, inverse problem, logarithmic convexity,
Ornstein-Uhlenbeck equation

###### 2020 Mathematics Subject Classification:

93B07, 35R30, 93B05, 35K65

##  1\. Introduction

Let $N\geq 1$ be an integer and let $\theta>0$ be a fixed time. We consider
the Ornstein-Uhlenbeck equation given by

$\displaystyle\begin{aligned} &\partial_{t}u=\Delta u+Bx\cdot\nabla u,\qquad
0<t<\theta,&&x\in\mathbb{R}^{N},\\\
&u\rvert_{t=0}=u_{0}(x),&&x\in\mathbb{R}^{N},\end{aligned}$ (1)

where $B$ is a real constant $N\times N$-matrix, $u_{0}\in
L^{2}\left(\mathbb{R}^{N}\right)$ is an unknown initial datum, and the dot
sign denotes the inner product in $\mathbb{R}^{N}$.

In the present paper, we are concerned with the following problem:

Inverse initial data problem. Determine the unknown initial datum $u_{0}$
belonging to an admissible set $\mathcal{I}$ from the measurement

$u\rvert_{(0,\theta)\times\omega},$

where $\omega\subset\mathbb{R}^{N}$ is an observation region that will be
given later. Two problems to be answered:

  * $\bullet$

Uniqueness: for two initial data $u_{0}$ and $u_{1}$, does the equality
$u(u_{0})=u(u_{1})$ in $\left(0,\theta\right)\times\omega$ imply $u_{0}=u_{1}$
in $L^{2}\left(\mathbb{R}^{N}\right)$?

  * $\bullet$

Stability: is it possible to estimate
$\|u_{0}-u_{1}\|_{L^{2}\left(\mathbb{R}^{N}\right)}$ by a suitable norm of the
quantity
$\left(u(u_{0})-u(u_{1})\right)\rvert_{\left(0,\theta\right)\times\omega}$?

In the parabolic framework, one expects a stability estimate of logarithmic
type, i.e.,

$\|u_{0}\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\frac{c}{|\log\|u\||^{\alpha}}$
(2)

for a constant $c>0$, some $\alpha\in(0,1]$, a suitable norm of
$u\rvert_{\omega}$, typically
$\|u\|_{L^{2}\left(0,\theta;L^{2}(\omega)\right)}$ or
$\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}$, and for all
$u_{0}\in\mathcal{I}$.

The estimate (2) is commonly referred to as conditional stability since it
depends on the admissible set $\mathcal{I}$. Note that the rate of conditional
stability depends on the choice of the admissible set. Also, conditional
stability is very useful when dealing with the numerical reconstruction of
initial data, see for instance [18, 33].

The Ornstein-Uhlenbeck equation (1) appears in several fields of research and
applications including stochastic processes, Malliavin calculus, theory of
quantum fields as well as control theory. We refer to [7, 8, 17, 20, 21, 22,
23, 24] and the references therein for more details. The associated Ornstein-
Uhlenbeck semigroup has been extensively studied in two spaces; the space
$L^{2}\left(\mathbb{R}^{N},\mathrm{d}x\right)$ (with the Lebesgue measure,
shortly denoted by $L^{2}\left(\mathbb{R}^{N}\right)$) and in
$L^{2}\left(\mathbb{R}^{N},\mathrm{d}\mu\right)$ with the so-called invariant
measure $\mu$. The existence of $\mu$ is equivalent to the fact that the
spectrum of the matrix $B$ lies in the left half-plane, i.e.,

$\sigma(B)\subset\mathbb{C}_{-}:=\\{\lambda\in\mathbb{C}\colon\mathrm{Re}\,\lambda<0\\}.$

The semigroup enjoys different properties with respect to the previous spaces.
It is a $C_{0}$-semigroup on $L^{2}\left(\mathbb{R}^{N},\mathrm{d}x\right)$
which is not analytic (it is not an eventually norm continuous semigroup),
unless $B=0$, whereas its realization in
$L^{2}\left(\mathbb{R}^{N},\mathrm{d}\mu\right)$ is an analytic
$C_{0}$-semigroup with angle depending on the matrix $B$.

As for inverse problems, parabolic equations with bounded coefficients have
been extensively studied, see for instance [14, 15, 18, 32]. Surprisingly,
there are only a few researches for parabolic equations with unbounded
coefficients, especially for the Ornstein-Uhlenbeck equation. We are only
aware of the paper by Lorenzi [19], where the author studied the
identification (existence and uniqueness) of a constant $\alpha\in\mathbb{R}$
such that $B=\alpha B_{0}$ in (1), where the matrix $B_{0}$ and some
prescribed data are known. This inverse problem was motivated by the
determination of the matrix $B$ since its properties play a crucial role in
the study of the Ornstein-Uhlenbeck equation.

Inverse problems are closely related to controllability and observability
questions. The link between the two theories was initially observed in the
case of a hyperbolic system in [28]. Then many stability estimates have been
proven for inverse problems by using observability inequalities. Such inverse
problems for parabolic systems are highly ill-posed in the sense of Hadamard,
so one cannot expect strong stability in general. In this context, the
logarithmic convexity method has been successfully applied to prove
logarithmic stability for initial data. Many works have been done for second-
order parabolic equations with bounded coefficients [14, 15, 31, 32], and more
generally for some abstract systems governed by analytic semigroups of angle
$\frac{\pi}{2}$ as in the self-adjoint case; see [12].

