On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis
In the paper, the equation of a vibrating non-homogeneous string, whose potential is not equal to a constant, is considered on a half-axis. The Neumann control of the class L∞ is considered at a point x = 0. The control problem is studied in the Sobolev spaces. The suffcient conditions for nullcontr...
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irk-123456789-1067262016-10-04T03:02:17Z On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis Khalina, K.S. In the paper, the equation of a vibrating non-homogeneous string, whose potential is not equal to a constant, is considered on a half-axis. The Neumann control of the class L∞ is considered at a point x = 0. The control problem is studied in the Sobolev spaces. The suffcient conditions for nullcontrollability and approximate null-controllability at a free time T > 0 are obtained for the given system. The controls solving these problems are found explicitly. Рассмотрено уравнение колебания неоднородной струны на полуоси с потенциалом, не равным константе. На левом конце рассмотрено управление типа Неймана из класса L∞. Задача управляемости изучена в пространствах Соболева. Для заданной системы получены достаточные условия 0-управляемости и ε-управляемости за свободное время T > 0. Управления, которые решают эти задачи, найдены в явном виде. 2012 Article On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis / K.S. Khalina // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 307-335. — Бібліогр.: 26 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106726 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the paper, the equation of a vibrating non-homogeneous string, whose potential is not equal to a constant, is considered on a half-axis. The Neumann control of the class L∞ is considered at a point x = 0. The control problem is studied in the Sobolev spaces. The suffcient conditions for nullcontrollability and approximate null-controllability at a free time T > 0 are obtained for the given system. The controls solving these problems are found explicitly. |
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Khalina, K.S. |
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Khalina, K.S. On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis Журнал математической физики, анализа, геометрии |
| author_facet |
Khalina, K.S. |
| author_sort |
Khalina, K.S. |
| title |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis |
| title_short |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis |
| title_full |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis |
| title_fullStr |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis |
| title_full_unstemmed |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis |
| title_sort |
on the neumann boundary controllability for the non-homogeneous string on a half-axis |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/106726 |
| citation_txt |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Half-Axis / K.S. Khalina // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 307-335. — Бібліогр.: 26 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khalinaks ontheneumannboundarycontrollabilityforthenonhomogeneousstringonahalfaxis |
| first_indexed |
2025-07-07T18:54:32Z |
| last_indexed |
2025-07-07T18:54:32Z |
| _version_ |
1837015470372290560 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 4, pp. 307�335
On the Neumann Boundary Controllability for the
Non-Homogeneous String on a Half-Axis
K.S. Khalina
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail: khalina@meta.ua
Received June 8, 2012, revised August 31, 2012
In the paper, the equation of a vibrating non-homogeneous string, whose
potential is not equal to a constant, is considered on a half-axis. The Neu-
mann control of the class L∞ is considered at a point x = 0. The control
problem is studied in the Sobolev spaces. The su�cient conditions for null-
controllability and approximate null-controllability at a free time T > 0 are
obtained for the given system. The controls solving these problems are found
explicitly.
Key words: wave equation, controllability problem, Neumann control,
Sobolev space, Sturm�Liouville equation, transformation operator.
Mathematics Subject Classi�cation 2010: 93B05, 35B37, 35L05, 34B24.
1. Introduction
In the paper, the controllability problems for a vibrating non-homogeneous
string on a half-axis are studied. The control system under consideration is
wtt(x, t) = wxx(x, t)− q(x)w(x, t), x ∈ (0, +∞), t ∈ (0, T ), (1.1)
wx(0, t) = u(t), t ∈ (0, T ), (1.2)
where T > 0, u ∈ L∞(0, T ) is a control, q is a potential under the conditions
q ∈ C[0,∞) ∩ L∞[0,∞),
∞∫
0
x|q(x)| dx < ∞. (1.3)
This control system is considered in the Sobolev spaces Hs
0 . A time T > 0 is not
�xed.
c© K.S. Khalina, 2012
K.S. Khalina
Controllability problems for hyperbolic partial di�erential equations were stud-
ied in a number of papers (see, e.g., [1�20]). The boundary controllability of the
wave equation on bounded domains in the context of Lp-controls (2 ≤ p ≤ ∞) is
well studied. Some results for a homogeneous string were obtained in [1�8] and
other papers. The results for a non-homogeneous string were obtained in [9�14].
