# On Boundary Exact Controllability of One-Dimensional Wave Equations with
Weak and Strong Interior Degeneration

Peter I. Kogut Department of Differential Equations, Oles Honchar Dnipro
National University, Gagarin av., 72, 49010 Dnipro, Ukraine<EMAIL_ADDRESS>Olha P. Kupenko<EMAIL_ADDRESS>Department of System Analysis and
Control, National Technical University ”Dnipro Polytechnics”, Yavornitsky av.,
19, 49005 Dnipro, Ukraine; Institute of Applied and System Analysis, Ihor
Sikorsky National Technical University of Ukraine “Kiev Polytechnical
Institute”, Peremogy av., 37, build. 35, 03056 Kiev, Ukraine  Günter Leugering
Department Mathematik Lehrstuhl II Universität Erlangen-Nürnberg Cauerstr. 11
D-91058 Erlangen, Germany<EMAIL_ADDRESS>
###### Abstract

In this paper we study exact boundary controllability for a linear wave
equation with strong and weak interior degeneration of the coefficient in the
principle part of the elliptic operator. The objective is to provide a well-
posedness analysis of the corresponding system and derive conditions for its
controllability through boundary actions. Passing to a relaxed version of the
original problem, we discuss existence and uniqueness of solutions, and using
the HUM method we derive conditions on the rate of degeneracy for both exact
boundary controllability and the lack thereof.

###### keywords:

Degenerate wave equation, boundary control, existence result, weighted Sobolev
spaces, exact controllability.

###### AMS:

35L80, 49J20, 49J45, 93C73.

##  1 Introduction

In this paper we discuss exact boundary controllability for one-dimensional
degenerate wave equations with a weak and strong interior degeneration in the
principle part of the elliptic operator. Let $[0,T]$ be a given time interval.
For simplicity, let $c$ and $d$ be a given pair of real numbers such that
$0\leq c<1<d\leq 2$. We set

$\Omega_{1}=[c,1),\quad\Omega_{2}=(1,d],\quad\Omega=(c,d),\quad\text{and}\quad\Omega_{0}=\Omega\setminus\\{1\\}.$

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given weight function with
properties

  1. (i)

$a(1)=0$, $a(x)>0$ for all
$x\in\overline{\Omega}\setminus\\{1\\}=\Omega_{1}\cup\Omega_{2}$, and there
exist subintervals $(x^{\ast}_{1},1)\subset\Omega_{1}$ and
$(1,x^{\ast}_{2})\subset\Omega_{2}$ such that $a(\cdot)$ is monotonically
decreasing on $(x^{\ast}_{1},1)$, monotonically increasing on
$(1,x^{\ast}_{2})$, and

$\displaystyle\sup_{x\in[x^{\ast}_{1},1)}\frac{(1-x)|a^{\prime}(x)|}{a(x)}=\lim_{x\nearrow
1}\frac{(1-x)|a^{\prime}(x)|}{a(x)}>0,$ (1)
$\displaystyle\sup_{x\in(1,x^{\ast}_{2}]}\frac{(x-1)|a^{\prime}(x)|}{a(x)}=\lim_{x\searrow
1}\frac{(x-1)|a^{\prime}(x)|}{a(x)}>0;$ (2)

  2. (ii)

$a\in C(\overline{\Omega})\cap C^{1}(\overline{\Omega}\setminus\\{1\\})$;

  3. (iii)

$\left(\sqrt{a}\right)_{x}\not\in L^{\infty}(\Omega)$ whereas
$\left(\sqrt{a}\right)^{-1}_{x}\in L^{\infty}(\Omega)$.

As an example of function $a:\overline{\Omega}\to\mathbb{R}_{+}$ with the
above indicated properties (i)–(iii), we may consider the following one:

$a(x)=\left\\{\begin{array}[]{ll}(1-x)^{2p_{1}},&\text{ if }\ x\in[c,1]\\\
(x-1)^{2p_{2}},&\text{ if }\ x\in(1,d],\end{array}\right.\quad\text{ with
$p_{1},p_{2}>0$}.$ (3)

It is easy to check that, in this case, properties (i)–(iii) hold with
$x_{1}^{\ast}=c$, $x_{2}^{\ast}=d$, and

$\displaystyle\sup_{x\in[x^{\ast}_{1},1)}\frac{(1-x)|a^{\prime}(x)|}{a(x)}=\lim_{x\nearrow
1}\frac{(1-x)|a^{\prime}(x)|}{a(x)}=2p_{1},$
$\displaystyle\sup_{x\in(1,x^{\ast}_{2}]}\frac{(x-1)|a^{\prime}(x)|}{a(x)}=\lim_{x\searrow
1}\frac{(x-1)|a^{\prime}(x)|}{a(x)}=2p_{2},$
$\displaystyle\left(\sqrt{a}\right)_{x}\not\in L^{\infty}(\Omega)\ \text{ and
}\ \left(\sqrt{a}\right)^{-1}_{x}\in L^{\infty}(\Omega)\ \text{ if $p_{1}$ and
$p_{2}$ are less than $1$}.$

We are concerned with the following controlled system

$\displaystyle y_{tt}-\left(a(x)y_{x}\right)_{x}=0\quad\text{in }\
(0,T)\times\Omega,$ (4) $\displaystyle y(t,c)=f_{c}(t),\quad
y(t,d)=f_{d}(t)\quad\text{on }\ (0,T),$ (5) $\displaystyle
y(0,\cdot)=y_{0},\quad y_{t}(0,\cdot)=y_{1}\quad\text{ in }\ \Omega,$ (6)
$\displaystyle f_{c},f_{d}\in\mathcal{F}_{ad}=L^{2}(0,T).$ (7)

Here, $y_{0}$, and $y_{1}$ are given functions, and $\mathcal{F}_{ad}$ stands
for the class of admissible controls.

The system (4)-(7) describes the dynamics of a linear elastic string with out-
of-the-plane displacement under the actions of boundary sources $f_{c},f_{d}$
acting on the system as controls through the Dirichlet boundary conditions at
$x=c$ and $x=d$. The coefficient $a(x)$ can be interpreted as the spatially
varying stiffness (modulus of elasticity) of the elastic string. In contrast
to the standard case that is widely studied in the literature (see, for
instance, [25]), where the stiffness is assumed to be positive and bounded
away from zero, we assume that the string $[c,d]$ has a defect at the internal
point $x_{0}=1$. In case the defect occurs at the endpoint $x_{0}=c$, the
problem has been investigated in [1]. In the latter case, the spatial operator
is related to a classical singular Sturm-Liouville-problem that has been
treated already by Weyl in [30]. Degeneration in that context is related to
the notions of limit-point and limit-cycle. In [1] the authors define

$\mu_{a}:=\sup\limits_{0<x\leq\ell}\frac{x|a^{\prime}(x)|}{a(x)}$ (8)

for the problem on the interval $[0,\ell]$. The problem is called weakly
damaged if $0\leq\mu_{a}<1$ in which case $\frac{1}{a}\in L^{1}$, and strongly
damaged in case $1\leq\mu_{a}<2$. The authors show, among other things, one-
sided boundary observability and consequently one-sided boundary exact
controllability if $\mu_{a}<2$ with an observability/controllability time
approaching $+\infty$ as $\mu_{a}\rightarrow 2$. Thus, for $\mu_{a}\geq 2$,
these properties are lost.

Using the Liouville transform (see [11] 1954), it is possible to transform the
system above into a homogeneous wave equation with singular potential on an
interval that tends to infinity if $\mu_{a}\geq 2$. See e.g. [12], where
controllability properties are investigated based on this transformation.
Working in the $L^{\infty}$-framework, the author obtains similar results as
in [1].

The authors of this article are not aware of any publication where in-span
degeneration of the wave equation is treated, in particular in the context of
controllability or observability. For the parabolic case see [5]. The main
question that we are going to discuss in this paper, therefore, is how the
defect at the internal point $x_{0}=1$ affects the transmission conditions at
the singular (damage) point and the corresponding solution of the system
(4)–(7) as well as its observability or controllability properties.

Such analysis could be important for many applications. In particular, for the
cloaking problem (building of devices that lead to lack of obeservability)
[15], the evolution of damage in materials [16], optimization problems for
elastic bodies arising in contact mechanics, coupled systems, composite
materials [17], where ’life-cycle-optimization’ appears as a challenge.

The indicated type of degeneracy raises a number of new questions related to
the well-posedness of the hyperbolic equations in suitable functional spaces
as well as new estimates for their solutions. Hence, new tools are necessary
for the analysis of the corresponding optimal control problems. It should be
emphasized here that boundary value problems for degenerate elliptic and
parabolic equations have received a lot of attention in the last years (see,
for instance, [5, 6, 9, 23, 24, 26]). As for the control issue for degenerate
wave equations, we already mentioned [1, 14] (see also [20] for the
sensitivity analysis of optimal control problems for wave equations in domain
with defects).

The purpose of this paper is to provide a qualitative analysis of system
(4)–(7), prove an exact controllability result, and investigate how the degree
of degeneracy in the principle coefficient $a(x)$ affects the system (4)–(7)
and its solution. In contrast to the recent results [1], where the authors
study controllability and observability for degenerate equation of the form
(4) with the degeneracy of (4) at the boundary $x=c=0$, we focus on the case
where the ’damaged’ point is internal. So, our core idea is to to pass from
the original initial-boundary value problem (4)–(7) to a relaxed version,
namely, to some transmission problem with appropriate compatibility conditions
at the ’damaged’ point. We show that these conditions play a crucial role and
essentially depend on the ’degree of degeneracy’ (for some generalization we
refer to [18]). In multi-dimensional situation, when the ’damage zone’ does
not split the domain $\Omega$ onto several disconnected subdomains, there is
an expectation that the boundary controllability of the corresponding
hyperbolic equation with interior degeneration could be true under some
suitable ’degree of degeneracy’ in the diffusion coefficients. Although, in
general, the problem of boundary controllability in multi-dimensional case
appears as a challenge. This issue will be considered in details in the
forthcoming paper.

In Section 2 we introduce a special class of weighted Sobolev spaces that are
associated with the original initial-boundary value problem. It allows us not
only to to study in detail some properties of their elements in the regions
which are in close vicinity to the ’damaged’ point, but also propose an
appropriate relaxation for to the initial-boundary value problems (4)–(6). In
Section 3, we mainly focus on the well-posedness of the proposed relaxation
for the original controlled system. It allows us to consider in Section 4 the
issues related to the boundary observability of degenerate wave equations. In
Section 5 we discuss the questions of exact and null boundary controllability
of the original degenerate system and the lack of these properties for strong
degeneration.

##  2 Preliminaries

To specify the original controlled system (4)–(7) and fix the main ideas, we
begin with some preliminaries and assumptions. Let
$a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given weight function with
properties (i)–(iii). For $u$ smooth, define the functional $\|\cdot\|_{a}$ as
follows

$\|u\|_{a}=\left(\int_{\Omega}\left[u^{2}+au_{x}^{2}\right]\,dx\right)^{1/2}.$

Let $W^{1,2}(\Omega)$ be the standard Sobolev space. We denote by
$H^{1}_{a}(\Omega)$, $H^{1}_{a,0}(\Omega)$, and $W^{1}_{a}(\Omega)$ the spaces
which are defined as follows

  * •

$H^{1}_{a}(\Omega)$ is the closure of the set $\varphi\in
C^{\infty}(\overline{\Omega})$ with respect to the $\|\cdot\|_{a}$-norm;

  * •

$H^{1}_{a,0}(\Omega)$ is the closure of the set $C^{\infty}_{c}(\Omega)$ with
respect to the $\|\cdot\|_{a}$-norm;

  * •

$W^{1}_{a}(\Omega)$ is the space of maps $u\in L^{2}(\Omega)$ with
distributional derivatives $u_{x}$ that satisfy $u_{x}\in
L^{2}(\Omega,a\,dx)\cap L^{1}(\Omega)$, where

$L^{2}(\Omega,a\,dx)=\left\\{v:\Omega\rightarrow\mathbb{R}\ :\ v\ \text{ is
measurable and }\ \int_{\Omega}av^{2}\,dx<+\infty\right\\}.$

First note that since $C^{\infty}_{c}(\Omega)\subset
C^{\infty}(\overline{\Omega})$, we have that $H^{1}_{a,0}(\Omega)\subset
H^{1}_{a}(\Omega)$. Moreover, due to compactness of the embedding

$H^{1}_{a}(\left(c,1-\varepsilon\right)\cup\left(1+\varepsilon,d\right))\hookrightarrow
C^{0,1}([c,1-\varepsilon]\cup[1+\varepsilon,d]),\quad\text{ for all
$\varepsilon>0$ small enough},$

we see that, if $y\in H^{1}_{a}(\Omega)$, then $y(\cdot)$ is an absolutely
continuous function in $\overline{\Omega}\setminus\\{1\\}$. So, the conditions
$y(c)=0$ and $y(d)=0$ are consistent for all $y\in H^{1}_{a,0}(\Omega)$.
Therefore, $H^{1}_{a,0}(\Omega)$ can be equivalently defined as the closed
subspace of $H^{1}_{a}(\Omega)$ such that

$H^{1}_{a,0}(\Omega):=\left\\{y\in H^{1}_{a}(\Omega)\ :\
y(c)=y(d)=0\right\\}.$

It is worth to notice that, unlike classical Sobolev space, the subspace of
smooth functions are not necessarily dense in $W^{1}_{a}(\Omega)$. So, for
’typical’ weight functions $a:\overline{\Omega}\rightarrow\mathbb{R}$ with
properties (i)–(iii) it is unknown whether the identity
$H^{1}_{a}(\Omega)=W^{1}_{a}(\Omega)$ is valid (for the corresponding examples
we refer to [8, 27]). Therefore, it would be plausible to suppose that
$H^{1}_{a}(\Omega)\subseteq W^{1}_{a}(\Omega)$, in general.