We emphasize that the logarithmic stability for initial data in a parabolic
equation with unbounded coefficients has not been studied before, as far as we
know. The prototype of such equations is given by the Ornstein-Uhlenbeck
equation on $L^{2}$-spaces, which contains an unbounded gradient term.
Recently, the authors of [3] have investigated the case of analytic semigroups
and established an abstract stability result. Consequently, the stability
problem for the Ornstein-Uhlenbeck equation has been recently resolved on
$L^{2}\left(\mathbb{R}^{N},\mathrm{d}\mu\right)$ with the invariant measure
when $\sigma(B)\subset\mathbb{C}_{-}$. Therein, the unbounded gradient
coefficient is handled thanks to the exponential decay of the measure density.
In that case, the semigroup is analytic and the angle of analyticity is
generally smaller than $\frac{\pi}{2}$. The developed approach does not work
on $L^{2}\left(\mathbb{R}^{N}\right)$ due to the lack of analyticity of the
semigroup. The logarithmic convexity estimate has been initially known for
self-adjoint operators since the work of Agmon and Nirenberg [1], and has been
generalized to analytic semigroups in [16] (see also [25]). In our non-
analytic framework, we exploit the explicit representation formula of the
semigroup to prove a logarithmic convexity estimate, and then deduce a
logarithmic stability for initial data using final state observability with
respect to the $L^{2}\left(\mathbb{R}^{N}\right)$-norm.

We mainly aim to clarify the relations between the final state observability,
logarithmic convexity and stability estimates for initial data in the
Ornstein-Uhlenbeck equation. Such a relation is by now fully understood in the
analytic framework of parabolic equations with bounded coefficients. We start
this article by gathering some useful facts on the Ornstein-Uhlenbeck equation
(Section 2). In Section 3, we first prove a logarithmic convexity estimate for
the Ornstein-Uhlenbeck equation (1) exploiting the explicit form of the
associated semigroup. Then we recall some recent results concerning the final
state observability of (1). Afterward in Section 4, we show that the
logarithmic convexity along with the observability imply some logarithmic
stability estimates for certain classes of initial data. These results allow
us in particular to extend some known results for the heat equation on
$\mathbb{R}^{N}$. Finally, Section 5 is devoted to some final comments and
open problems related to the stability estimates for initial data in parabolic
equations with unbounded coefficients.

##  2\. Miscellaneous facts on the Ornstein-Uhlenbeck semigroup

We summarize some useful facts about the Ornstein-Uhlenbeck semigroup on
$L^{2}\left(\mathbb{R}^{N}\right)$ for future use. We refer to [22, 24] and
the references therein for detailed proofs.

Denote by $I$ the identity operator and by $\|\cdot\|$ the standard norm in
$L^{2}\left(\mathbb{R}^{N}\right)$. The Ornstein-Uhlenbeck operator is the sum
of the diffusion term and the drift term.

###  2.1. The diffusion part

Let us first consider the diffusion term. It is well known that the Laplace
operator $\Delta$ on $L^{2}\left(\mathbb{R}^{N}\right)$ with maximal domain

$\displaystyle D(\Delta)$ $\displaystyle:=\left\\{u\in
L^{2}\left(\mathbb{R}^{N}\right):\Delta u\in
L^{2}\left(\mathbb{R}^{N}\right)\right\\}$
$\displaystyle=H^{2}\left(\mathbb{R}^{N}\right),$

is self-adjoint and generates the heat $C_{0}$-semigroup which is explicitly
defined by $U(t)\colon L^{2}\left(\mathbb{R}^{N}\right)\rightarrow
L^{2}\left(\mathbb{R}^{N}\right)$,

$\displaystyle U(0)$ $\displaystyle=I,$ $\displaystyle(U(t)f)(x)$
$\displaystyle=\frac{1}{(4\pi
t)^{N/2}}\int_{\mathbb{R}^{N}}\mathrm{e}^{-\frac{|x-y|^{2}}{4t}}f(y)\,\mathrm{d}y,\qquad
t>0,\quad x\in\mathbb{R}^{N}$

for every $f\in L^{2}\left(\mathbb{R}^{N}\right)$, where $|\cdot|$ denotes the
Euclidean norm on $\mathbb{R}^{N}$. We clearly see that

$U(t)f=h_{t}*f,$

where

$h_{t}(y)=\frac{1}{(4\pi t)^{N/2}}\mathrm{e}^{-\frac{|y|^{2}}{4t}},\quad
t>0,\quad y\in\mathbb{R}^{N}.$

Note that the heat semigroup $\left(U(t)\right)_{t\geq 0}$ is analytic and
contractive on $L^{2}\left(\mathbb{R}^{N}\right)$.

###  2.2. The drift part

Let $B$ be the drift matrix in (1). We consider the drift operator

$\mathcal{L}=Bx\cdot\nabla=\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}},$

with its maximal domain

$D(\mathcal{L})=\left\\{u\in L^{2}\left(\mathbb{R}^{N}\right):\mathcal{L}u\in
L^{2}\left(\mathbb{R}^{N}\right)\right\\},$

where $\mathcal{L}u$ is understood in the sense of distributions. The operator
$\left(\mathcal{L},D(\mathcal{L})\right)$ is closed in
$L^{2}\left(\mathbb{R}^{N}\right)$ and the space
$C_{c}^{\infty}(\mathbb{R}^{N})$ is a core of $\mathcal{L}$. The operator
$\mathcal{L}$ generates a $C_{0}$-group $\left(S(t)\right)_{t\in\mathbb{R}}$
given by

$\left(S(t)f\right)(x)=f\left(\mathrm{e}^{tB}x\right),\quad
t\in\mathbb{R},\quad x\in\mathbb{R}^{N}$ (3)

for $f\in L^{2}\left(\mathbb{R}^{N}\right)$. Moreover,

$\|S(t)f\|=\mathrm{e}^{-\frac{t}{2}\operatorname{tr}(B)}\|f\|,\qquad
t\in\mathbb{R}$ (4)

for every $f\in L^{2}\left(\mathbb{R}^{N}\right)$.