It should be noted that only L∞-controls can be implemented practically.
The controllability problems for the wave equation on unbounded domains
have not been studied as extensively as on bounded domains. The boundary
controllability of the wave equation on a half-axis in the context of L∞-controls
was studied in [15�20]. In particular, the controllability for a homogeneous string
with the Dirichlet control was investigated in [15, 16], and with the Neumann
control in [17]. In [18] and [19], the controllability for a non-homogeneous string
was studied for the case when q ≡ const ≥ 0. In [18], a time T > 0 was �xed.
In [19], both cases with �xed and free time were studied. The Neumann control
was considered in [18], and the Dirichlet control was considered in [19]. In [20],
the controllability for a non-homogeneous string was studied for the case when
the potential q was not generally speaking a constant. A control system was
considered in the class of functions with bounded supports, and a time T > 0 was
�xed in [20]. The case of the Dirichlet control and the case of the Neumann control
were studied there. In papers [15�20], the control systems were considered in the
Sobolev spaces Hs
0 . The necessary and su�cient conditions for null-controllability
and approximate null-controllability were obtained. The controls solving these
problems were found explicitly.
In the present paper, unlike in [15�19], the potential q is not a constant, which
makes the studying of controllability problems more complicated. To solve these
problems, we apply the transformation operators for the Sturm�Liouville equation
that do not change a solution asymptotic at in�nity. We extend these operators
to the Sobolev spaces and prove their continuity under conditions (1.3). Notice
that in contrast to [20], in the present paper a time T > 0 is free and there are no
restrictions on the functions supports, but stronger restrictions on the potential
q are required. We prove that the application of the transformation operator to
the control system with q 6= const reduces it to the similar control system studied
in [17] with q ≡ 0. The converse is also correct: the application of the inverse
transformation operator to the control system with q ≡ 0 reduces it to the control
system with q 6= const. A one-to-one correspondence between the solutions of
these systems is proved. Moreover, the control u of the system is transformed to
the control p of the system with q ≡ 0. We also prove that if a state of the control
system with q ≡ 0 is approximately null-controllable, then a state of the control
system with q 6= const is approximately null-controllable. All the above makes
it possible to study the control system under consideration by using the results
obtained in [17].
308 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Thus, in the paper, the su�cient conditions for null-controllability and ap-
proximate null-controllability are obtained for the given control system at a free
time. There is obtained the explicit formula for the control depending on the
initial state of the given system and on the control p of the system with q ≡ 0.
It should be noticed that the su�cient conditions obtained for null-controllability
and approximate null-controllability of the system are also necessary when a time
T > 0 is �xed and a control system is considered in the class of functions with
bounded supports.
2. Notation and the Problem De�nition
Consider control system (1.1), (1.2) with the initial conditions
w(x, 0) = V0
0(x), wt(x, 0) = V0
1(x), x ∈ (0, +∞). (2.1)
The aim of the paper is to study the null-controllability and approximate null-
controllability problems for system (1.1), (1.2), (2.1), namely, to �nd the control of
the class L∞(0, T ) which transfers a semi-in�nite string from the given initial state
to the origin and to a given neighborhood of the origin at time T . In addition,
time T is free and may depend on the neighborhood.
Introduce the spaces used in the paper. Let S be the Schwartz space [21],
S = {ϕ ∈ C∞(R) : ∀m, l ∈ N∪{0} ∃Cml > 0 : ∀x ∈ R
∣∣∣ϕ(m)(x)(1 + |x|2)l
∣∣∣ ≤ Cml},
and let S′ be the dual space. Denote by Hs
l (s, l ∈ R) the Sobolev spaces [22,
Chap. 1]
Hs
l =
{
f ∈ S′ : (1 + x2)l/2(1 + |D|2)s/2f ∈ L2(R)
}
,
‖f‖s
l =
+∞∫
−∞
∣∣∣(1 + x2)l/2(1 + |D|2)s/2f(x)
∣∣∣
2
dx
1/2
,
where D = −id/dx. The norm |||f |||sl =
(
(‖f0‖s
l )
2 +
(
‖f1‖s−1
l
)2
)1/2
is used for
f =
(
f0
f1
)
∈ Hs
l × Hs−1
l . A distribution f ∈ S′ is said to be odd if (f, ϕ(x)) =
−(f, ϕ(−x)), ϕ ∈ S. A distribution f ∈ S′ is said to be even if (f, ϕ(x)) =
(f, ϕ(−x)), ϕ ∈ S.