We also have $H^{1}_{a,0}(\Omega)\subseteq W^{1}_{a,0}(\Omega)$, where
$W^{1}_{a,0}(\Omega)=\left\\{y\in W^{1}_{a}(\Omega)\ :\ y(c)=y(d)=0\right\\}$.

Setting

$\displaystyle\mu_{1,a}:=\sup_{x\in[x^{\ast}_{1},1)}\frac{(1-x)|a^{\prime}(x)|}{a(x)},\quad\mu_{2,a}:=\sup_{x\in(1,x^{\ast}_{2}]}\frac{(x-1)|a^{\prime}(x)|}{a(x)},$
(9)

we deduce from properties (i)–(ii) that there exist constants
$\kappa_{1,a}\geq 1$ and $\kappa_{2,a}\geq 1$ such that

$\sup_{x\in\Omega_{1}}\frac{(1-x)|a^{\prime}(x)|}{a(x)}=\kappa_{1,a}\mu_{1,a},\quad\sup_{x\in\Omega_{2}}\frac{(x-1)|a^{\prime}(x)|}{a(x)}=\kappa_{2,a}\mu_{2,a}.$
(10)

Then the common characteristic of the weight functions
$a:\overline{\Omega}\rightarrow\mathbb{R}$ with properties (i)–(iii) can be
summarizing as follows (see see Theorems 3.1 and 3.2 in [19] for comparison).

######  Proposition 1.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii). Then

$\displaystyle a(x)\geq
a(c)\frac{(1-x)^{\kappa_{1,a}\mu_{1,a}}}{(1-c)^{\kappa_{1,a}\mu_{1,a}}},\quad\forall\,x\in[c,1]\subset[0,1],$
(11) $\displaystyle a(x)\geq
a(d)\frac{(x-1)^{\kappa_{2,a}\mu_{2,a}}}{(d-1)^{\kappa_{2,a}\mu_{2,a}}},\quad\forall\,x\in[1,d]\subset[1,2].$
(12)

###### Proof.

Making use of representation (10), we get

$(1-x)a^{\prime}(x)\stackrel{{\scriptstyle\text{by
\eqref{1.1b}}}}{{\geq}}-\kappa_{1,a}\mu_{1,a}a(x),\quad\forall\,x\in\Omega_{1}.$
(13)

Integrating this inequality over $[c,x]\subset[c,1)$, where $c>0$ by the
initial assumptions,

$\int_{c}^{x}\frac{a^{\prime}}{a}\,ds\geq-\kappa_{1,a}\mu_{1,a}\int_{c}^{x}\frac{1}{1-s}\,ds,\quad\forall\,x\in[c,1),$

we arrive at the inequality (11). Arguing in a similar manner, we have

$(x-1)a^{\prime}(x)\stackrel{{\scriptstyle\text{by
\eqref{1.1b}}}}{{\leq}}\kappa_{2,a}\mu_{2,a}a(x),\quad\forall\,x\in(1,d].$
(14)

Therefore,

$\int_{x}^{d}\frac{a^{\prime}}{a}\,ds\leq\kappa_{2,a}\mu_{2,a}\int_{x}^{d}\frac{1}{s-1}\,ds,\quad\forall\,x\in(1,d].$

From this and the fact that $d<2$ we deduce (12). ∎

Arguing in a similar manner, it is easy to establish the following
inequalities.

$\displaystyle a(x)\geq
a(x_{1}^{\ast})\frac{(1-x)^{\mu_{1,a}}}{(1-x_{1}^{\ast})^{\mu_{1,a}}},\quad\forall\,x\in[x_{1}^{\ast},1]\subset[c,1],$
(15) $\displaystyle a(x)\geq
a(x_{2}^{\ast})\frac{(x-1)^{\mu_{2,a}}}{(x_{2}^{\ast}-1)^{\mu_{2,a}}},\quad\forall\,x\in[1,x_{2}^{\ast}]\subset[1,d].$
(16)

The next result is crucial for our further consideration and it explores some
remarkable properties of the weight functions
$a:\overline{\Omega}\rightarrow\mathbb{R}$ satisfying conditions (i)–(iii).

######  Proposition 2.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii). Then the following assertions hold true:

$\displaystyle 0<\mu_{1,a}<2\quad\text{and}\quad 0<\mu_{2,a}<2.$ (17)

###### Proof.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given function with
properties (i)–(iii). Setting

$k:=\left\|\left(\sqrt{a}\right)_{x}^{-1}\right\|^{2}_{L^{\infty}(\Omega)}\quad\text{
and }\quad\widehat{a}(x):=ka(x)\ \text{ for all $x\in\Omega$},$

we see that the function $\widehat{a}:\overline{\Omega}\rightarrow\mathbb{R}$
possesses all properties (i)–(iii) and the direct calculations show that
$\mu_{i,a}=\mu_{i,\widehat{a}}$ for $i=1,2$. Moreover, in this case, we have

$\left\|\left(\sqrt{\widehat{a}}\right)_{x}^{-1}\right\|_{L^{\infty}(\Omega)}=\frac{1}{\sqrt{k}}\left\|\left(\sqrt{a}\right)_{x}^{-1}\right\|_{L^{\infty}(\Omega)}=1.$

Hence, without loss of generality, we can suppose that the function
$a:\overline{\Omega}\rightarrow\mathbb{R}$ is such that

$\left\|\left(\sqrt{a}\right)_{x}^{-1}\right\|_{L^{\infty}(\Omega)}\leq 1.$
(18)

As a consequence of this condition, we have

$\sup_{x\in[c,1)}\frac{2\sqrt{a(x)}}{|a^{\prime}(x)|}\leq
1\quad\text{and}\quad\sup_{x\in(1,d]}\frac{2\sqrt{a(x)}}{|a^{\prime}(x)|}\leq
1.$ (19)

So, we can suppose that

$2\sqrt{a(x)}\leq|a^{\prime}(x)|,\quad\forall\,x\in[x^{\ast}_{1},x^{\ast}_{2}].$
(20)

Since $a(\cdot)$ is a monotonically decreasing function on $(x^{\ast}_{1},1)$
and $a(\cdot)$ is a monotonically increasing function on $(1,x^{\ast}_{2})$,
it follows from (20) that

$a^{\prime}(x)\leq-2\sqrt{a(x)},\quad\forall\,x\in[x^{\ast}_{1},1)\quad\text{and}\quad
a^{\prime}(x)\geq 2\sqrt{a(x)},\quad\forall\,x\in(1,x^{\ast}_{2}].$

Then, after integration, we obtain

$\displaystyle\int_{x}^{1}\frac{a^{\prime}(s)}{\sqrt{a(s)}}\,ds\leq-2(1-x),\quad\forall\,x\in[x^{\ast}_{1},1),$
(21) $\displaystyle\int_{1}^{x}\frac{a^{\prime}(s)}{\sqrt{a(s)}}\,ds\geq
2(x-1),\quad\forall\,x\in(1,x^{\ast}_{2}].$ (22)

Taking into account that $a(1)=0$, we deduce from (21)–(22) that

$\sqrt{a(x)}\geq
1-x,\quad\forall\,x\in[x^{\ast}_{1},1)\quad\text{and}\quad\sqrt{a(x)}\geq
x-1,\quad\forall\,x\in(1,x^{\ast}_{2}],$

and, as a consequence of these inequalities, we have

$a(x)\geq(x-1)^{2},\quad\forall\,x\in[x^{\ast}_{1},x^{\ast}_{2}].$ (23)

Utilizing the monotonicity property of $a(\cdot)$ around the point $1$ and the
fact that $\left(\sqrt{a}\right)_{x}\not\in L^{\infty}(\Omega)$, we deduce
from (23) that there exists a positive value $\gamma\in(0,2)$ such that

$a(x)=\mathcal{O}\left(|x-1|^{2-\gamma}\right)\quad\text{ in
$[x^{\ast}_{1},x^{\ast}_{2}]$},$ (24)

that is, $a(x)\thicksim|x-1|^{2-\gamma}$ near the degeneration point $1$.
Therefore, in view of representation (9), we finally have

$\displaystyle\mu_{1,a}$
$\displaystyle:=\sup_{x\in[x^{\ast}_{1},1)}\frac{|x-1||a^{\prime}(x)|}{a(x)}=2-\gamma<2,$
(25) $\displaystyle\mu_{2,a}$
$\displaystyle:=\sup_{x\in(1,x^{\ast}_{2}]}\frac{|x-1||a^{\prime}(x)|}{a(x)}=2-\gamma<2.$
(26)

∎

Let $V^{1}_{a,0}(\Omega)$ be some intermediate space with
$H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. Our next intention is to show that due to the properties
(i)–(iii) of the weight function $a:\overline{\Omega}\rightarrow\mathbb{R}$,
$V^{1}_{a,0}(\Omega)$ is a Hilbert space with respect to the scalar product

$\left<u,v\right>_{H^{1}_{a,0}(\Omega)}=\int_{\Omega}a(x)u^{\prime}(x)v^{\prime}(x)\,dx,\quad\forall\,u,v\in
V^{1}_{a,0}(\Omega).$ (27)

To do so, it is enough to establish some version of Friedrichs’s inequality.
Following in many aspects [1, 19], we will do it in two different manners (see
Lemmas 3 and 4 below)

######  Lemma 3.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given function with
properties (i)–(iii). Let $V^{1}_{a,0}(\Omega)$ be some intermediate space
with $H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. Let $u$ be an arbitrary element of
$V^{1}_{a,0}(\Omega)$. Then the following inequality

$\|u\|_{L^{2}(\Omega)}\leq
D_{a}\|u^{\prime}\|_{L^{2}(\Omega,a\,dx)}=D_{a}\left(\int_{\Omega}a(x)u_{x}^{2}(x)\,dx\right)^{1/2}$
(28)

holds true for all $u\in V^{1}_{a,0}(\Omega)$, where

$\displaystyle D^{2}_{a}$ $\displaystyle=\max\\{D_{1,a},D_{2,a}\\},$ (29)
$\displaystyle D_{1,a}$
$\displaystyle=\frac{\left(x_{1}^{\ast}-c\right)\left(2-x_{1}^{\ast}-c\right)}{2\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)}+\frac{\left(1-x_{1}^{\ast}\right)^{2}}{a(x_{1}^{\ast})\left(2-\mu_{1,a}\right)},$
(30) $\displaystyle D_{2,a}$
$\displaystyle=\frac{\left(d-x_{2}^{\ast}\right)\left(d+x_{2}^{\ast}-2\right)}{2\min\limits_{x\in[x_{2}^{\ast},d]}a(x)}+\frac{\left(x_{2}^{\ast}-1\right)^{2}}{a(x_{2}^{\ast})\left(2-\mu_{2,a}\right)}.$
(31)

###### Proof.

Let $u$ be a given element of $V^{1}_{a,0}(\Omega)$. Then, for any
$x\in[c,1)$, we have

$\displaystyle|u(x)|=\left|\int_{c}^{x}u^{\prime}(s)\,ds\right|$
$\displaystyle=\left|\int_{c}^{x}\sqrt{a(s)}u^{\prime}(s)\frac{1}{\sqrt{a(s)}}\,ds\right|\leq\|u^{\prime}\|_{L^{2}(\Omega,a\,dx)}\left(\int_{c}^{x}\frac{ds}{a(s)}\right)^{\frac{1}{2}},$

where $L^{2}(\Omega,a\,dx)$ stands for the weighted Lebesgue space endowed
with the norm
$\|f\|_{L^{2}(\Omega,a\,dx)}=\left(\int_{\Omega}f^{2}a\,dx\right)^{1/2}$.