###  2.3. The Ornstein-Uhlenbeck semigroup

Let $B^{*}$ denote the transpose matrix of $B$. We introduce the matrices

$Q_{t}=\int_{0}^{t}\mathrm{e}^{sB}\,\mathrm{e}^{sB^{*}}\,\mathrm{d}s,\qquad
t>0,$

which are symmetric and positive definite ($Q_{0}$ is the null matrix). The
Ornstein-Uhlenbeck semigroup $T(t)\colon
L^{2}\left(\mathbb{R}^{N}\right)\rightarrow L^{2}\left(\mathbb{R}^{N}\right)$
is given by Kolmogorov’s formula

$\displaystyle T(0)$ $\displaystyle=I,$ $\displaystyle(T(t)f)(x)$
$\displaystyle=\frac{1}{\sqrt{(4\pi)^{N}\operatorname{det}Q_{t}}}\int_{\mathbb{R}^{N}}\mathrm{e}^{-\frac{1}{4}\left\langle
Q_{t}^{-1}y,y\right\rangle}f\left(\mathrm{e}^{tB}x-y\right)\mathrm{d}y,\quad
t>0,\quad x\in\mathbb{R}^{N}$

for every $f\in L^{2}\left(\mathbb{R}^{N}\right)$. This can be written as

$T(t)f=S(t)\left(g_{t}*f\right),$ (5)

where

$g_{t}(y)=\frac{1}{\sqrt{(4\pi)^{N}\operatorname{det}Q_{t}}}\mathrm{e}^{-\frac{1}{4}\left\langle
Q_{t}^{-1}y,y\right\rangle},\quad t>0,\quad y\in\mathbb{R}^{N}.$ (6)

Let us define the maximal domain of $\mathcal{A}$ by

$D\left(\mathcal{A}\right)=\left\\{u\in L^{2}\left(\mathbb{R}^{N}\right)\cap
H_{\mathrm{loc}}^{2}\left(\mathbb{R}^{N}\right):\mathcal{A}u\in
L^{2}\left(\mathbb{R}^{N}\right)\right\\}.$

The space $C_{c}^{\infty}\left(\mathbb{R}^{N}\right)$ is a core for
$\mathcal{A}$. More precisely, $D\left(\mathcal{A}\right)$ is characterized by

$D\left(\mathcal{A}\right)=\left\\{u\in
H^{2}\left(\mathbb{R}^{N}\right):Bx\cdot\nabla u\in
L^{2}\left(\mathbb{R}^{N}\right)\right\\}=D(\Delta)\cap D(\mathcal{L}).$

It is known that $\left(\mathcal{A},D\left(\mathcal{A}\right)\right)$
generates a $C_{0}$-semigroup on $L^{2}\left(\mathbb{R}^{N}\right)$ given by
$\left(T(t)\right)_{t\geq 0}$ which satisfies the estimate

$\|T(t)\|\leq\mathrm{e}^{-\frac{t}{2}\operatorname{tr}(B)},\qquad t\geq 0,$

where $\|\cdot\|$ also denotes the operator norm. The $C_{0}$-semigroup
$\left(T(t)\right)_{t\geq 0}$ is not analytic in
$L^{2}\left(\mathbb{R}^{N}\right)$ (e.g. it is not an eventually norm
continuous semigroup), except for $B=0$.

##  3\. Logarithmic convexity and observability

###  3.1. Logarithmic convexity estimates

The logarithmic convexity method is one of the well-known approaches that had
been widely used to prove conditional stability for improperly posed problems
such as backward parabolic equations as well as inverse initial data problems.
The interested reader can be referred to [1, 2, 13, 16, 27] and the references
therein.

Next we give an abstract definition to a logarithmic convexity estimate. Let
$\left(H,\langle\cdot,\cdot\rangle\right)$ be a Hilbert space with
corresponding norm $\|\cdot\|$. Let $\left(\mathrm{e}^{tA}\right)_{t\geq 0}$
be a $C_{0}$-semigroup on $H$ associated with its generator $A\colon
D(A)\subset H\rightarrow H$. Inspired by [1, 16], we introduce the following
definition.

######  Definition 3.1.

We say that the $C_{0}$-semigroup $\left(\mathrm{e}^{tA}\right)_{t\geq 0}$
satisfies a logarithmic convexity estimate for $\theta>0$ if there exists a
constant $\kappa_{\theta}\geq 1$ and a function
$w\colon(0,\theta)\rightarrow(0,1)$, $w(0)=0$ and $w(\theta)=1$, so that the
following estimate holds

$\left\|\mathrm{e}^{tA}u\right\|\leq\kappa_{\theta}\|u\|^{1-w(t)}\left\|\mathrm{e}^{\theta
A}u\right\|^{w(t)}$ (7)

for all $t\in[0,\theta]$ and all $u\in H$.

It is known that the logarithmic convexity holds for self-adjoint bounded
above operators with $\kappa_{\theta}=1$ and $w(t)=\dfrac{t}{\theta}$, see
e.g. [12, Section 2]. If the operator is subordinated to its symmetric part,
there is a logarithmic convexity result in [13, Theorem 3.1.3], where
$w(t)=\dfrac{1-\mathrm{e}^{-ct}}{1-\mathrm{e}^{-c\theta}}$ for some constant
$c>0$. More generally, it holds for analytic semigroups with a more general
function $w(t)$. We refer to [3] for more details.