We also use the following subspaces of the Sobolev spaces (s, l ∈ R):
Hs
l,o = {f ∈ Hs
l : f is odd}, Hs
l,e = {f ∈ Hs
l : f is even},
Hs
l,p = Hs
l,p ×Hs−1
l,p , p = o, e.
Obviously, if f ∈ Hs
l,o, then f ′ ∈ Hs−1
l,e and if f ∈ Hs
l,e, then f ′ ∈ Hs−1
l,o .
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 309
K.S. Khalina
We assume that V0 =
(
V0
0
V0
1
)
∈ H1
0,e. The solutions of system (1.1), (1.2),
(2.1) are considered in H1
0,e.
Denote by Ω : S′ → S′ and Ξ : S′ → S′, D(Ω) = D(Ξ) = S′ the operators
(Ωf)(x) = f(x)− f(−x) and (Ξf)(x) = f(x) + f(−x), f ∈ S′. Notice that these
operators coincide with the odd and the even extension operators, respectively,
for functions f ∈ S′ such that supp f ⊂ (0,∞). Assume that q is de�ned on R
and q ≡ 0 on (−∞, 0). Denote Q = Ξq, V(·, t) = Ξw(·, t), t ∈ (0, T ). Evidently,
V(·, t) ∈ H1
0,e, t ∈ (0, T ).
Let w be the solution of control problem (1.1), (1.2), (2.1). It is easy to see
that V is the solution of the problem
Vtt(x, t) = Vxx(x, t)−Q(x)V(x, t)− 2u(t)δ(x), x ∈ R, t ∈ (0, T ), (2.2)
V(x, 0) = V0
0(x), Vt(x, 0) = V0
1(x), x ∈ R. (2.3)
Consider some steering conditions for (2.2), (2.3):
V(x, T ) = VT
0 (x), Vt(x, T ) = VT
1 (x), x ∈ R, (2.4)
where VT =
(
VT
0
VT
1
)
∈ H1
0,e. Let T > 0. For a given V0 ∈ H1
0,e, denote by Re
T (V0)
a set of the states VT ∈ H1
0,e for which there exists a control u ∈ L∞(0, T ) such
that problem (2.2)�(2.4) has a unique solution in H1
0,e.
De�nition 2.1. A state V0 ∈ H1
0,e is called null-controllable with respect to
system (2.2), (2.3) if 0 belongs to
⋃
T>0Re
T (V0), and it is called approximately
null-controllable with respect to system (2.2), (2.3) if 0 belongs to the closure of⋃
T>0Re
T (V0) in H1
0,e.
To study the controllability problems for system (2.2), (2.3), we use the trans-
formation operators for the Sturm�Liouville equation that do not change a so-
lution asymptotic at in�nity. These operators were studied, e.g., in [23, Chap.
3]. In the present paper, the operators are extended to Hs
0,e, s = 1, 0 and proved
to be continuous (see Sec. 5). Determine the operators M, M−1 : H0
0,e → H0
0,e,
D(M) = D(M−1) = H0
0,e by the formulas
(Mf)(x) = f(x) +
∞∫
|x|
M(|x|, t)f(t)dt, x ∈ R, (2.5)
(M−1g)(x) = g(x) +
∞∫
|x|
N(|x|, t)g(t)dt, x ∈ R, (2.6)
310 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
where f, g ∈ H0
0,e, M(ξ, η) and N(ξ, η) are the kernels of the operators, (ξ, η) ∈
(0,∞) × (0,∞). The properties of the kernels as well as the method used to
�nd them are described at the beginning of Sec 5. In Lemma 5.2, we prove
that the operators are continuous from H0
0,e to H0
0,e, and R(M) = R(M−1) =
H0
0,e. Consider the restrictions of the operators M and M−1 to H1
0,e, D(M) =
D(M−1) = H1
0,e. In Lemma 5.3, we prove that they are continuous from H1
0,e to
H1
0,e, and R(M) = R(M−1) = H1
0,e. In Lemma 5.4, the formulas for the adjoint
operators M∗, (M−1)∗, D(M∗) = D((M−1)∗) = H0
0,e are obtained and they are
proved to be continuous from H0
0,e to H0
0,e, and R(M∗) = R((M−1)∗) = H0
0,e.