From this and estimate (15), by Fubini’s theorem, we obtain

$\displaystyle\|$ $\displaystyle
u\|^{2}_{L^{2}(\Omega_{1})}\leq\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}\int_{c}^{1}\int_{c}^{x}\frac{ds}{a(s)}\,dx=\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}\int_{c}^{1}\int_{s}^{1}\,dx\frac{ds}{a(s)}$
$\displaystyle=\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}\left[\int_{c}^{x_{1}^{\ast}}\frac{1-s}{a(s)}\,ds+\int_{x_{1}^{\ast}}^{1}\frac{1-s}{a(s)}\,ds\right]$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1d}}}}{{\leq}}\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}\left[\frac{1}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)}\int_{c}^{x_{1}^{\ast}}(1-s)\,ds+\frac{(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})}\int_{x_{1}^{\ast}}^{1}(1-s)^{1-\mu_{1,a}}\,ds\right]$
$\displaystyle=\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}\left[\frac{\left(x_{1}^{\ast}-c\right)\left(2-x_{1}^{\ast}-c\right)}{2\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)}+\frac{\left(1-x_{1}^{\ast}\right)^{2}}{a(x_{1}^{\ast})\left(2-\mu_{1,a}\right)}\right].$
(32)

Arguing in a similar manner, for any $x\in(1,d]$, we have

$|u(x)|=\left|\int_{x}^{d}u^{\prime}(s)\,ds\right|\leq\|u^{\prime}\|_{L^{2}(\Omega,a\,dx)}\left(\int_{x}^{d}\frac{ds}{a(s)}\right)^{\frac{1}{2}}.$

Then, estimate (16) and Fubini’s theorem lead us to the following chain of
estimates

$\displaystyle\|$ $\displaystyle
u\|^{2}_{L^{2}(\Omega_{2})}\leq\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}\int_{1}^{d}\int_{x}^{d}\frac{ds}{a(s)}\,dx=\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}\int_{1}^{d}\int_{1}^{s}\,dx\frac{ds}{a(s)}$
$\displaystyle=\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}\left[\int_{1}^{x_{2}^{\ast}}\frac{s-1}{a(s)}\,ds+\int_{x_{2}^{\ast}}^{d}\frac{s-1}{a(s)}\,ds\right]$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1e}}}}{{\leq}}\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}\left[\frac{1}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)}\int_{x_{2}^{\ast}}^{d}(s-1)\,ds+\frac{(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\int_{1}^{x_{2}^{\ast}}(s-1)^{1-\mu_{2,a}}\,ds\right]$
$\displaystyle\leq\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}\left[\frac{\left(d-x_{2}^{\ast}\right)\left(d+x_{2}^{\ast}-2\right)}{2\min\limits_{x\in[x_{2}^{\ast},d]}a(x)}+\frac{\left(x_{2}^{\ast}-1\right)^{2}}{a(x_{2}^{\ast})\left(2-\mu_{2,a}\right)}\right].$
(33)

It remains to notice that due to Proposition 2, the obtained estimates (32)
and (33) are consistent. ∎

######  Lemma 4.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given function with
properties (i)–(iii). Let $V^{1}_{a,0}(\Omega)$ be some intermediate space
with $H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. Then the estimate

$\displaystyle\|u\|_{L^{2}(\Omega)}\leq
C_{a}\|u^{\prime}\|_{L^{2}(\Omega,a\,dx)}=C_{a}\left(\int_{\Omega}a(x)u_{x}^{2}(x)\,dx\right)^{1/2}$
(34)

holds true for all $u\in V^{1}_{a,0}(\Omega)$ with

$C_{a}^{2}=4\max\left\\{\frac{(1-c)^{\mu_{1,a}}}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)},\frac{(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})},\frac{(d-1)^{\mu_{2,a}}}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)},\frac{(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\right\\}$
(35)

In particular, if $x_{1}^{\ast}=c$ and $x_{2}^{\ast}=d$, then (34) can be
specified as follows

$\displaystyle\|u\|^{2}_{L^{2}(\Omega)}\leq
4\max\left\\{\frac{(1-c)^{\mu_{1,a}}}{a(c)},\frac{(d-1)^{\mu_{2,a}}}{a(d)}\right\\}\|u^{\prime}\|^{2}_{L^{2}(\Omega,a\,dx)}$
(36)

###### Proof.

Let $u$ be a given element of $V^{1}_{a,0}(\Omega)$. Taking into account the
following transformation

$\displaystyle\int_{c}^{x}(1-s)u^{\prime}(s)u(s)\,ds$
$\displaystyle=\frac{1}{2}\int_{c}^{x}(1-s)\frac{d}{ds}u^{2}(s)\,dx=\frac{1}{2}(1-x)u^{2}(x)+\frac{1}{2}\int_{0}^{x}u^{2}(s)\,dx,$
(37)

which is valid for all $x\in[c,1)$, we observe that

$\displaystyle 0$
$\displaystyle\leq\int_{c}^{x}\left[(1-s)u^{\prime}(s)-\frac{1}{2}u(s)\right]^{2}\,ds$
$\displaystyle=\int_{c}^{x}\left[(1-s)^{2}\left[u^{\prime}(s)\right]^{2}+\frac{1}{4}u^{2}(s)-(1-s)u^{\prime}(s)u(s)\right]\,ds$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{2.7}}}}{{=}}\int_{c}^{x}\left[(1-s)^{2}\left[u^{\prime}(s)\right]^{2}-\frac{1}{4}u^{2}(s)\right]\,ds-\frac{1}{2}(1-x)u^{2}(x),\quad\forall\,x\in[c,1).$

From this, we deduce that

$\displaystyle\int_{c}^{x}u^{2}(s)\,ds$ $\displaystyle\leq
4\int_{c}^{x}(1-s)^{2}\left[u^{\prime}(s)\right]^{2}\,ds$
$\displaystyle=4\int_{c}^{x_{1}^{\ast}}(1-s)^{2}\left[u^{\prime}(s)\right]^{2}\,ds+4\int_{x_{1}^{\ast}}^{x}(1-s)^{2}\left[u^{\prime}(s)\right]^{2}\,ds$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.4a}}}}{{\leq}}4(1-c)^{\mu_{1,a}}\int_{c}^{x_{1}^{\ast}}\left[u^{\prime}(s)\right]^{2}\,ds+4\int_{x_{1}^{\ast}}^{x}(1-s)^{\mu_{1,a}}\left[u^{\prime}(s)\right]^{2}\,ds$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1d}}}}{{\leq}}\frac{4(1-c)^{\mu_{1,a}}}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)}\int_{c}^{x_{1}^{\ast}}a(s)\left[u^{\prime}(s)\right]^{2}\,ds+\frac{4(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})}\int_{x_{1}^{\ast}}^{x}a(s)\left[u^{\prime}(s)\right]^{2}\,ds$
$\displaystyle\leq\max\left\\{\frac{4(1-c)^{\mu_{1,a}}}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)},\frac{4(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})}\right\\}\int_{c}^{x}a(s)\left[u^{\prime}(s)\right]^{2}\,ds,\quad\forall\,x\in[x_{1}^{\ast},1).$

Taking the limit as $x\nearrow 1$ in the last relation, we arrive at the
estimate

$\|u\|^{2}_{L^{2}(\Omega_{1})}\leq\max\left\\{\frac{4(1-c)^{\mu_{1,a}}}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)},\frac{4(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})}\right\\}\|u^{\prime}\|^{2}_{L^{2}(\Omega_{1},a\,dx)}.$
(38)

By analogy with the previous case, we make use of the following transformation
which is valid for each $x\in[1,d]$.

$\int_{x}^{d}(s-1)u(s)u^{\prime}(s)\,ds=\frac{1}{2}\int_{x}^{d}(s-1)\frac{d}{ds}u^{2}(s)\,ds=\frac{1}{2}\left[-(x-1)u^{2}(x)-\int_{x}^{d}u^{2}(s)\,ds\right].$
(39)

Then

$\displaystyle 0$
$\displaystyle\leq\int_{x}^{d}\left[(s-1)u^{\prime}(s)+\frac{1}{2}u(s)\right]^{2}\,ds$
$\displaystyle=\int_{x}^{d}\left[(s-1)^{2}\left[u^{\prime}(s)\right]^{2}+\frac{1}{4}u^{2}(s)+(s-1)u(s)u^{\prime}(s)\right]\,ds$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{2.9}}}}{{=}}\int_{x}^{d}\left[(s-1)^{2}\left[u^{\prime}(s)\right]^{2}-\frac{1}{4}u^{2}(s)\right]\,ds-\frac{1}{2}(x-1)u^{2}(x).$

Since $(x-1)u(x)\geq 0$ for all $x\in[1,d]$, it follows that

$\displaystyle\int_{x}^{d}$ $\displaystyle u^{2}(s)\,ds\leq
4\int_{x}^{x_{2}^{\ast}}(s-1)^{2}\left[u^{\prime}(s)\right]^{2}\,ds+4\int_{x_{2}^{\ast}}^{d}(s-1)^{2}\left[u^{\prime}(s)\right]^{2}\,ds$
$\displaystyle\stackrel{{\scriptstyle\text{by \eqref{1.0.1e},
\eqref{1.0.4a}}}}{{\leq}}\left[\frac{4(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\int_{x}^{x_{2}^{\ast}}a(s)\left[u^{\prime}(s)\right]^{2}\,ds+\frac{4(d-1)^{\mu_{2,a}}}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)}\int_{x_{2}^{\ast}}^{d}a(s)\left[u^{\prime}(s)\right]^{2}\,ds\right]$
$\displaystyle\leq\max\left\\{\frac{4(d-1)^{\mu_{2,a}}}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)},\frac{4(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\right\\}\int_{x}^{d}a(s)\left[u^{\prime}(s)\right]^{2}\,ds,\quad\forall\,x\in(1,x_{2}^{\ast}].$

As a result, passing to the limit in the last relation as $x\searrow 1$, we
arrive at the inequality

$\|u\|^{2}_{L^{2}(\Omega_{2})}\leq\max\left\\{\frac{4(d-1)^{\mu_{2,a}}}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)},\frac{4(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\right\\}\|u^{\prime}\|^{2}_{L^{2}(\Omega_{2},a\,dx)}.$
(40)

Thus, the announced estimate (34) is a direct consequence of (38) and (40). ∎

As a direct consequence of Lemmas 3 and 4, we have the following result.

######  Theorem 5 (Friedrichs’s inequality).

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given function with
properties (i)–(iii). Let $u$ be an arbitrary element of some intermediate
space $V^{1}_{a,0}(\Omega)$. Then

$\|u\|_{L^{2}(\Omega)}\leq\min\\{D_{a},C_{a}\\}\left(\int_{\Omega}a(x)u_{x}^{2}(x)\,dx\right)^{1/2}\\\
=\min\left\\{\max\left\\{\sqrt{D_{1,a}},\sqrt{D_{2,a}}\right\\},C_{a}\right\\}\left(\int_{\Omega}a(x)u_{x}^{2}(x)\,dx\right)^{1/2},\quad\forall\,u\in
V^{1}_{a,0}(\Omega),$ (41)

where the constants $D_{1,a}$, $D_{2,a}$ and $C_{a}$ are defined in (30),
(31), and (35), respectively.

Now we can give the following conclusion.

######  Corollary 6.

Under the assumptions of Theorem 5, $V^{1}_{a,0}(\Omega)$ is a Hilbert space
with respect to the scalar product

$\left<u,v\right>_{V^{1}_{a,0}(\Omega)}=\int_{\Omega}a(x)u^{\prime}(x)v^{\prime}(x)\,dx,\quad\forall\,u,v\in
V^{1}_{a,0}(\Omega).$ (42)

In what follows, we will distinguish two possible cases for the weight
function $a:\overline{\Omega}\rightarrow\mathbb{R}$. Namely, we say that we
deal with

  * •

a weak degeneration in (4) if $a:\overline{\Omega}\rightarrow\mathbb{R}$
satisfies properties (i)–(iii) and $1/a\in L^{1}(\Omega)$;

  * •

a strong degeneration in (4) if $a:\overline{\Omega}\rightarrow\mathbb{R}$
satisfies properties (i)–(iii) and $1/a\not\in L^{1}(\Omega)$.

Starting with the weak degenerate case, we note that due to the continuous
embedding $W^{1,1}(\Omega)\hookrightarrow C(\overline{\Omega})$ and the
estimates

$\displaystyle\int_{\Omega}|y|\,dx\leq|\Omega|^{1/2}\left(\int_{\Omega}|y|^{2}\,dx\right)^{1/2}\leq\sqrt{|\Omega|}\,\|y\|_{a},$
$\displaystyle\int_{\Omega}|y_{x}|\,dx\leq\left(\int_{\Omega}|y_{x}|^{2}a\,dx\right)^{1/2}\left(\int_{\Omega}a^{-1}\,dx\right)^{1/2}\leq
C\|y\|_{a},$

we have the following result (we refer to [1, Proposition 2.5] for the
details).

######  Theorem 7.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii) and $1/a\in L^{1}(\Omega)$. Then
$H^{1}_{a}(\Omega)=W^{1}_{a}(\Omega)$, $W^{1,2}(\Omega)\hookrightarrow
W^{1}_{a}(\Omega)$, $W^{1}_{a}(\Omega)\hookrightarrow W^{1,1}(\Omega)$,
$W^{1}_{a}(\Omega)\hookrightarrow\hookrightarrow L^{1}(\Omega)$ compactly, and
$W^{1}_{a}(\Omega)$ is continuously embedded into the class of absolutely
continuous functions on $\overline{\Omega}$, so

$\displaystyle\lim_{x\nearrow 1}y(x)=\lim_{x\searrow
1}y(x),\quad|y(1)|<+\infty,\quad\forall\,y\in H^{1}_{a}(\Omega),$ (43)
$\displaystyle\lim_{x\nearrow 1}\sqrt{a(x)}y(x)=\lim_{x\searrow
1}\sqrt{a(x)}y(x)=0,\quad\forall\,y\in H^{1}_{a}(\Omega).$ (44)

In addition, if $y$ is an arbitrary element of the space

$W^{2}_{a}(\Omega):=\left\\{y\in W^{1}_{a}(\Omega)\ :\ ay_{x}\in
W^{1,2}(\Omega)\right\\},$ (45)

then the following transmission condition

$\lim_{x\nearrow 1}a(x)y_{x}(x)=\lim_{x\searrow
1}a(x)y_{x}(x)=L,\quad\text{with }\ |L|<+\infty,$ (46)

holds true.

However, the situation changes drastically if we deal with strong degeneration
in (4). Indeed, let us consider the following example. Let $c=0$, $d=2$, and

$y(x)=\left\\{\begin{array}[]{ll}|x-1|^{-\frac{1}{4}}-1,&\text{ if }\
x\in(0,1),\\\ |x-1|^{\frac{1}{2}},&\text{ if }\ x\in[1,2).\end{array}\right.$

Setting $a(x)=|x-1|^{7/4}$, we see that properties (i)–(iii) hold true.
Moreover, in this case we have $1/a\not\in L^{1}(\Omega)$. Then, in spite of
the fact that the function $y:\Omega\rightarrow\mathbb{R}$ has a discontinuity
of the second kind at $x^{0}=1$, a direct calculations show that

$\|y\|^{2}_{a}=\int_{0}^{2}\left[y^{2}(x)+a(x)y_{x}^{2}(x)\right]\,dx<+\infty$

whereas

$a(x)y_{x}(x)=\left\\{\begin{array}[]{ll}-\frac{1}{4}|x-1|^{\frac{1}{2}},&\text{
if }\ x\in(0,1),\\\ +\frac{1}{2}|x-1|^{\frac{5}{4}},&\text{ if }\
x\in[1,2).\end{array}\right.$

So, transmission conditions (46) for the given function $y$ with finite
$H^{1}_{a}$-norm can be specified as follows

$\lim_{x\nearrow 1}a(x)y_{x}(x)=\lim_{x\searrow 1}a(x)y_{x}(x)=0.$ (47)

In fact, we have the following result (see [1, Proposition 2.5] for
comparison).