######  Remark 1.

Note that the estimate (7) implies the backward uniqueness for the semigroup
$\left(\mathrm{e}^{tA}\right)_{t\geq 0}$, namely the following property: if
$\mathrm{e}^{\theta A}u=0$ for some $u\in H$, then $u=0$. Furthermore, the
logarithmic convexity can be seen as a stability estimate for the backward
uniqueness. More precisely, if one assumes a priori bound on the initial data
$\|u\|\leq R$ for some positive constant $R$, the norm of the corresponding
solution $\left\|\mathrm{e}^{tA}u\right\|$, $t\in(0,\theta)$ is small whenever
$\left\|\mathrm{e}^{\theta A}u\right\|$ is small.

Since the Laplace operator with domain $H^{2}\left(\mathbb{R}^{N}\right)$ is
self-adjoint and negative on $L^{2}\left(\mathbb{R}^{N}\right)$, the heat
semigroup $\left(U(t)\right)_{t\geq 0}$ satisfies the logarithmic convexity

$\|U(t)f\|\leq\|f\|^{1-\frac{t}{\theta}}\|U(\theta)f\|^{\frac{t}{\theta}},\qquad
f\in L^{2}\left(\mathbb{R}^{N}\right),\qquad t\in[0,\theta].$

In fact, the function $t\mapsto\|U(t)f\|^{2}$ is even log-convex in this case.
The same holds for the drift $C_{0}$-group
$\left(S(t)\right)_{t\in\mathbb{R}}$. Indeed, using (4), we see that

$\|S(t)f\|=\|f\|^{1-\frac{t}{\theta}}\|S(\theta)f\|^{\frac{t}{\theta}},\qquad
f\in L^{2}\left(\mathbb{R}^{N}\right),\qquad t\in[0,\theta].$

Next we prove a logarithmic convexity estimate for the Ornstein-Uhlenbeck
semigroup $\left(T(t)\right)_{t\geq 0}$.

######  Proposition 1.

There exist constants $c=c(\theta)\in(0,1]$ and $\kappa_{\theta}\geq 1$ such
that the following estimate holds

$\|T(t)f\|\leq\kappa_{\theta}\|f\|^{1-c\frac{t}{\theta}}\|T(\theta)f\|^{c\frac{t}{\theta}},\qquad
f\in L^{2}\left(\mathbb{R}^{N}\right),\qquad t\in[0,\theta].$ (8)

###### Proof.

Using (4) and (5), the estimate (8) is implied by the following inequality

$\|g_{t}*f\|\leq\|f\|^{1-c\frac{t}{\theta}}\|g_{\theta}*f\|^{c\frac{t}{\theta}},\qquad
f\in L^{2}\left(\mathbb{R}^{N}\right),\qquad t\in(0,\theta],$ (9)

with $\kappa_{\theta}=\mathrm{e}^{\frac{|\mathrm{tr}(B)|}{2}(1-c)\theta}$.
Invoking the Fourier transform denoted by $\widehat{f}$ for any $f\in
L^{2}\left(\mathbb{R}^{N}\right)$, we obtain the identities
$\widehat{g}_{t}(\xi)=\mathrm{e}^{-\left\langle Q_{t}\xi,\xi\right\rangle}$
and $\widehat{g_{t}*f}(\xi)=\mathrm{e}^{-\left\langle
Q_{t}\xi,\xi\right\rangle}\widehat{f}(\xi)$, $\xi\in\mathbb{R}^{N}$. We first
claim that there exists a constant $c_{\theta}>0$ such that

$\left\langle Q_{t}\xi,\xi\right\rangle\geq
c_{\theta}\frac{t}{\theta}\left\langle
Q_{\theta}\xi,\xi\right\rangle\qquad\forall
t\in[0,\theta],\quad\forall\xi\in\mathbb{R}^{N}.$ (10)

Indeed, since $\frac{Q_{t}}{t}\to I_{N}$ (the identity matrix) as $t\to 0$, by
continuity and positivity of the function

$\beta(t,\xi)=\begin{cases}\displaystyle\frac{1}{t}\left\langle
Q_{t}\xi,\xi\right\rangle&\mbox{ if }t>0,\\\ \left|\xi\right|^{2}&\mbox{ if
}t=0,\end{cases}$

in the compact set $[0,\theta]\times\mathbb{S}^{N-1}$
($\mathbb{S}^{N-1}=\\{\xi\in\mathbb{R}^{N}:|\xi|=1\\}$), there exist positive
constants $c_{1}$ and $c_{2}$ (dependent on $\theta$) such that

$c_{1}t\leq\left\langle Q_{t}\xi,\xi\right\rangle\leq c_{2}t\qquad\forall
t\in[0,\theta],\quad\forall\xi\in\mathbb{S}^{N-1}.$

Thus,

$c_{1}t|\xi|^{2}\leq\left\langle Q_{t}\xi,\xi\right\rangle\leq
c_{2}t|\xi|^{2}\qquad\forall t\in[0,\theta],\quad\forall\xi\in\mathbb{R}^{N}.$

This inequality implies (10) with $c:=c_{\theta}=\frac{c_{1}}{c2}\leq 1$. The
inequality (10) entails that

$\mathrm{e}^{-2\left\langle
Q_{t}\xi,\xi\right\rangle}|\widehat{f}(\xi)|^{2}\leq|\widehat{f}(\xi)|^{2\left(1-c\frac{t}{\theta}\right)}\left(\mathrm{e}^{-2\left\langle
Q_{\theta}\xi,\xi\right\rangle}|\widehat{f}(\xi)|^{2}\right)^{c\frac{t}{\theta}}.$

Let $t\in(0,\theta)$ be fixed. Applying the Hölder inequality for
$p=\frac{\theta}{\theta-ct}$, $q=\frac{\theta}{ct}$ and the functions
$F(\xi)=|\widehat{f}(\xi)|^{2\left(1-c\frac{t}{\theta}\right)}$,
$G(\xi)=\left(\mathrm{e}^{-2\left\langle
Q_{\theta}\xi,\xi\right\rangle}|\widehat{f}(\xi)|^{2}\right)^{c\frac{t}{\theta}}$,
we obtain (9). This completes the proof of (8). ∎

######  Remark 2.

Proposition 1, aside from being of independent interest, implies the backward
uniqueness property for the Ornstein-Uhlenbeck semigroup. This property plays
an important role in some control problems.