Consider the restrictions of M∗, (M−1)∗ to H1
0,e, D(M∗) = D((M−1)∗) = H1
0,e.
In Lemma 5.5, M∗, (M−1)∗ are proved to be continuous from H1
0,e to H1
0,e, and
R(M∗) = R((M−1)∗) = H1
0,e. Therefore, we can extend the operators M, M−1 to
H−1
0,e by the rule
(Mf, ψ) = (f, M∗ψ), (2.7)
(M−1g, ϕ) = (g, (M−1)∗ϕ), (2.8)
where f, g ∈ H−1
0,e , ϕ,ψ ∈ H1
0,e, D(M) = D(M−1) = H−1
0,e . In Lemma 5.5, we
establish that these operators are continuous from H−1
0,e to H−1
0,e , and R(M) =
R(M−1) = H−1
0,e .
3. Null- and Approximate Null-Controllability Conditions
Consider the auxiliary control system with Q ≡ 0,
Vtt(x, t) = Vxx(x, t)− 2p(t)δ(x), x ∈ R, t ∈ (0, T ), (3.1)
V(x, 0) = V0
0 (x), Vt(x, 0) = V0
1 (x), x ∈ R, (3.2)
with some steering conditions
V(x, T ) = VT
0 (x), Vt(x, T ) = VT
1 (x), x ∈ R, (3.3)
where V(·, t) ∈ H1
0,e, V0 =
(V0
0
V0
1
)
∈ H1
0,e, VT =
(VT
0
VT
1
)
∈ H1
0,e, p ∈ L∞(0, T ) is
a control. Let T > 0. For a given V0 ∈ H1
0,e, denote by Ze
T (V0) a set of the
states VT ∈ H1
0,e for which there exists a control p ∈ L∞(0, T ) such that problem
(3.1)�(3.3) has a unique solution in H1
0,e.
De�nition 3.1. A state V0 ∈ H1
0,e is called null-controllable with respect to
system (3.1), (3.2) if 0 belongs to
⋃
T>0Ze
T (V0), and it is called approximately
null-controllable with respect to system (3.1), (3.2) if 0 belongs to the closure of⋃
T>0Ze
T (V0) in H1
0,e.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 311
K.S. Khalina
The controllability problems for system (3.1), (3.2) were well studied in [17].
The following assertions are special cases of the results obtained in [17]:
Statement 3.1 (Fardigola, [17]). A solution of system (3.1), (3.2) is described
by the formula
(V(·, t)
Vt(·, t)
)
= E(·, t) ∗
[
V0 −
(
∂−1ΩP t
ΞP t
)]
, t ∈ (0, T ), (3.4)
where P t(x) = p(x)[H(x)−H(x− t)], ∂−1ΩP t(x) =
∫ x
−∞ΩP t(ξ)dξ,
E(x, t) =
1
2
(
δ(x + t) + δ(x− t) 1
2(sign(x + t)− sign(x− t))
δ′(x + t)− δ′(x− t) δ(x + t) + δ(x− t)
)
, x, t ∈ R.
Theorem 3.1 (Fardigola, [17]). A state V0 ∈ H1
0,e is approximately null-
controllable with respect to system (3.1), (3.2) i� the conditions below hold
V0
1 ∈ L∞(R), (3.5)
V0
1 = signx(V0
0 )′. (3.6)
Under these conditions there exists a sequence {Tn}∞n=1 such that Tn|V0
0 (Tn)|2 → 0
as n → ∞. For this sequence the controls pn(t) = V0
1 (t) a.e. on (0, Tn), n ∈ N,
solve the approximate null-controllability problem for system (3.1), (3.2).