######  Theorem 8.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii) and $1/a\not\in L^{1}(\Omega)$. Let $V^{1}_{a}(\Omega)$
be some intermediate space with $H^{1}_{a}(\Omega)\subseteq
V^{1}_{a}(\Omega)\subseteq W^{1}_{a}(\Omega)$. Then the following assertions
hold true:

$\displaystyle\lim_{x\nearrow 1}|x-1|y^{2}(x)=\lim_{x\searrow
1}|x-1|y^{2}(x)=0,\quad\forall\,y\in V^{1}_{a}(\Omega),$ (48)
$\displaystyle\exists\,x_{i}\in\Omega_{i},\ i=1,2,\text{ such that }\
y(x)=o\left(|x-1|^{-\frac{1}{2}}\right)\ \text{for a.a. }\ x\in(x_{1},x_{2}),$
(49) $\displaystyle\lim_{x\nearrow 1}a(x)y_{x}(x)=\lim_{x\searrow
1}a(x)y_{x}(x)=0,\quad\forall\,y\in V^{2}_{a}(\Omega),$ (50)
$\displaystyle\lim_{x\nearrow 1}|x-1|a(x)y_{x}(x)^{2}=\lim_{x\searrow
1}|x-1|a(x)y_{x}(x)^{2}=0,\quad\forall\,y\in V^{2}_{a}(\Omega),$ (51)
$\displaystyle\lim_{x\nearrow 1}a(x)\varphi_{x}(x)y(x)=\lim_{x\searrow
1}a(x)\varphi_{x}(x)y(x)=0,\quad\forall\,y\in V^{1}_{a}(\Omega),\
\forall\,\varphi\in V^{2}_{a}(\Omega),$ (52)

where the small symbol $o$ stands for the Bachmann-Landau asymptotic notation,
and

$V^{2}_{a}(\Omega):=\left\\{y\in V^{1}_{a}(\Omega)\ :\ ay_{x}\in
W^{1,2}(\Omega)\right\\}.$

###### Proof.

Let $y\in V^{1}_{a}(\Omega)$. Without loss of generality, we assume (for the
simplicity) that $x_{1}^{\ast}=c$ and $x_{2}^{\ast}=d$. Let us show that the
function

$v(x)=\left\\{\begin{array}[]{ll}(1-x)y^{2}(x),&c\leq x<1,\\\ 0,&x=1,\\\
(x-1)y^{2}(x),&1<x\leq d\end{array}\right.$

is continuous on $\overline{\Omega}$. Indeed, $v$ is locally absolutely
continuous in $\overline{\Omega}\setminus\\{1\\}$ and

$v_{x}=\mathrm{sign}\,(x-1)\,y^{2}(x)+2|x-1|y(x)y_{x}(x),\quad\text{a.e. in }\
\overline{\Omega}.$

Since $y\in L^{2}(\Omega)$ and

$\displaystyle\int_{\Omega}|x-1|^{2}$ $\displaystyle
y^{2}_{x}(x)\,dx\stackrel{{\scriptstyle\text{by
\eqref{1.0.4a}}}}{{\leq}}\int_{\Omega_{1}}|x-1|^{\mu_{1,a}}y^{2}_{x}(x)\,dx+\int_{\Omega_{2}}|x-1|^{\mu_{2,a}}y^{2}_{x}(x)\,dx$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1d}--\eqref{1.0.1e}}}}{{\leq}}\frac{(1-c)^{\mu_{1,a}}}{a(c)}\int_{\Omega_{1}}a(x)y^{2}_{x}(x)\,dx+\frac{(d-1)^{\mu_{2,a}}}{a(d)}\int_{\Omega_{2}}a(x)y^{2}_{x}(x)\,dx$
$\displaystyle\leq\max\left\\{\frac{(1-c)^{\mu_{1,a}}}{a(c)},\frac{(d-1)^{\mu_{2,a}}}{a(d)}\right\\}\|y\|^{2}_{a},$
(53)

it follows that $v_{x}\in L^{1}(\Omega)$. Hence, $v$ is an absolutely
continuous functions and, as a consequence, the limits $\lim_{x\nearrow
1}|x-1|y^{2}(x)=\lim_{x\searrow 1}|x-1|y^{2}(x)=L$ do exist and must vanish,
for otherwise $y^{2}(x)\sim L/|x-1|$ (near the point $x_{0}=1$) would be not
integrable. So, we come into conflict with the initial condition: $y\in
L^{2}(\Omega)$. From this and the fact that $a(x)=O(|x-1|)$ in some
neighborhood of $x=1$, we immediately deduce properties (48)–(49).

To prove the equality (50), it is enough to observe that the function
$a(x)y_{x}(x)$ with $y\in V^{2}_{a}(\Omega)$ is absolutely continuous. Hence,
the limits $\lim_{x\nearrow 1}a(x)y_{x}(x)=\lim_{x\searrow 1}a(x)y_{x}(x)=L$
do exist and must vanish, for otherwise $a(x)y_{x}(x)^{2}\sim L^{2}/a(x)$
(near the point $x_{0}=1$) would be not integrable. So, we come into conflict
with the initial condition: $y\in V^{1}_{a}(\Omega)$.

To establish property (51), we set

$v(x)=\left\\{\begin{array}[]{ll}(1-x)a(x)y^{2}_{x}(x),&c\leq x<1,\\\
0,&x=1,\\\ (x-1)a(x)y^{2}_{x}(x),&1<x\leq d,\end{array}\right.$

where $y$ is an arbitrary element of $V^{2}_{a}(\Omega)$. Then $v(x)$ is
continuous on $\Omega$. Indeed, $v$ is locally absolutely continuous in
$\overline{\Omega}\setminus\\{1\\}$ and

$\displaystyle v_{x}(x)$
$\displaystyle=\mathrm{sign}\,(x-1)\,a(x)y^{2}_{x}(x)+2|x-1|y_{x}(x)\left(a(x)y_{x}(x)\right)_{x}$
$\displaystyle\quad-|x-1|a_{x}(x)y^{2}_{x}(x)=I_{1}(x)+I_{2}(x)+I_{3}(x),\quad\text{a.e.
in }\ \overline{\Omega}.$

Since $y\in V^{1}_{a}(\Omega)$, it follows that $I_{1}\in L^{1}(\Omega)$. The
same conclusion is true for the second term $I_{2}$. Indeed, in view of
estimate (53), we have

$\displaystyle\|I_{2}\|_{L^{1}(\Omega)}$ $\displaystyle\leq
2\left(\int_{\Omega}|x-1|^{2}y_{x}(x)^{2}\,dx\right)^{\frac{1}{2}}\left(\int_{\Omega}\left(a(x)y_{x}(x)\right)_{x}^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.1.2}}}}{{\leq}}2\sqrt{\max\left\\{\frac{(1-c)^{\mu_{1,a}}}{a(c)},\frac{(d-1)^{\mu_{2,a}}}{a(d)}\right\\}}\|y\|_{a}\|(ay_{x})_{x}\|_{L^{2}(\Omega)}<+\infty.$

As for the third term, we see that

$\displaystyle\|I_{3}\|_{L^{1}(\Omega)}$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1d}--\eqref{1.0.1e}}}}{{\leq}}{\mu_{1,a}}\int_{\Omega_{1}}a(x)y^{2}_{x}(x)\,dx+{\mu_{2,a}}\int_{\Omega_{2}}a(x)y^{2}_{x}(x)\,dx$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.4a}}}}{{\leq}}2\int_{\Omega}a(x)y^{2}_{x}(x)\,dx\leq
2\|y\|^{2}_{a}<+\infty.$

So, $v(x)$ is absolutely continuous in $\overline{\Omega}$. As a consequence,
we see that the limits $\lim_{x\nearrow
1}|x-1|a(x)y_{x}(x)^{2}=\lim_{x\searrow 1}|x-1|a(x)y_{x}(x)^{2}=L$ do exist
and must vanish, for otherwise $a(x)y_{x}(x)^{2}\sim L/|x-1|$ (near the point
$x_{0}=1$) would be not integrable.

It remains to prove relation (52). We do it by proving that the function

$v(x)=\left\\{\begin{array}[]{ll}a(x)\varphi_{x}(x)y(x),&c\leq x<1,\\\
0,&x=1,\\\ a(x)\varphi_{x}(x)y(x),&1<x\leq d\end{array}\right.$

is continuous on $\overline{\Omega}$. This follows by the arguments as above,
because

$\displaystyle\|v_{x}\|_{L^{1}(\Omega)}$
$\displaystyle\leq\int_{\Omega}|\sqrt{a}y_{x}||\sqrt{a}\varphi_{x}|\,dx+\int_{\Omega}|y||(a\varphi_{x})_{x}|\,dx$
$\displaystyle\leq\|y\|_{a}\|\varphi\|_{a}+\|y\|_{L^{2}(\Omega)}\|(a\varphi_{x})_{x}\|_{L^{2}(\Omega)}<+\infty,$

and, therefore, $v$ is absolutely continuous in $\overline{\Omega}$. Thus, we
see that the limits

$\lim_{x\nearrow 1}a(x)\varphi_{x}(x)y(x)=\lim_{x\searrow
1}a(x)\varphi_{x}(x)y(x)=L$ (54)

do exist. To conclude the proof, we show that $L=0$. Indeed, in view of the
property (50), we have

$a(x)|\varphi_{x}(x)|=\left|\int_{1}^{x}\left(a\varphi_{x}\right)_{x}\,dx\right|\leq\sqrt{|x-1|}\|(a\varphi_{x})_{x}\|_{L^{2}(\Omega)},\
\forall\,x\in\Omega_{0},\ \forall\,\varphi\in V^{2}_{a}(\Omega).$ (55)

Hence, if we assume that $L\neq 0$, then, in a small neighborhood
$\mathcal{U}(1)$ of $x=1$, for any functions $y\in V^{1}_{a}(\Omega)$ and
$\varphi\in V^{2}_{a}(\Omega)$, we have

$\displaystyle\frac{L}{2}\stackrel{{\scriptstyle\text{by
\eqref{A.1}}}}{{\leq}}a(x)|\varphi_{x}(x)||y(x)|$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{A.2}}}}{{\leq}}\sqrt{|x-1|}|y(x)|\|(a\varphi_{x})_{x}\|_{L^{2}(\Omega)}$
$\displaystyle=\frac{\mathrm{Const}}{2}\,\sqrt{|x-1|}|y(x)|,\
\forall\,x\in\mathcal{U}(1).$

From this we deduce that

$\frac{L}{\sqrt{|x-1|}}\leq\mathrm{Const}\,|y(x)|,\
\forall\,x\in\mathcal{U}(1).$

However, since $y$ is an $L^{2}(\Omega)$-function, this relations becomes
inconsistent. Thus, $L=0$. ∎

The main technical difficulty related to the problem (4)–(6) comes from the
degeneration effect at the point $x_{0}=1$. Therefore, taking now into account
Theorems 7 and 8, we specify the original initial-boundary value problem
(4)–(6) in the form of the following transmission problem:

$y_{tt}-\left(a(x)y_{x}\right)_{x}=0\quad\text{in }\ (0,T)\times(c,1)\
\text{and }\ (0,T)\times(1,d),$ (56)

with the initial conditions

$y(0,\cdot)=y_{0},\quad y_{t}(0,\cdot)=y_{1}\quad\text{ in }\ \Omega,$ (57)

the boundary conditions

$y(t,c)=f_{c}(t),\quad y(t,d)=f_{d}(t)\quad\text{on }\ (0,T),$ (58)

and the transmission conditions:

  1. (I)

For the case $1/a\in L^{1}(\Omega)$

$\displaystyle\lim_{x\nearrow 1}y(t,x)=\lim_{x\searrow 1}y(t,x),\quad 0<t<T,$
(59) $\displaystyle\lim_{x\nearrow 1}a(x)y_{x}(t,x)=\lim_{x\searrow
1}a(x)y_{x}(t,x),\quad 0<t<T;$ (60)

  2. (II)

For the case $1/a\not\in L^{1}(\Omega)$

$\displaystyle\lim_{x\nearrow 1}a(x)\varphi_{x}(x)y(t,x)=0=\lim_{x\searrow
1}a(x)\varphi_{x}(x)y(t,x),\ \forall\,\varphi\in V^{1}_{a}(\Omega),\ 0<t<T,$
(61) $\displaystyle\lim_{x\nearrow 1}a(x)y_{x}(t,x)=0=\lim_{x\searrow
1}a(x)y_{x}(t,x),\quad 0<t<T.$ (62)

Since transmission conditions (60)–(62) were substantiated in Theorems 7 and 8
if only $y(t,\cdot)\in V^{2}_{a}(\Omega)$ and $\varphi\in V^{2}_{a}(\Omega)$
(which mainly corresponds to the case of classical solutions), it is
reasonable to consider the transmission problems (56)–(62) as a relaxed
version of the original problem (4)–(6).