###  3.2. Observability estimate

In this subsection, we discuss some recent results on the final state
observability for the system (1). We refer to [29, 34] for the general theory.

In [7], the authors have shown the final state observability of the system (1)
at any positive time $\theta$ whenever the observation region
$\omega\subset\mathbb{R}^{N}$ is a nonempty open set satisfying the geometric
condition

$\exists\delta,r>0,\forall y\in\mathbb{R}^{N},\exists
y^{\prime}\in\omega,\quad B\left(y^{\prime},r\right)\subset\omega\text{ and
}\left|y-y^{\prime}\right|<\delta.$ (11)

This condition was known as a sufficient condition ensuring the final state
observability of the heat equation on $\mathbb{R}^{N}$ at any positive time
[26]. The later property has been recently characterized by the thickness of
the set $\omega$ [10, 30]:

######  Definition 3.2.

Let $\gamma\in(0,1]$ and
$a=\left(a_{1},\ldots,a_{N}\right)\in\left(\mathbb{R}_{+}^{*}\right)^{N}$. Let
us denote by
$\mathcal{C}=\left[0,a_{1}\right]\times\ldots\times\left[0,a_{N}\right]$.

  * •

A measurable set $\omega\subset\mathbb{R}^{N}$ is said to be
$(\gamma,a)$-thick if

$\left|\omega\cap\left(x+\mathcal{C}\right)\right|\geq\gamma\prod_{j=1}^{N}a_{j}\qquad\forall
x\in\mathbb{R}^{N},$

where $\left|E\right|$ denotes the Lebesgue measure of a measurable set
$E\subset\mathbb{R}^{N}$.

  * •

A measurable set $\omega\subset\mathbb{R}^{N}$ is thick if there exist
$\gamma\in(0,1]$ and $a\in\left(\mathbb{R}_{+}^{*}\right)^{N}$ such that
$\omega$ is $(\gamma,a)$-thick.

  * •

A measurable set $\omega\subset\mathbb{R}^{N}$ is $\gamma$-thick at scale
$L>0$ if $\omega$ is $(\gamma,a)$-thick and
$a=(L,\ldots,L)\in\left(\mathbb{R}_{+}^{*}\right)^{N}$.

We emphasize that the notion of thickness is weaker than the condition (11).
Furthermore, the thickness of the observation set $\omega$ turns out to be a
sufficient condition that ensures the final state observability of several
parabolic equations at any positive time, including the system (1).

Next we state the observability inequality of system (1) from thick sets.

######  Proposition 2.

Let $\theta>0$ be fixed and let $\omega\subset\mathbb{R}^{N}$ be a thick set.
Consider $u$ the mild solution of (1). Then there exists a positive constant
$\kappa_{\theta}$ such that for all $u_{0}\in
L^{2}\left(\mathbb{R}^{N}\right)$, we have

$\|u(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{2}\leq\kappa_{\theta}^{2}\int_{0}^{\theta}\|u(t,\cdot)\|_{L^{2}(\omega)}^{2}\,\mathrm{d}t.$
(12)

The above result has been recently proven in [4] for the possibly degenerate
fractional Ornstein-Uhlenbeck equation under a Kalman rank condition. It
improves its counterpart in [7] regarding the Ornstein-Uhlenbeck equation for
observation sets $\omega$ satisfying (11) which has been considered in [3].

##  4\. Logarithmic stability for a class of initial data

We introduce the set of admissible initial data:

$\mathcal{I}_{R}=\left\\{u_{0}\in
D(\mathcal{A})\colon\|u_{0}\|_{D(\mathcal{A})}\leq R\right\\}$

for a fixed constant $R>0$.

######  Theorem 4.1.

Let $\theta>0$ be fixed and let $\omega\subset\mathbb{R}^{N}$ be a thick set.
There exist positive constants $C$ and $C_{1}$ depending on
$(N,\theta,\omega,R)$ such that, for all $u_{0}\in\mathcal{I}_{R}$,

$\|u_{0}\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\frac{-C}{\log\left(C_{1}\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}\right)}$
(13)

for $\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}$ sufficiently small,
where $u$ is the solution of system (1).

###### Proof.

Let $z=u_{t}$ and apply (12) to $z$, we obtain

$\|z(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
C\|z\|_{L^{2}\left(0,\theta;L^{2}(\omega)\right)}.$ (14)

Applying the logarithmic convexity estimate to $z$ (Proposition 1), we have

$\left\|z(t,\cdot)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\kappa_{\theta}R^{1-c\frac{t}{\theta}}\left\|z(\theta,\cdot)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{c\frac{t}{\theta}},\quad
0\leq t\leq\theta.$ (15)

Since

$u(0,\cdot)=-\int_{0}^{\theta}z(\tau,\cdot)\mathrm{d}\tau+u(\theta,\cdot),$

by (14) and (15) we have

$\displaystyle\|u(0,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}$
$\displaystyle\leq\int_{0}^{\theta}\left\|z(\tau,\cdot)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}\mathrm{d}\tau+\|u(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}$
$\displaystyle\leq
C\int_{0}^{\frac{\theta}{c}}\left\|z(\theta,\cdot)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{\frac{c\tau}{\theta}}\mathrm{d}\tau+C\|u\|_{L^{2}\left(0,\theta;L^{2}(\omega)\right)}\quad(c\in(0,1])$
$\displaystyle\leq
C\frac{\theta}{c}\frac{\|z(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}-1}{\log\|z(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}}+C\|u\|_{L^{2}\left(0,\theta;L^{2}(\omega)\right)}$
$\displaystyle\leq C\left(\frac{E-1}{\log E}+E\right),$

where we denoted
$E:=\|z(\theta,\cdot)\|_{L^{2}\left(\mathbb{R}^{N}\right)}+C\|u\|_{L^{2}\left(0,\theta;L^{2}(\omega)\right)}$,
and $C$ is a constant that varies from line to line. By (14), when the norm
$\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}$ is sufficiently small, we
obtain

$0<E\leq C_{1}\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}<1$ (16)

for some constant $C_{1}>0$. Using the inequality
$\dfrac{\tau-1}{\log\tau}+\tau\leq-\dfrac{1+\mathrm{e}^{-2}}{\log\tau}$ for
$0<\tau<1$, with (14) we obtain

$\displaystyle\|u_{0}\|_{L^{2}\left(\mathbb{R}^{N}\right)}$
$\displaystyle\leq\frac{-C}{\log\left(C_{1}\|u\|_{H^{1}\left(0,\theta;L^{2}(\omega)\right)}\right)}.$

∎

At this level, some comments should be made on the particular case of the heat
equation (when $B=0$): Theorem 4.1 enables us to derive a logarithmic
stability result for a class of initial data of the heat equation posed in
$\mathbb{R}^{N}$. A similar result was shown in [31] for the heat equation on
a bounded domain $\Omega\subset\mathbb{R}^{N}$ with homogeneous Dirichlet
boundary conditions on $\partial\Omega$.

Since the heat semigroup is analytic on $L^{2}\left(\mathbb{R}^{N}\right)$, we
can show an improved logarithmic stability result for a large class of initial
data

$\mathcal{I}_{\varepsilon,R}:=\left\\{u_{0}\in
H^{2\varepsilon}\left(\mathbb{R}^{N}\right)\colon\|u_{0}\|_{H^{2\varepsilon}\left(\mathbb{R}^{N}\right)}\leq
R\right\\}$

for fixed $\varepsilon\in(0,1)$ and $R>0$. Although the used techniques are
quite classical, we will give a full proof for the reader convenience.

######  Theorem 4.2.

Let $\theta>0$ be fixed and let $\omega\subset\mathbb{R}^{N}$ be a thick set.
Assume also that $p\in\left(1,\dfrac{1}{1-\varepsilon}\right)$ and
$s\in\left(0,1-\dfrac{1}{p}\right)$. Then there exists a positive constant
$K(\varepsilon,R,\theta,\kappa_{\theta},p,s)$ such that, for all
$u_{0}\in\mathcal{I}_{\varepsilon,R}$, we have

$\|u_{0}\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
K\left(\frac{\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}^{p}-1}{\log\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}}\right)^{\frac{s}{p}},$
(17)

where $u$ is the solution of system (1) with $B=0$. Moreover, if
$\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}<1$, then

$\|u_{0}\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
K\left(-\log\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}\right)^{-\frac{s}{p}}.$ (18)

###### Proof.

Since $u_{0}\in\mathcal{I}_{\varepsilon,R}$, we deduce by the logarithmic
convexity that

$\|u(t)\|_{\left(\mathbb{R}^{N}\right)}\leq
R^{\frac{\theta-t}{\theta}}\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{\frac{t}{\theta}}\quad\left(0\leq
t\leq\theta\right).$

Assume $p>1$. By integrating the above relation between $0$ and $\theta$, we
obtain

$\displaystyle\int_{0}^{\theta}\|u(t)\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}\mathrm{d}t$
$\displaystyle\leq
R^{p}\int_{0}^{\theta}\mathrm{e}^{p\frac{t}{\theta}\log\left(R^{-1}\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}\right)}\mathrm{d}t$
$\displaystyle\leq\theta\frac{\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-R^{p}}{\log\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-\log
R^{p}},$

and thus

$\|u\|_{L^{p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)}\leq\theta^{\frac{1}{p}}\left(\frac{\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-R^{p}}{\log\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-\log
R^{p}}\right)^{\frac{1}{p}}.$ (19)

On the other hand, by the semigroup representation of the solution $u$, we
have

$u_{t}=-(-\Delta)^{1-\varepsilon}\mathrm{e}^{t\Delta}(-\Delta)^{\varepsilon}u_{0}.$

Moreover, since $\Delta$ is the generator of an analytic semigroup on
$L^{2}\left(\mathbb{R}^{N}\right)$, we have

$\left\|u_{t}(t)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\frac{1}{t^{1-\varepsilon}}\left\|u_{0}\right\|_{H^{2\varepsilon}\left(\mathbb{R}^{N}\right)}.$

We deduce from the above inequality that for
$p\in\left(1,\dfrac{1}{1-\varepsilon}\right)$,

$\left\|u_{t}\right\|_{L^{p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)}\leq
R\frac{\theta^{\frac{1}{p}-(1-\varepsilon)}}{(1-p(1-\varepsilon))^{\frac{1}{p}}}.$

Using
$\|u(t)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\|u(0)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
R$ for all $t$, we also have

$\|u\|_{L^{p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)}\leq
R\theta^{\frac{1}{p}},$

and therefore

$\|u\|_{W^{1,p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)}\leq
R\theta^{\frac{1}{p}}\left(1+\frac{1}{\theta^{(1-\varepsilon)}(1-p(1-\varepsilon))^{\frac{1}{p}}}\right).$
(20)

Combining (19) and (20), and using a Sobolev interpolation, we obtain for all
$0<s<1$ that