######  Remark 2.1.

It is clear that the proposed relaxation is only a matter of regularity of
some solutions. We refer to the recent papers [2, 3], where the authors
consider a particular case of the problem (4)–(6) with
$a(x)=\mathrm{const}\,|x-1|^{\alpha}$ for $\alpha\in[1,2)$, and they show that
this problems is ill-posed and admits many solutions, but only one of them
satisfies transmission conditions (59)–(60) and has a continuously
differentiable flux at $x=1$. As for the rest ones, they satisfy transmission
conditions in the form (61)–(62). So, the passage to the relaxed version does
not change the original problem (4)–(6).

##  3 On well-posedness of the degenerate transmission problems

In this section we recall the main results of semi-group theory concerning
weak and classical notions of solutions for differential operator equation.
Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a given function with
properties (i)–(iii). Let $V^{1}_{a,0}(\Omega)$ be some intermediate space
with $H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. Such space can be constructed as the
$\|\cdot\|_{a}$-closure of a linear span of $H^{1}_{a,0}(\Omega)$ with any
element $y^{\ast}\in W^{1}_{a,0}(\Omega)\setminus H^{1}_{a,0}(\Omega)$. We
associate with it the Hilbert space
$\mathcal{H}_{a}:=V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$ and endow it with
the scalar product (see [1] for comparison)

$\left<\left[u\atop
v\right],\left[\widetilde{u}\atop\widetilde{v}\right]\right>_{\mathcal{H}_{a}}=\int_{\Omega}v(x)\widetilde{v}(x)\,dx+\int_{\Omega}a(x)u_{x}(x)\widetilde{u}_{x}(x)\,dx.$

We define the unbounded operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}_{a}\rightarrow\mathcal{H}_{a}$,
associated with the problem (56)–(62) provided $f_{c}(t),f_{d}(t)\equiv 0$, as
follows

$\displaystyle\mathcal{A}\left[u\atop
v\right]=\left[v\atop\left(au_{x}\right)_{x}\right].$ (63)

and either

$D(\mathcal{A})=\left\\{\left[u\atop v\right]\in W^{2}_{a}(\Omega)\times
V^{1}_{a,0}(\Omega)\ :\ \begin{array}[]{c}\displaystyle\lim_{x\nearrow
1}u(x)=\lim_{x\searrow 1}u(x),\\\ \displaystyle\lim_{x\nearrow
1}a(x)u_{x}(x)=\lim_{x\searrow 1}a(x)u_{x}(x),\\\ u(d)=0\end{array}\right\\}$
(64)

if $1/a\in L^{1}(\Omega)$, or

$D(\mathcal{A})=\left\\{\left[u\atop v\right]\in W^{2}_{a}(\Omega)\times
V^{1}_{a,0}(\Omega)\ :\ \begin{array}[]{c}\displaystyle\lim_{x\nearrow
1}a\varphi_{x}u=0=\lim_{x\searrow 1}a\varphi_{x}u,\ \forall\,\varphi\in
H^{2}_{a}(\Omega),\\\ \displaystyle\lim_{x\nearrow
1}a(x)u_{x}(x)=0=\lim_{x\searrow 1}a(x)u_{x}(x),\\\
u(d)=0\end{array}\right\\}$ (65)

provided $1/a\not\in L^{1}(\Omega)$.

Arguing as in [13, Section II.2], it can be shown that $D(\mathcal{A})$ is a
dense subset of $\mathcal{H}_{a}$.

######  Lemma 9.

$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}_{a}\rightarrow\mathcal{H}_{a}$
is the generator of a contraction semi-group in $\mathcal{H}_{a}$.

###### Proof.

It is well-known that if $H$ is a Hilbert space and $B:D(B)\subset
H\rightarrow H$ is a densely defined linear operator such that both $B$ and
$B^{\ast}$ are dissipative, i.e.,

$\left<Bu,u\right>_{H}\leq
0\quad\text{and}\quad\left<u,B^{\ast}u\right>_{H}\leq 0\quad\forall\,u\in
D(B),$

then $B$ generates a strongly continuous semi-group of contraction operators
[22, p. 686]. Let us show that the operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}_{a}\rightarrow\mathcal{H}_{a}$
satisfies the above mentioned properties.

Indeed, since the inclusion $\mathcal{A}\left[u\atop
v\right]\in\mathcal{H}_{a}$ is obvious for all $\left[u\atop v\right]\in
D(\mathcal{A})$, it remains to check whether the properties

$\left<\mathcal{A}\left[u\atop v\right],\left[u\atop
v\right]\right>_{\mathcal{H}_{a}}\leq 0,\quad\text{and}\quad\left<\left[u\atop
v\right],\left(\mathcal{A}\right)^{\ast}\left[u\atop
v\right]\right>_{\mathcal{H}_{a}}\leq 0\quad\forall\,\left[u\atop v\right]\in
D(\mathcal{A})$ (66)

hold true. We do it for the case (II), $1/a\not\in L^{1}(\Omega)$, because the
case (I) can be considered in a similar manner. The first inequality in (66)
immediately follows from the definition of the set $D(\mathcal{A})$ and the
following relations

$\left<\mathcal{A}\left[u\atop v\right],\left[u\atop
v\right]\right>_{\mathcal{H}_{a}}=\left<\left[v\atop\left(au_{x}\right)_{x}\right],\left[u\atop
v\right]\right>_{\mathcal{H}_{a}}=\sum_{i=1}^{2}\int_{\Omega_{i}}\left(au_{x}\right)_{x}v\,dx+\sum_{i=1}^{2}\int_{\Omega_{i}}av_{x}u_{x}\,dx\\\
=\lim_{x\nearrow
1}\left[\int_{c}^{x}\left(au_{s}\right)_{s}v\,ds+\int_{c}^{x}av_{s}u_{s}\,ds\right]+\lim_{x\searrow
1}\left[\int_{x}^{d}\left(au_{s}\right)_{s}v\,ds+\int_{x}^{d}av_{s}u_{s}\,ds\right]\\\
=\left[\lim_{x\nearrow 1}a(x)u_{x}(x)v(x)\right]-\left[\lim_{x\searrow
1}a(x)u_{x}(x)v(x)\right]=0,\quad\text{for all }\ \left[u\atop v\right]\in
D(\mathcal{A})$ (67)

by the transmission conditions.

Taking into account the equality

$\left<\mathcal{A}\left[u\atop
v\right],\left[\widetilde{u}\atop\widetilde{v}\right]\right>_{\mathcal{H}_{a}}=\left<\left[u\atop
v\right],\mathcal{A}^{\ast}\left[\widetilde{u}\atop\widetilde{v}\right]\right>_{\mathcal{H}_{a}},\quad\left[u\atop
v\right],\left[\widetilde{u}\atop\widetilde{v}\right]\in D(\mathcal{A}),$

we see that

$\left<\mathcal{A}\left[u\atop
v\right],\left[\widetilde{u}\atop\widetilde{v}\right]\right>_{\mathcal{H}_{a}}=\left<\left[v\atop\left(au_{x}\right)_{x}\right],\left[\widetilde{u}\atop\widetilde{v}\right]\right>_{\mathcal{H}_{a}}=\sum_{i=1}^{2}\int_{\Omega_{i}}\left(au_{x}\right)_{x}\widetilde{v}\,dx+\sum_{i=1}^{2}\int_{\Omega_{i}}av_{x}\widetilde{u}_{x}\,dx\\\
=\lim_{x\nearrow
1}\left[\int_{c}^{x}\left(au_{s}\right)_{s}\widetilde{v}\,ds+\int_{c}^{x}av_{s}\widetilde{u}_{s}\,ds\right]+\lim_{x\searrow
1}\left[\int_{x}^{d}\left(au_{s}\right)_{s}\widetilde{v}\,ds+\int_{x}^{d}av_{s}\widetilde{u}_{s}\,ds\right]\\\
=\lim_{x\nearrow
1}\left[-\int_{c}^{x}au_{s}\widetilde{v}_{s}\,ds-\int_{c}^{x}v\left(a\widetilde{u}_{s}\right)_{s}\,ds\right]+\lim_{x\searrow
1}\left[-\int_{x}^{d}au_{s}\widetilde{v}_{s}\,ds-\int_{x}^{d}v\left(a\widetilde{u}_{s}\right)_{s}\,ds\right]\\\
+\left[\lim_{x\nearrow 1}a(x)u_{x}(x)\widetilde{v}(x)-\lim_{x\searrow
1}a(x)u_{x}(x)\widetilde{v}(x)\right]\\\ +\left[\lim_{x\nearrow
1}a(x)\widetilde{u}_{x}(x)v(x)-\lim_{x\searrow
1}a(x)\widetilde{u}_{x}(x)v(x)\right]\\\ \stackrel{{\scriptstyle\text{by
t.c.}}}{{=}}-\int_{\Omega}\left(a\widetilde{u}_{x}\right)_{x}v\,dx-\int_{\Omega}a\widetilde{v}_{x}u_{x}\,dx=\left<\left[u\atop
v\right],\left[-\widetilde{v}\atop-\left(a\widetilde{u}_{x}\right)_{x}\right]\right>_{\mathcal{H}_{a}}.$

Hence,
$\mathcal{A}^{\ast}\left[\widetilde{u}\atop\widetilde{v}\right]=\left[-\widetilde{v}\atop-\left(a\widetilde{u}_{x}\right)_{x}\right]$,
and arguing as in (67), we deduce that $\mathcal{A}^{\ast}$ is a dissipative
operator. Thus,
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}_{a}\rightarrow\mathcal{H}_{a}$
generates a strongly continuous semi-group of contraction operators. ∎

For further convenience, let us denote this semi-group by $e^{\mathcal{A}t}$.
Then for any $U_{0}=\left[u_{0}\atop v_{0}\right]\in\mathcal{H}_{a}$, the
representation $U(t)=e^{\mathcal{A}t}U_{0}$ gives the so-called
$V^{1}_{a}$-mild solution of the Cauchy problem

$\left\\{\begin{array}[]{rcl}\displaystyle\frac{d}{dt}U(t)&=&\mathcal{A}U(t),\quad
t>0,\\\ U(0)&=&U_{0}.\end{array}\right.$ (68)

When $U_{0}\in D(\mathcal{A})$, the solution $U(t)=e^{\mathcal{A}t}U_{0}$ is
classical in the sense that

$U(\cdot)\in C^{1}([0,\infty);\mathcal{H}_{a})\cap
C([0,\infty);D(\mathcal{A}))$

and equation (68) holds on $[0,\infty)$.

Thus, in view of the above consideration, we say that, for given $y_{0}\in
V^{1}_{a,0}(\Omega)$ and $y_{1}\in L^{2}(\Omega)$, the function

$y\in C^{1}([0,T];L^{2}(\Omega))\cap C([0,T];V^{1}_{a,0}(\Omega))$

is the $V^{1}_{a}$-mild solution of problem

$\displaystyle y_{tt}-\left(a(x)y_{x}\right)_{x}=0\quad\text{in }\
(0,T)\times\Omega_{i},\ i=1,2,$ (69) $\displaystyle y(t,c)=0,\quad
y(t,d)=0,\quad t\in(0,T),$ (70) $\displaystyle y(0,x)=y_{0}(x),\quad
y_{t}(0,x)=y_{1}(x),\quad x\in\Omega,$ (71) with the transmission conditions
(59)–(60) or (61)–(62), (72)

if $\left[y(t)\atop v(t)\right]=e^{\mathcal{A}t}\left[y_{0}\atop y_{1}\right]$
for all $t\in[0,T]$. By the aforementioned regularity result for
$e^{\mathcal{A}t}$, if

$\left[y_{0}\atop y_{1}\right]\in W^{2}_{a}(\Omega)\times
V^{1}_{a,0}(\Omega),$

then $y$ is the $V^{1}_{a}$-classical solution of (69)–(72) meaning that

$y\in C^{2}([0,T];L^{2}(\Omega))\cap C^{1}([0,T];V^{1}_{a,0}(\Omega))\cap
C([0,T];W^{2}_{a}(\Omega))$

and the equation (69) is satisfied for all $t\in[0,T]$ and a.e.
$x\in\Omega_{0}$.