$\displaystyle\|u\|_{W^{1-s,p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)}\leq
CR^{1-s}\theta^{\frac{1}{p}}\left(1+\frac{1}{\theta^{(1-\varepsilon)}(1-p(1-\varepsilon))^{\frac{1}{p}}}\right)^{1-s}$
$\displaystyle\times\left(\frac{\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-R^{p}}{\log\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-\log
R^{p}}\right)^{\frac{s}{p}}.$

Using the Sobolev embedding

$W^{1-s,p}\left(0,\theta;L^{2}\left(\mathbb{R}^{N}\right)\right)\subset
C\left(\left[0,\theta\right];L^{2}\left(\mathbb{R}^{N}\right)\right)$

for $(1-s)p>1$, there exists $K=K\left(R,\theta,p,\varepsilon,s\right)>0$ such
that

$\|u(0)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
K\left(\frac{\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-R^{p}}{\log\left\|u\left(\theta\right)\right\|_{L^{2}\left(\mathbb{R}^{N}\right)}^{p}-\log
R^{p}}\right)^{\frac{s}{p}}.$

Using the observability inequality (12),

$\|u(\theta)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq\kappa_{\theta}\|u\|_{L^{2}(0,\theta;L^{2}(\omega))},$

we deduce that

$\|u(0)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
K\left(\frac{\kappa_{\theta}^{p}\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}^{p}-R^{p}}{\log\kappa_{\theta}^{p}\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}^{p}-\log
R^{p}}\right)^{\frac{s}{p}}.$

Then considering both cases $\kappa_{\theta}\geq R$, $\kappa_{\theta}<R$, and
using again the concavity of the logarithm function, we deduce that

$\|u(0)\|_{L^{2}\left(\mathbb{R}^{N}\right)}\leq
K\left(\frac{\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}^{p}-1}{\log\|u\|_{L^{2}(0,\theta;L^{2}(\omega))}}\right)^{\frac{s}{p}}.$

This ends the proof. ∎

######  Remark 3.

We emphasize that in Theorem 4.2 we have considered observation sets
$\omega\subset\mathbb{R}^{N}$ that are thick. Such a result improves [5,
Theorem 1.1] in terms of observation where the authors consider open sets
$\omega$ such that $\mathbb{R}^{N}\setminus\omega$ is bounded, which is far to
be optimal.

##  5\. Comments and open problems

We have investigated the interplay between observability and inverse problems
for a class of equations which represents a prototype of parabolic equations
with unbounded coefficients (of gradient type) via the logarithmic convexity
estimate. More precisely, we have proven that the observability inequality
along with the logarithmic convexity imply conditional logarithmic stability
of initial data.

In this paper, we have considered a case including a non-analytic semigroup
which has not been studied in the literature within this context. Also,
stability estimates for initial data for parabolic equations with unbounded
coefficients have not been considered before up to our knowledge, except in
[3] where analyticity have played a crucial role.

In the particular (but important) case of heat equation, i.e. the case without
the drift term, that is $B=0$, we can improve some interesting results
obtained in the recent paper [5, 6] in terms of the observability regions
$\omega\subset\mathbb{R}^{N}$ for the reconstruction of the initial
temperatures. More precisely, we can consider sharp observation regions
$\omega$ given by thick sets instead of considering sets such that the
unobserved region $\mathbb{R}^{N}\setminus\omega$ is bounded. Furthermore, for
the general case $B\neq 0$, the results can be improved as far as the
observation region $\omega$ is sharpened.

In the present paper, we have proven a logarithmic convexity estimate for the
Ornstein-Uhlenbeck equation leveraging the explicit representation formula of
the corresponding semigroup. This raises the following problems:

  1. (1)

Can one prove a logarithmic convexity estimate in the absence of an explicit
formula for the associated semigroup as in the analytic case? For instance,
can we consider more general Ornstein-Uhlenbeck operators as
$\mathcal{A}=\mathrm{div}(Q\nabla)-Rx\cdot x+Bx\cdot\nabla,$ where $Q=Q^{*}$
and $R=R^{*}$ are real constant $N\times N$-matrices which are positive
semidefinite? Under suitable assumptions on the matrices, $B,Q$ and $R$, the
observability inequality holds true, see [9, Subsection 6.4]. Note that this
case also covers some situations where the observation region $\omega$ is not
thick.

  2. (2)

Given the perturbation argument in Section 2, it is natural to ask whether the
logarithmic convexity is preserved when a main abstract operator $\mathcal{A}$
is perturbed by an operator $\mathcal{B}$. This is true for instance in the
analytic case, when $\mathcal{A}$ is perturbed by a bounded operator or more
generally a relatively $\mathcal{A}$-bounded operator, since analyticity is
preserved in this case, see e.g. [11, Theorem 2.10, p. 176].

## Acknowledgment

The first named author would like to thank Giorgio Metafune for a fruitful
discussion with invaluable comments.