The energy of a $V^{1}_{a}$-mild solution $y$ of (69)–(72) is the continuous
function defined by

$E_{y}(t)=\frac{1}{2}\int_{\Omega_{0}}\left[y^{2}_{t}(t,x)+a(x)y^{2}_{x}(t,x)\right]\,dx,\quad\forall\,t\geq
0.$

######  Proposition 10.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii), and let $y$ be a $V^{1}_{a}$-mild solution of (69)–(72).
Then

$E_{y}(t)=E_{y}(0),\quad\forall\,t\geq 0.$ (73)

###### Proof.

Suppose, first, that $y$ is a $V^{1}_{a}$-classical solution of (69)–(72).
Then, multiplying the equation by $y_{t}$ and integrating by parts, in view of
the transmission conditions (59)–(60) or (61)–(62), we obtain

$\displaystyle 0=\int_{\Omega_{0}}$ $\displaystyle
y_{t}(t,x)y_{tt}(t,x)\,dx-\sum_{i=1}^{2}\int_{\Omega_{i}}y_{t}(t,x)\left(a(x)y_{x}(t,x)\right)_{x}\,dx$
$\displaystyle=\int_{\Omega_{0}}\left[y_{t}(t,x)y_{tt}(t,x)+a(x)y_{x}(t,x)y_{xt}(t,x)\right]\,dx$
$\displaystyle\quad-\left[y_{t}(t,x)a(x)y_{x}(t,x)\right]_{x=c}^{x=1}-\left[y_{t}(t,x)a(x)y_{x}(t,x)\right]_{x=1}^{x=d}$
$\displaystyle=\frac{d}{dt}E_{y}(t)-y_{t}(t,1)\left(\lim_{x\nearrow
1}\left[a(x)y_{x}(t,x)\right]-\lim_{x\searrow
1}\left[a(x)y_{x}(t,x)\right]\right),$

where the last term vanishes because of the transmission conditions. Thus, we
conclude that the energy of the $V^{1}_{a}$-classical solution $y$ is
constant. The same conclusion can be extended to any $V^{1}_{a}$-mild solution
by approximation arguments. ∎

##  4 On Boundary Observability

For a given weight function $a:\overline{\Omega}\rightarrow\mathbb{R}$ with
properties (i)–(iii), we fix some intermediate space $V^{1}_{a,0}(\Omega)$
with $H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. We say that the system (69)–(72) is boundary
$V^{1}_{a}$-observable (via the normal derivative at $x=c$ and $x=d$) in time
$T>0$ if there exists a constant $C_{T}>0$ such that for any $y_{0}\in
V^{1}_{a,0}(\Omega)$ and $y_{1}\in L^{2}(\Omega)$ the $V^{1}_{a}$-mild
solution of (69)–(72) satisfies the estimate

$\int_{0}^{T}y_{x}^{2}(t,c)\,dt+\int_{0}^{T}y_{x}^{2}(t,d)\,dt\geq
C_{T}\,E_{y}(0).$ (74)

Any constant satisfying (74) is called an $V^{1}_{a}$-observability constant
for (69)–(72) in time $T$. We denoted the supremum of all
$V^{1}_{a}$-observability constants for (69)–(72) by $C_{T}$.

######  Lemma 11.

For any $V^{1}_{a}$-mild solution $y(t,x)$ of (69)–(72), we have that
$y_{x}(\cdot,c)\in L^{2}(0,T)$ and $y_{x}(\cdot,d)\in L^{2}(0,T)$ for any
$T>0$, and

$\displaystyle
a(c)\int_{0}^{T}y_{x}^{2}(t,c)\,dt\leq\frac{1}{1-c}\left[\max\\{1,C_{a}^{2}\\}+2T+2T\max\limits_{i=1,2}\left\\{\kappa_{i,a}\mu_{i,a}\right\\}\right]E_{y}(0),$
(75) $\displaystyle
a(d)\int_{0}^{T}y_{x}^{2}(t,d)\,dt\leq\frac{1}{d-1}\left[\max\\{1,C_{a}^{2}\\}+2T+2T\max\limits_{i=1,2}\left\\{\kappa_{i,a}\mu_{i,a}\right\\}\right]E_{y}(0),$
(76)

where the constant $C_{a}$ is defined by (35). Moreover,

$(1-c)a(c)\int_{0}^{T}y^{2}_{x}(t,c)\,dt+(d-1)a(d)\int_{0}^{T}y^{2}_{x}(t,d)\,dt\\\
=2\int_{\Omega_{0}}\left[(x-1)y_{x}(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx\\\
+\int_{0}^{T}\int_{\Omega_{0}}\left(y_{t}^{2}(t,x)+\left[1-\frac{(x-1)a_{x}(x)}{a(x)}\right]a(x)y^{2}_{x}(t,x)\right)\,dx\,dt.$
(77)

###### Proof.

To begin with, we assume that $\left[y_{0}\atop y_{1}\right]\in
W^{2}_{a}(\Omega)\times V^{1}_{a,0}(\Omega)$, that is, $y$ given by the
formula $\left[y(t)\atop v(t)\right]=e^{\mathcal{A}t}\left[y_{0}\atop
y_{1}\right]$ is a $V^{1}_{a}$-classical solution of the problem (69)–(72).
Following in many aspects [1, Lemma 3.2], we multiply equation (69) by
$(x-1)y_{x}$. Integrating over $(0,T)\times\Omega_{0}$, we obtain

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}\int_{\Omega_{0}}(x-1)y_{x}(t,x)\left(y_{tt}(t,x)-\left(a(x)y_{x}(t,x)\right)_{x}\right)\,dx\,dt$
$\displaystyle=\int_{\Omega_{0}}\left[(x-1)y_{x}(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx-\int_{0}^{T}\int_{\Omega_{0}}(x-1)y_{tx}(t,x)y_{t}(t,x)\,dx\,dt$
$\displaystyle\quad-\int_{0}^{T}\left[(x-1)a(x)y^{2}_{x}(t,x)\right]^{x=d}_{x=c}\,dt+\int_{0}^{T}\int_{\Omega_{0}}\Big{(}a(x)y_{x}+(x-1)a(x)y_{xx}\Big{)}y_{x}\,dx\,dt$
$\displaystyle=\int_{\Omega_{0}}\left[(x-1)y_{x}(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx-\int_{0}^{T}\left[(x-1)a(x)y^{2}_{x}(t,x)\right]^{x=d}_{x=c}\,dt$
$\displaystyle\quad-\int_{0}^{T}\int_{\Omega_{0}}\left((x-1)\left[\frac{y^{2}_{t}}{2}\right]_{x}-(x-1)a(x)\left[\frac{y^{2}_{x}}{2}\right]_{x}-a(x)y_{x}^{2}\right)\,dx\,dt$
(78)

After integration of the last two term, we have

$\displaystyle\int_{0}^{T}\int_{\Omega_{0}}$
$\displaystyle(x-1)\left[\frac{y^{2}_{t}(t,x)}{2}\right]_{x}\,dx\,dt=-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{0}}y^{2}_{t}(t,x)\,dx\,dt$
$\displaystyle+\frac{1}{2}\int_{0}^{T}\left[(x-1)y^{2}_{t}(t,x)\right]_{x=c}^{x=1}\,dt+\frac{1}{2}\int_{0}^{T}\left[(x-1)y^{2}_{t}(t,x)\right]_{x=1}^{x=d}\,dt$
$\displaystyle\stackrel{{\scriptstyle\text{by \eqref{1.1.1a},
\eqref{2.5.b}}}}{{=}}-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{0}}y^{2}_{t}(t,x)\,dx\,dt,$
(79) $\displaystyle\int_{0}^{T}\int_{\Omega_{0}}$
$\displaystyle(x-1)a(x)\left[\frac{y^{2}_{x}(t,x)}{2}\right]_{x}\,dx\,dt=-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{0}}\left[(x-1)a(x)\right]_{x}y^{2}_{x}(t,x)\,dx\,dt$
$\displaystyle\quad+\frac{1}{2}\int_{0}^{T}\left[(x-1)a(x)y^{2}_{x}(t,x)\right]_{x=c}^{x=1}\,dt+\frac{1}{2}\int_{0}^{T}\left[(x-1)a(x)y^{2}_{x}(t,x)\right]_{x=1}^{x=d}\,dt$
$\displaystyle\stackrel{{\scriptstyle\text{by \eqref{1.1.1e},
\eqref{2.5.b}}}}{{=}}\frac{(1-c)a(c)}{2}\int_{0}^{T}y^{2}_{x}(t,c)\,dt+\frac{(d-1)a(d)}{2}\int_{0}^{T}y^{2}_{x}(t,d)\,dt$
$\displaystyle\quad-\frac{1}{2}\int_{0}^{T}\int_{\Omega_{0}}\left[(x-1)a(x)\right]_{x}y^{2}_{x}(t,x)\,dx\,dt.$
(80)

As a result, the identity (77) follows by inserting (79) and (80) into (78).
To deduce the estimate (75), it is enough to notice that

$\displaystyle\left|\int_{\Omega_{0}}(x-1)y_{x}(t,x)y_{t}(t,x)\,dx\right|\leq\frac{1}{2}\int_{\Omega_{0}}\left[y^{2}_{t}(t,x)+\frac{(x-1)^{2}}{a(x)}a(x)y^{2}_{x}(t,x)\right]\,dx$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1d}--\eqref{1.0.1e}}}}{{\leq}}E_{y}(0)\max\left\\{1,\frac{(1-c)^{\mu_{1,a}}}{\min\limits_{x\in[c,x_{1}^{\ast}]}a(x)},\frac{(1-x_{1}^{\ast})^{\mu_{1,a}}}{a(x_{1}^{\ast})},\frac{(d-1)^{\mu_{2,a}}}{\min\limits_{x\in[x_{2}^{\ast},d]}a(x)},\frac{(x_{2}^{\ast}-1)^{\mu_{2,a}}}{a(x_{2}^{\ast})}\right\\}$
$\displaystyle\stackrel{{\scriptstyle\text{by
\eqref{1.0.1dd}}}}{{=}}\frac{\max\\{4,C_{a}^{2}\\}}{4}E_{y}(0),$ (81)
$\displaystyle\left[1-\frac{(x-1)a_{x}(x)}{a(x)}\right]\stackrel{{\scriptstyle\text{by
\eqref{1.0.A1},
\eqref{1.0.A2}}}}{{\leq}}1+\max\left\\{\kappa_{1,a}\mu_{1,a},\kappa_{2,a}\mu_{2,a}\right\\}\quad\text{in
}\ \Omega,$

and the energy $E_{y}(t)$ is constant.

In order to extend relations (75) and (77) to the $V^{1}_{a}$-mild solution
associated with the initial data $y_{0}\in V^{1}_{a,0}(\Omega)$ and $y_{1}\in
L^{2}(\Omega)$, it suffices to approximate such data by $\left[y^{k}_{0}\atop
y^{k}_{1}\right]\in W^{2}_{a}(\Omega)\times V^{1}_{a,0}(\Omega)$ and use
estimate (75) to show that the normal derivatives of the corresponding
classical solutions give a Cauchy sequence in $L^{2}(0,T)$. ∎

######  Lemma 12.

For any $V^{1}_{a}$-mild solution $y(t,x)$ of (69)–(72) we have: for each
$T>0$,

$\int_{0}^{T}\int_{\Omega_{0}}\Big{[}a(x)y^{2}_{x}(t,x)-y_{t}^{2}(t,x)\Big{]}\,dx\,dt+\int_{\Omega_{0}}\left[y(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx=0.$
(82)

###### Proof.

Let $y$ be a $V^{1}_{a}$-classical solution of (69)–(72). Then, multiplying
equation (69) by $y$ and integrating over $(0,T)\times\Omega_{0}$, we obtain

$\displaystyle 0$
$\displaystyle=\int_{0}^{T}\int_{\Omega_{0}}y(t,x)\left[y_{tt}(t,x)-\left(a(x)y_{x}(t,x)\right)_{x}\right]\,dx\,dt=\left[\int_{\Omega_{0}}y(t,x)y_{t}(t,x)\,dx\right]_{t=0}^{t=T}$
$\displaystyle\quad-\int_{0}^{T}\int_{\Omega_{0}}y_{t}^{2}(t,x)\,dx\,dt-\int_{0}^{T}\left[a(x)y_{x}(t,x)y(t,x)\right]_{x=c}^{x=1}\,dt$
$\displaystyle\quad-\int_{0}^{T}\left[a(x)y_{x}(t,x)y(t,x)\right]_{x=1}^{x=d}\,dt+\int_{0}^{T}\int_{\Omega_{0}}a(x)y^{2}_{x}(t,x)\,dx\,dt.$

Since

$\int_{0}^{T}\left[a(x)y_{x}(t,x)y(t,x)\right]_{x=c}^{x=1}\,dt+\int_{0}^{T}\left[a(x)y_{x}(t,x)y(t,x)\right]_{x=1}^{x=d}\,dt\\\
\stackrel{{\scriptstyle\text{by
\eqref{2.5.b}}}}{{=}}\int_{0}^{T}\left[\lim_{x\nearrow
1}a(x)y_{x}(t,x)y(t,x)-\lim_{x\searrow 1}a(x)y_{x}(t,x)y(t,x)\right]\,dt=0$

by the transmission conditions (59)–(60) or (61)–(62), the announced equality
(82) follows from the above identity. Then the approximation arguments allow
to extend this conclusion to $V^{1}_{a}$-mild solutions. ∎

######  Theorem 13.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii). In addition, we assume that

$\displaystyle\frac{d\ln
a(x)}{dx}\geq\frac{d\ln(1-x)^{\mu_{1,a}}}{dx},\quad\forall\,x\in[c,x_{1}^{\ast}],$
(83) $\displaystyle\frac{d\ln
a(x)}{dx}\leq\frac{d\ln(x-1)^{\mu_{2,a}}}{dx},\quad\forall\,x\in[x_{2}^{\ast},d].$
(84)

Let $y$ be a $V^{1}_{a}$-mild solution of (69)–(72). Then, for every $T>0$,
the estimate

$(1-c)a(c)\int_{0}^{T}y^{2}_{x}(t,c)\,dt+(d-1)a(d)\int_{0}^{T}y^{2}_{x}(t,d)\,dt\\\
\geq\left[\left(2-\max\\{\mu_{1,a},\mu_{2,a}\\}\right)T-\max\\{4,C_{a}^{2}\\}-\min\\{D_{a},C_{a}\\}\max\left\\{\mu_{1,a},\mu_{2,a}\right\\}\right]E_{y}(0)$
(85)

holds true with $C_{a}$ and $D_{a}$ are given by relations (35) and (29),
respectively.

###### Proof.

Since the case of mild solutions can be recovered by approximation arguments,
we restrict ourself by assumptions that $y$ is a $V^{1}_{a}$-classical
solution of the problem (69)–(72). Then adding to the right hand side of (77)
the left side of (82) multiplied by

$B_{a}:=\frac{1}{2}\max\left\\{\mu_{1,a},\mu_{2,a}\right\\},$

we obtain

$(1-c)a(c)\int_{0}^{T}y^{2}_{x}(t,c)\,dt+(d-1)a(d)\int_{0}^{T}y^{2}_{x}(t,d)\,dt\\\
=2\int_{\Omega_{0}}\left[(x-1)y_{x}(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx+B_{a}\int_{\Omega_{0}}\left[y(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx\\\
+\int_{0}^{T}\int_{\Omega_{0}}\left(1-B_{a}\right)y_{t}^{2}(t,x)\,dx\,dt\\\
+\int_{0}^{T}\int_{\Omega_{0}}\left(\left[1+B_{a}-\frac{(x-1)a_{x}(x)}{a(x)}\right]a(x)y^{2}_{x}(t,x)\right)\,dx\,dt=I_{1}+I_{2}+I_{3}+I_{4}.$

Since

$\displaystyle-\frac{(x-1)a_{x}(x)}{a(x)}\geq-\frac{|x-1||a_{x}(x)|}{a(x)}\geq-\max\\{\mu_{1,a},\mu_{2,a}\\}\quad\text{in
}\ [x_{1}^{\ast},x_{2}^{\ast}],$
$\displaystyle-\frac{(x-1)a_{x}(x)}{a(x)}\stackrel{{\scriptstyle\text{by
\eqref{4.0.1}}}}{{\geq}}-\mu_{1,a}\quad\forall\,x\in[c,x_{1}^{\ast}],$
$\displaystyle-\frac{(x-1)a_{x}(x)}{a(x)}\stackrel{{\scriptstyle\text{by
\eqref{4.0.2}}}}{{\geq}}-\mu_{2,a}\quad\forall\,x\in[x_{2}^{\ast},d],$

it follows that

$I_{3}+I_{4}\geq\left(2-\max\\{\mu_{1,a},\mu_{2,a}\\}\right)\int_{0}^{T}E_{y}(0)\,dt=\left(2-\max\\{\mu_{1,a},\mu_{2,a}\\}\right)TE_{y}(0).$

Taking into account that

$I_{1}=2\int_{\Omega_{0}}\left[(x-1)y_{x}(t,x)y_{t}(t,x)\right]_{t=0}^{t=T}\,dx\stackrel{{\scriptstyle\text{by
\eqref{4.1.5.1}}}}{{\geq}}-\max\\{4,C_{a}^{2}\\}E_{y}(0),$

and

$\displaystyle\left|\int_{\Omega_{0}}y(t,x)y_{t}(t,x)\,dx\right|$
$\displaystyle\leq\frac{1}{2}\int_{\Omega_{0}}\left(\frac{1}{\min\\{D_{a},C_{a}\\}}y^{2}(t,x)+\min\\{D_{a},C_{a}\\}y_{t}^{2}(t,x)\right)\,dx$
$\displaystyle\leq\min\\{D_{a},C_{a}\\}E_{y}(0),$

where $\min\\{D_{a},C_{a}\\}$ is Poincaré’s constant in (41), we see that

$I_{2}\geq-2B_{a}\min\\{D_{a},C_{a}\\}E_{y}(0)=-\min\\{D_{a},C_{a}\\}\max\left\\{\mu_{1,a},\mu_{2,a}\right\\}E_{y}(0).$

Thus, the announced estimate (85) is proven. ∎

Due to Theorem 13, the observability constant $C_{T}$ (see inequality (74))
for the problem (69)–(72) in time $T$ can be derived from (85). Namely,

$C_{T}=\frac{1}{\max\\{(1-c)a(c),(d-1)a(d)\\}}\\\
\times\left[\left(2-\max\\{\mu_{1,a},\mu_{2,a}\\}\right)T-\max\\{4,C_{a}^{2}\\}-\min\\{D_{a},C_{a}\\}\max\left\\{\mu_{1,a},\mu_{2,a}\right\\}\right].$

As for the minimal time $T_{a}>0$ when the system (69)–(72) becomes boundary
$V^{1}_{a}$-observable in time $T>T_{a}$, it can be defined as follows

$T_{a}:=\frac{1}{\left(2-\max\\{\mu_{1,a},\mu_{2,a}\\}\right)}\left[\max\\{4,C_{a}^{2}\\}+\min\\{D_{a},C_{a}\\}\max\left\\{\mu_{1,a},\mu_{2,a}\right\\}\right].$
(86)

######  Example 4.1.

Setting $c=0$, $d=2$, and $a(x)=|x-1|^{p}$, we see that the initial
assumptions (i)–(iii) holds true with $p\in(0,2)$, and

$x_{1}^{\ast}=0,\quad
x_{2}^{\ast}=2,\quad\text{and}\quad\mu_{1,a}=\mu_{2,a}=p.$

Then we deduce from (29) and (35) that $C_{a}^{2}=4$ and
$D_{a}^{2}=\frac{1}{2-p}$. Since

$\min\\{D_{a},C_{a}\\}=\min\left\\{\sqrt{\frac{1}{2-p}},2\right\\}=\left\\{\begin{array}[]{ll}\sqrt{\frac{1}{2-p}},&\
\text{if }\ p\in(0,\frac{7}{4}),\\\ 2,&\ \text{if }\
p\in[\frac{7}{4},2),\end{array}\right.,$

it follows from (86) that

$\displaystyle T_{a}$
$\displaystyle=\displaystyle\frac{p+4\sqrt{2-p}}{(2-p)\sqrt{2-p}},\quad\text{for
all }\ p\in\left(0,\frac{7}{4}\right),$ $\displaystyle T_{a}$
$\displaystyle=\displaystyle 2\left(\frac{2+p}{2-p}\right),\quad\text{for all
}\ p\in\left[\frac{7}{4},2\right).$

It is worth to notice here that $T_{a}\searrow 2$ as $p\searrow 0$. In this
case, the damage effect at the middle point $x=1$ disappears and $T_{a}$
coincides with the classical observability time for the wave equations on the
two connected planar strings (see [10, Section 4.4]). At the same time we
aarive at the blow up of the observability time if $p\nearrow 2$.

##  5 On Boundary Null Controllability

In this section the problem of boundary controllability of the degenerate wave
equation is studied. The control is assumed to act at the boundary points
$x=c$ and $x=d$ through the Dirichlet conditions. So, we consider the
following degenerate control system

$\displaystyle y_{tt}-\left(a(x)y_{x}\right)_{x}=0\quad\text{in }\
(0,+\infty)\times\Omega_{i},\ i=1,2,$ (87) $\displaystyle
y(t,c)=f_{c}(t),\quad y(t,d)=f_{d}(t),\quad t\in(0,+\infty),$ (88)
$\displaystyle y(0,x)=y_{0}(x),\quad y_{t}(0,x)=y_{1}(x),\quad x\in\Omega,$
(89) with the transmission conditions (59)–(60) or (61)–(62), (90)

where $f_{c},f_{d}\in L^{2}(0,T)$ are the controls.

By analogy with the previous section, for a given weight function
$a:\overline{\Omega}\rightarrow\mathbb{R}$ with properties (i)–(iii), we fix
some intermediate space $V^{1}_{a,0}(\Omega)$ with
$H^{1}_{a,0}(\Omega)\subseteq V^{1}_{a,0}(\Omega)\subseteq
W^{1}_{a,0}(\Omega)$. Let $V^{-1}_{a}(\Omega)$ be the dual space to
$V^{1}_{a,0}(\Omega)$ with respect to the pivot space $L^{2}(\Omega)$. In
order to make a precise definition of the solution to the boundary value
problem (87)–(90), where $f_{c},f_{d}\in L^{2}(0,T)$ are the controls, and
indicate its characteristic properties, we notice that by Theorem 5 (see also
(42)) the operator $A_{a}:D(A_{a})\subset L^{2}(\Omega)\rightarrow
L^{2}(\Omega)$, where $A_{a}(y)=-(ay_{x})_{x}$ and $D(A_{a})=\left\\{y\in
V^{1}_{a,0}(\Omega)\ :\ ay_{x}\in W^{1,2}(\Omega)\right\\}$, is an isomorphism
from $V^{1}_{a,0}(\Omega)$ onto $V^{-1}_{a}(\Omega)$. In particular,
$V^{-1}_{a}(\Omega)=A_{a}\left(V^{1}_{a,0}(\Omega)\right)$.

######  Definition 14.

System (87)–(90) is boundary null controllable in time $T>0$ if, for every
initial data $y_{0}\in L^{2}(\Omega)$, and $y_{1}\in V^{-1}_{a}(\Omega)$, the
set of reachable states $(y(T),y_{t}(T))$, where $y$ is a solution of
(87)–(90) with $f_{c},f_{d}\in L^{2}(0,T)$, contains the element $(0,0)$.

######  Definition 15.

System (87)–(90) is boundary exactly controllable in time $T>0$ if, for every
initial data $y_{0}\in L^{2}(\Omega)$, and $y_{1}\in V^{-1}_{a}(\Omega)$, the
set of reachable states $(y(T),y_{t}(T))$, coincides with $L^{2}(\Omega)\times
V^{-1}_{a}(\Omega)$.

######  Remark 5.1.

Arguing as in Proposition 2.2.1 in [28], and utilizing the linearity and
reversibility properties of system (87)–(90), it can be shown that this system
is exactly controllable through the boundary Dirichlet conditions at $x=c$ and
$x=d$ if and only if it is null controllable.

Following the standard approach and utilizing the transmission conditions, we
define the solution of controlled system (87)–(90) by transposition.

######  Definition 16.

Let $f_{c},f_{d}\in L^{2}(0,T)$, $y_{0}\in L^{2}(\Omega)$, and $y_{1}\in
V^{-1}_{a}(\Omega)$ be given distributions. We say that $y$ is a
$V^{1}_{a}$-solution by transposition of the problem (87)–(90) if

$y\in C^{1}\left([0,\infty);V^{-1}_{a}(\Omega)]\right)\cap
C\left([0,\infty);L^{2}(\Omega)\right)$

satisfies for all $T>0$ and all $w^{0}_{T}\in V^{1}_{a,0}(\Omega)$ and
$w^{1}_{T}\in L^{2}(\Omega)$ the following equality

$\displaystyle\left<y_{t}(T),w^{0}_{T}\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}$
$\displaystyle-\int_{\Omega}y(T)w^{1}_{T}\,dx$
$\displaystyle=\left<y_{1},w(0)\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}-\int_{\Omega}y_{0}w_{t}(0)\,dx$
$\displaystyle\quad-a(d)\int_{0}^{T}f_{d}(t)w_{x}(t,d)\,dt+a(c)\int_{0}^{T}f_{c}(t)w_{x}(t,c)\,dt,$
(91)

where $w$ is the solution of the backward homogeneous equation

$w_{tt}-(a(x)w_{x})_{x}=0\quad\text{in }\ (0,+\infty)\times\Omega_{i},\ i=1,2$
(92)

with the final conditions

$w(T)=w^{0}_{T},\quad w_{t}(T)=w^{1}_{T}\quad\text{ in }\ \Omega,$ (93)

the boundary conditions

$w(t,c)=0,\quad w(t,d)=0\quad\text{on }\ (0,T),$ (94)

and the transmission conditions:

  1. (I)

For the case $1/a\in L^{1}(\Omega)$

$\displaystyle\lim_{x\nearrow 1}w(t)=\lim_{x\searrow 1}w(t),\quad 0<t<T,$ (95)
$\displaystyle\lim_{x\nearrow 1}aw_{x}(t)=\lim_{x\searrow 1}aw_{x}(t),\quad
0<t<T;$ (96)

  2. (II)

For the case $1/a\not\in L^{1}(\Omega)$

$\displaystyle\lim_{x\nearrow 1}a\varphi_{x}w(t)=0=\lim_{x\searrow
1}a\varphi_{x}w(t),\ \forall\,\varphi\in V^{1}_{a,0}(\Omega),\ 0<t<T,$ (97)
$\displaystyle\lim_{x\nearrow 1}aw_{x}(t)=0=\lim_{x\searrow 1}aw_{x}(t),\quad
0<t<T.$ (98)

Following the results of Section 3 and making the change of variable
$u(t,x)=w(T-t,x)$, we see that the backward problem (92)–(98) admits a unique
$V_{a}^{1}$-mild solution $w\in C^{1}\left([0,T];L^{2}(\Omega)]\right)\cap
C\left([0,T];V^{1}_{a,0}(\Omega)\right)$ for each $T>0$. Moreover, arguing as
in Lemma 11, it can be shown that there exists a constant $C>0$ such that

$\int_{0}^{T}w_{x}^{2}(t,c)\,dt+\int_{0}^{T}w_{x}^{2}(t,d)\,dt\geq
C\,E_{w}(T),$ (99)

where

$E_{w}(t)=\frac{1}{2}\int_{\Omega_{0}}\left[w^{2}_{t}(t,x)+a(x)w^{2}_{x}(t,x)\right]\,dx=E_{w}(T),\quad\forall\,t\in[0,T],$

is the energy of a $V_{a}^{1}$-mild solution $w$ and it is conserved through
time. Since

$E_{w}(T)=\frac{1}{2}\left[\|w^{1}_{T}\|^{2}_{L^{2}(\Omega)}+\|w^{0}_{T}\|^{2}_{V^{1}_{a,0}(\Omega)}\right],$
(100)

it follows that a $V_{a}^{1}$-mild solution $w$ of (92)–(98) depends
continuously on the data $(w^{0}_{T},w^{1}_{T})\in V^{1}_{a,0}(\Omega)\times
L^{2}(\Omega)$, and, therefore, the right hand side of (91) defines a
continuous linear form with respect to $(w^{0}_{T},w^{1}_{T})\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$ $T>0$. Thus, a $V_{a}^{1}$-solution
$y$ by transposition of (87)–(90) is unique in
$C^{1}\left([0,\infty);V^{-1}_{a}(\Omega)]\right)\cap
C\left([0,\infty);L^{2}(\Omega)\right)$. The following theorem is a
consequence of the classical results of existence and uniquencess of solutions
of nonhomogeneous evolution equations. Full details can be found in [21] and
[29].