## References

  * [1] S. Agmon and L. Nirenberg,  Properties of solutions of ordinary differential equations in Banach space,  _Comm. Pure Appl. Math._ 16 (1963), 121–239.
  * [2] E. M. Ait Ben Hassi, S. E. Chorfi and L. Maniar,  An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions,  _J. Inverse Ill-Posed Probl._ 30 (2022), 363–378.
  * [3] E. M. Ait Ben Hassi, S. E. Chorfi and L. Maniar,  Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation,  _Discrete Contin. Dyn. Syst. - S_ (2023), doi: 10.3934/dcdss.2022212.
  * [4] P. Alphonse and J. Bernier,  Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability,  _Bull. Sci. math._ 165 (2020), 102914.
  * [5] P. Arratia, M. Courdurier, E. Cueva, A. Osses and B. Palacios,  Lipschitz stability for backward heat equation with application to fluorescence microscopy,  _SIAM J. Math. Anal._ 53 (2021), 5948–5978.
  * [6] P. Arratia, M. Courdurier, E. Cueva, A. Osses and B. Palacios,  Lipschitz stability for backward heat equation with application to fluorescence microscopy,  (2023) arXiv: 2012.02002.
  * [7] K. Beauchard and K. Pravda-Starov,  Null-controllability of hypoelliptic quadratic differential equations,  _J. Éc. Polytech. Math._ 5 (2018), 1–43.
  * [8] R. Chill, E. Fašangová, G. Metafune and D. Pallara,  The sector of analyticity of the Ornstein-Uhlenbeck semigroup on $L^{p}$ spaces with respect to invariant measure,  _J. London Math. Soc._ 3 (2005), 703–722.
  * [9] A. Dicke, A. Seelmann and I. Veselić,  Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density,  (2022) arXiv: 2201.02370.
  * [10] M. Egidi and I. Veselić,  Sharp geometric condition for null-controllability of the heat equation on $\mathbb{R}^{d}$ and consistent estimates on the control cost,  _Arch. Math._ 111 (2018), 85–99.
  * [11] K.-J. Engel and R. Nagel,  _One-Parameter Semigroups for Linear Evolution Equations_ ,  Grad. Texts in Math. 194, Springer, New York, 2000.
  * [12] G. García and T. Takahashi,  Inverse problem and null-controllability for parabolic systems,  _J. Inverse Ill-Posed Probl._ 19 (2011), 379–405.
  * [13] V. Isakov,  _Inverse Problems for Partial Differential Equations_ ,  third edition, Springer, 2017.
  * [14] M. V. Klibanov,  Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data,  _Inverse Problems_ 22 (2006), 495–514.
  * [15] M. V. Klibanov and A. V. Tikhonravov,  Estimates of initial conditions of parabolic equations and inequalities in infinite domains via lateral Cauchy data,  _J. Diff. Equ._ 237 (2007), 198–224.
  * [16] S. G. Krein and O. I. Prozorovskaya,  Analytic semigroups and incorrect problems for evolutionary equations,  _Dokl. Akad. Nauk SSSR_ , 133 (1960), 277–280.
  * [17] J. Le Rousseau and I. Moyano,  Null-controllability of the Kolmogorov equation in the whole phase space,  _J. Diff. Equ._ 260 (2016), 3193–3233.
  * [18] J. Li, M. Yamamoto and J. Zou,  Conditional stability and numerical reconstruction of initial temperature,  _Commun. Pure Appl. Anal._ 8 (2009), 361–382.
  * [19] L. Lorenzi,  An identification problem for the Ornstein-Uhlenbeck operator,  _J. Inverse Ill-Posed Probl._ 19 (2011), 293–326.
  * [20] A. Lunardi,  On the Ornstein-Uhlenbeck operator in $L^{2}$ spaces with respect to invariant measures,  _Trans. Amer. Math. Soc._ 349 (1997), 155–169.
  * [21] A. Lunardi, G. Metafune and D. Pallara,  The Ornstein-Uhlenbeck semigroup in finite dimension,  _Phil. Trans. R. Soc. A_ 378 (2020), 20200217.
  * [22] G. Metafune,  $L^{p}$-spectrum of Ornstein-Uhlenbeck operators,  _Ann. Scuola Norm. Sup. Pisa_ 30 (2001), 97–124.
  * [23] G. Metafune, D. Pallara and E. Priola,  Spectrum of Ornstein-Uhlenbeck operators in $L^{p}$ spaces with respect to invariant measures,  _J. Funct. Anal._ 196 (2002), 40–60.
  * [24] G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt,  The domain of the Ornstein-Uhlenbeck operator on an $L^{p}$-space with invariant measure,  _Ann. Scuola Norm. Sup. Pisa Cl. Sci._ 1 (2002), 471–485.
  * [25] K. Miller,  Logarithmic convexity results for holomorphic semigroups,  _Pacific J. Math._ 58 (1975), 549–551.
  * [26] L. Miller,  Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds,  _Math. Res. Lett._ 12 (2005) 37–47.
  * [27] L. E. Payne,  _Improperly Posed Problems in Partial Differential Equations_ ,  SIAM, Philadelphia, 1975.
  * [28] J.-P. Puel and M. Yamamoto,  On a global estimate in a linear inverse hyperbolic problem,  _Inverse Problems_ 12 (1996), 995–1002.
  * [29] M. Tucsnak and G. Weiss,  _Observation and Control for Operator Semigroups_ ,  Birkhäuser Adv. Texts, Basel, 2009.
  * [30] G. Wang, M. Wang, C. Zhang and Y. Zhang,  Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^{n}$,  _J. Math. Pures Appl._ 126 (2019), 144–194.
  * [31] D. Xu and M. Yamamoto,  Stability estimates in state-estimation for a heat process,  _Proc. Second ISAAC Congress_ , International Society for Analysis, Applications and Computation, vol 7. Springer, Boston (2000), 193–198.
  * [32] M. Yamamoto and J. Zou,  Conditional stability in reconstruction of initial temperatures,  _Proc. Fifth ISAAC Congress_ , World Scientific, New Jersey (2008), 1385–1390.
  * [33] M. Yamamoto and J. Zou,  Simultaneous reconstruction of the initial temperature and heat radiative coefficient,  _Inverse Problems_ 17 (2001), 1181–1202.
  * [34] J. Zabczyk,  _Mathematical Control Theory: An Introduction_ ,  Systems & Control: Foundations & Applications, Springer Nature Switzerland, 2nd ed., 2020.