######  Theorem 17.

For any $f_{c},f_{d}\in L^{2}(0,T)$ and $(y_{0},y_{1})\in L^{2}(\Omega)\times
V^{-1}_{a}(\Omega)$ transmission problem (87)–(90) has a unique
$V_{a}^{1}$-solution defined by transposition

$\left(y,y_{t}\right)\in C\left([0,T];L^{2}(\Omega)\times
V^{-1}_{a}(\Omega)\right).$

Moreover, the map
$\left(y_{0},y_{1},f_{c},f_{d}\right)\mapsto\left\\{y,y_{t}\right\\}$ is
linear and there exists a constant $C(T)>0$ such that

$\|y\|_{L^{\infty}(0,T;L^{2}(\Omega))}+\|y_{t}\|_{L^{\infty}(0,T;V^{-1}_{a}(\Omega))}\\\
\leq
C(T)\left[\|y_{0}\|_{L^{2}(\Omega)}+\|y_{1}\|_{V^{-1}_{a}(\Omega)}+\|f_{c}\|_{L^{2}(0,T)}+\|f_{d}\|_{L^{2}(0,T)}\right].$

We are now in a position to prove the main result of this section.

######  Theorem 18.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function satisfying
properties (i)–(iii) and (83)–(84). Let $T_{a}$ be a value defined as in (86).
Then, for every $T>T_{a}$ and for any $(y_{0},y_{1})\in L^{2}(\Omega)\times
V^{-1}_{a}(\Omega)$, there exists a pair of controls $f_{c},f_{d}\in
L^{2}(0,T)$ such that the $V^{1}_{a}$-solution of (87)–(90) (in the sense of
transposition) satisfies condition $\left(y(T),y_{t}(T)\right)\equiv(0,0)$,
i.e. the system (87)–(90) is boundary null controllable in time $T>T_{a}$.

###### Proof.

Let $\left[y_{0}\atop y_{1}\right]\in L^{2}(\Omega)\times
V^{-1}_{a}(\Omega),\quad\left[w^{0}_{T}\atop w^{1}_{T}\right]$,
$\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$ be arbitrary pairs. Let $w$ and
$\widehat{w}$ be $V^{1}_{a}$-mild solutions of the backward problem (92)–(98)
with final conditions $\left[w^{0}_{T}\atop w^{1}_{T}\right]$ and
$\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]$, respectively. Let
us define the bilinear form $\Lambda$ on $V^{1}_{a,0}(\Omega)\times
L^{2}(\Omega)$ as follows

$\displaystyle\Lambda\left(\left[w^{0}_{T}\atop
w^{1}_{T}\right],\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\right):=a(c)\int_{0}^{T}w_{x}(t,c)\widehat{w}_{x}(t,c)\,dt+a(d)\int_{0}^{T}w_{x}(t,d)\widehat{w}_{x}(t,d)\,dt,$
$\displaystyle\forall\,\left[w^{0}_{T}\atop w^{1}_{T}\right],\
\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega).$

Then, in view of estimate (99) and representation (100), we deduce that the
bilinear form $\Lambda:\left[V^{1}_{a,0}(\Omega)\times
L^{2}(\Omega)\right]^{2}\to\mathbb{R}$ is continuous. Moreover, due to Theorem
13 and observability inequality (85), this form is coercive on
$V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$ provided $T>T_{a}$. Thus, by the
Lax-Milgram Lemma, variational problem

$\Lambda\left(\left[w^{0}_{T}\atop
w^{1}_{T}\right],\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\right)=\left<y_{1},\widehat{w}(0)\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}-\int_{\Omega}y_{0}\widehat{w}_{t}(0)\,dx,\
\forall\,\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$

has a unique solution $\left[w^{0}_{T}\atop w^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$. Then setting $f_{c}=-w_{x}(t,c)$,
$f_{d}=w_{x}(t,d)$, and $T>T_{a}$, where $w\in
C^{1}\left([0,T];L^{2}(\Omega)]\right)\cap
C\left([0,T];V^{1}_{a,0}(\Omega)\right)$ is a $V_{a}^{1}$-mild solution of the
backward problem (92)–(98) with $\left[w^{0}_{T}\atop w^{1}_{T}\right]$ as the
final data, we see that

$a(d)\int_{0}^{T}f_{d}(t)\widehat{w}_{x}(t,d)\,dt-a(c)\int_{0}^{T}f_{c}(t)\widehat{w}_{x}(t,c)\,dt\\\
=a(c)\int_{0}^{T}w_{x}(t,c)\widehat{w}_{x}(t,c)\,dt+a(d)\int_{0}^{T}w_{x}(t,d)\widehat{w}_{x}(t,d)\,dt=\Lambda\left(\left[w^{0}_{T}\atop
w^{1}_{T}\right],\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\right)\\\
=\left<y_{1},\widehat{w}(0)\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}-\int_{\Omega}y_{0}\widehat{w}_{t}(0)\,dx,\quad\forall\,\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega).$ (101)

On the other hand, if $y$ is the $V_{a}^{1}$-solution by transposition of the
problem (87)–(90), then equality (91) implies that, for all
$\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$, we have

$a(d)\int_{0}^{T}f_{d}(t)\widehat{w}_{x}(t,d)\,dt-a(c)\int_{0}^{T}f_{c}(t)\widehat{w}_{x}(t,c)\,dt=\left<y_{1},\widehat{w}(0)\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}\\\
-\int_{\Omega}y_{0}\widehat{w}_{t}(0)\,dx-\left<y_{t}(T),\widehat{w}^{0}_{T}\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}+\int_{\Omega}y(T)\widehat{w}^{1}_{T}\,dx.$
(102)

Comparing the last relations (101)–(102), we obtain

$-\left<y_{t}(T),\widehat{w}^{0}_{T}\right>_{V^{-1}_{a}(\Omega);V^{1}_{a,0}(\Omega)}+\int_{\Omega}y(T)\widehat{w}^{1}_{T}\,dx=0,\quad\forall\,\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega).$

From this we finally deduce that $\left(y(T),y_{t}(T)\right)\equiv(0,0)$, i.e.
the system (87)–(90) is boundary null controllable in time $T>T_{a}$. ∎

As an obvious consequence of this result, we can give the following its
generalization to the case when only one boundary point $x=d$ is controlled.
So, formally setting $f_{c}(t)\equiv 0$ in (88), we consider the boundary null
controllability problem for the system (87)–(90) with only one control
$f_{d}$. Then arguing as in the proof of Theorem 18, we arrive at the
following conclusion.

######  Theorem 19.

Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a weight function such that
it satisfies properties (i)–(iii) and (83)–(84). Assume that the bilinear form

$\displaystyle\Lambda\left(\left[w^{0}_{T}\atop
w^{1}_{T}\right],\left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\right):=a(d)\int_{0}^{T}w_{x}(t,d)\widehat{w}_{x}(t,d)\,dt,\quad\forall\,\left[w^{0}_{T}\atop
w^{1}_{T}\right],\ \left[\widehat{w}^{0}_{T}\atop\widehat{w}^{1}_{T}\right]\in
V^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$

is coercive on $H^{1}_{a,0}(\Omega)\times L^{2}(\Omega)$, i.e., there exists a
positive constant $C_{1}>0$ such that

$\Lambda\left(\left[w^{0}_{T}\atop w^{1}_{T}\right],\left[w^{0}_{T}\atop
w^{1}_{T}\right]\right)\geq
C_{1}\left[\|w^{0}_{T}\|^{2}_{V^{1}_{a,0}(\Omega)}+\|w^{1}_{T}\|^{2}_{L^{2}(\Omega)}\right],\
\forall\,\left[w^{0}_{T}\atop w^{1}_{T}\right]\in V^{1}_{a,0}(\Omega)\times
L^{2}(\Omega).$

Then there exists a control $f_{d}\in L^{2}(0,T)$ such that the system
(87)–(90) with $f_{c}(t)\equiv 0$ is one side boundary null controllable in
time $T$.

## Acknowledgment

The first and third author acknowledge financial support by the DFG under
contract Le595/31-1: Sustainable Optimal Controls for Nonlinear Partial
Differential Equations with Applications. The first author was also partially
supported by National Research Foundation of Ukraine (Grant No. 2020.02/0066).

## References

  * [1] F. Alabau-Boussouira, P. Cannarsa, G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (3) (2017), 2052–2087.
  * [2] V.L. Borsch, P.I. Kogut, G. Leugering, On an initial boundary-value problem for 1D hyperbolic equation with interior degeneracy: Series solutions with the continuously differentiable fluxes, Journal of Optimization, Differential Equations, and Their Applications (JODEA), 28 (1) (2020), 1–42.
  * [3] V.L. Borsch, P.I. Kogut, The exact bounded solution to an initial boundary value problem for 1D hyperbolic equation with interior degeneracy. I. Separation of variables, Journal of Optimization, Differential Equations, and Their Applications (JODEA), 28 (2) (2020), 2–20.
  * [4] P. Cannarsa, G. Florida, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Communication in Appl. and Industrial Math., 2(2)(2008), 1–16.
  * [5] P. Cannarsa, R. Ferretti, P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim.  57, (2), pp. 900–924.
  * [6] S. Čanić, E.H. Kim, A class of quasilinear degenerate elliptic problems, J. Differential Equations., 189(2003), 71–98.
  * [7] F.L. Chernous’ko, N.V. Banichuk, _Variational Problems of Mechanics and Control,_ Nauka, Moscow, 1973 (in Russian).
  * [8] V. Chiadó Piat , F. Serra Cassano, _Some remarks about the density of smooth functions in weighted Sobolev spaces_ , J. Convex Analysis, No. 2, 1(1994), 135–142.
  * [9] M. Colombo, A. Figalli, Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., 101(2014), 94–117.
  * [10] R. Dáger and E. Zuazua, Wave Progagation, Observation and Control in $1$–d Flexible Multi-Structures, Series: Mathématiques and Applications, Vol. 50, Springer-Verlag, Berlin, 2006.
  * [11] A. C. Eringen, The finite Sturm-Liouville transform, Quart. J. Math. Oxford Ser. (2) 5 (1954), 120–129.
  * [12] L.V. Fardigola, Transformation operators in control problems for a degenerate wave equation with variable coefficients, (Ukrainian) ; translated from Ukrain. Mat. Zh. 70 (2018), no. 8, 1128–1142 Ukrainian Math. J. 70 (2019), no. 8, 1300–1318
  * [13] Ch. Gaevski, K. Greger, K. Zacharias, _Nonlinear operator equations and operator differential equations_ , Mir, Moskow, 1978 (in Russian).
  * [14] M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52(2014), 2037-–2054.
  * [15] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51(2009), 3–33.
  * [16] P.I. Kogut, G. Leugering, Optimal and approximate boundary control of an elastic body with quasistatic evolution of damage, Mathematical Methods in the Applied Sciences, 38(2015), 2739–2760.
  * [17] P.I. Kogut, G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkhäuser Verlag, Boston, 2011.
  * [18] P.I. Kogut, O.P. Kupenko, G. Leugering, Well-posedness and boundary observability of strongly degenerate hyperbolic systems on star-shaped planar network, Pure and Applied Functional Analysis, accepted for publication (2020).
  * [19] P.I. Kogut, O.P. Kupenko, G. Leugering, Y. Wang, A note on weighted Sobolev spaces related to weakly and strongly degenerate differential operators, Journal of Optimization, Differential Equations, and Their Applications (JODEA), 27 (2) (2019), 1–22.
  * [20] I. Lasiecka, J. Sokolowski, _Sensitivity analysis of optimal control problems for wave equations,_ SIAM J. Control Optim., 29 (1991), 1128-–1149.
  * [21] J.-L. Lions, E. Magenes, _Problèmes aux Limites non Homogènes et Applications_ , Vol.1, Dunon, Paris, 1968.
  * [22] G. Lumer, R.S. Phillips, Dissipative operators in a Banach space, Pacific Journal of Mathematics, 11 (2) (1961), 679–698.
  * [23] G. Marinoschi, R. M. Mininni, S. Romanelli, An optimal control problem in coefficients for a strongly degenerate parabolic equation with interior degeneracy, J. Optim. Theory Appl, 173(2017), 56–77.
  * [24] V.A. Rukavishnikov, E.I. Rukavishnikova, Dirichlet problem with degeneration of the input data on the boundary of the domain, Differential Equations, 52(5)(2016), 681–685.
  * [25] Russell, David L., Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory,  J. Math. Anal. Appl. 409 (1972), 336–368.
  * [26] F. Santambrogio, V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation, Nonlinear Anal., 73(2010), 3832–3841.
  * [27] V.V. Zhikov, _On Lavrentiev phenomenon_ ,  Russian J. Math. Phys., No. 2,  3(1994), 249–269.
  * [28] E. Zuazua, _Controllability of Partial Differential Equations_ , 3rd cycle. Castro Urdiales (Espagne), 2006.
  * [29] E. Zuazua, _Controlabilidad exacta y estabilización la ecuación de ondas_ , Textos de métodos matemáticos 23, Universidad Federal do Rio de Janeiro, 1990.
  * [30] H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen, Mathematische Annalen, 68, no. 2, pp. 220–269, 1910.