Theorem 3.2 (Fardigola, [17]). A state V0 ∈ H1
0,e is null-controllable with
respect to system (3.1), (3.2) i� conditions (3.5), (3.6) hold and there exists T > 0
such that suppV0
1 ⊂ (−T, T ). Under these conditions the control solving the null-
controllability problem for system (3.1), (3.2) is of the form p = V0
1 a.e. on (0, T ).
We �rst prove an auxiliary lemma for system (3.1), (3.2).
Lemma 3.1. Let V(x, t) be the solution of (3.1), (3.2). Then Vx(+0, t) = p(t),
t ∈ (0, T ).
P r o o f. From (3.4) it follows that
V(x, t) =
1
2
{
V0
0 (x + t) + V0
0 (x− t) + Ṽ0
1 (x + t)− Ṽ0
1 (x− t)− (∂−1ΩP t)(x + t)
− (∂−1ΩP t)(x− t)− (∂−1ΞP t)(x + t) + (∂−1ΞP t)(x− t)
}
, (3.7)
where x ∈ R, t ∈ (0, T ), Ṽ0
1 ∈ H1
0,o such that (Ṽ0
1 )′ = V0
1 . Di�erentiating (3.7)
with respect to x, we obtain
Vx(x, t) =
1
2
{
(V0
0 )′(x + t) + (V0
0 )′(x− t) + V0
1 (x + t)− V0
1 (x− t)− 2P t(x + t)
+ 2P t(−x + t)
}
, x ∈ R, t ∈ (0, T ). (3.8)
312 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Therefore,
Vx(+0, t) =
1
2
lim
x→+0
{
(V0
0 )′(x + t) + (V0
0 )′(x− t) + V0
1 (x + t)− V0
1 (x− t)
−2P t(x + t) + 2P t(−x + t)
}
, x ∈ R, t ∈ (0, T ).
For any f ∈ L2(R) we may set limx→+0 f(x) = limx→0 f(|x|). Hence, taking into
account the supports of P t(x + t) and P t(−x + t), we obtain
Vx(+0, t) =
1
2
{
(V0
0 )′(t) + (V0
0 )′(−t) + V0
1 (t)− V0
1 (−t) + 2p(t)
}
, t ∈ (0, T ).
We remark that the values (V0
0 )′(t), V0
1 (t), (V0
0 )′(−t), and V0
1 (−t) exist a.e. on
(0, T ), whereas (V0
0 )′, V0
1 are locally integrable. Taking into account that (V0
0 )′ is
odd and V0
1 is even, we obtain the assertion of the lemma. The lemma is proved.
Theorem 3.3. Let V(x, t) be the solution of (3.1), (3.2). Let V(·, t) = MV(·, t),
t ∈ (0, T ), V0
j = MV0
j , j = 0, 1. Determine the function u by the formula
u(t) = p(t) +
∞∫
0
Mx(0, ξ)V(ξ, t)dξ − 1
2
V(0, t)
∞∫
0
q(ξ)dξ, t ∈ (0, T ), (3.9)
where V(ξ, t) is de�ned by (3.7), p is the control of system (3.1), (3.2). Then
V(x, t) is the solution of system (2.2), (2.3) with the control u determined by
(3.9).
Pr o o f. Apply the operator M to system (3.1), (3.2). Thus, conditions (2.3)
hold immediately, and equation (3.1) takes the form
MVtt(·, t) = MVxx(·, t)− 2p(t)Mδ, t ∈ (0, T ). (3.10)
Using (5.29), it is easy to get (Mδ, ψ) = (M∗ψ)(0) = ψ(0) = (δ, ψ) for any even
ψ ∈ S. Hence, Mδ = δ. Due to Lemma 5.6, equation (3.10) takes the form
d2
dt2
MV(·, t) =
d2
dx2
MV(·, t)−QMV(·, t)− 2δ
∞∫
0
Mx(0, ξ)V(ξ, t)dξ
+ δV(0, t)
∞∫
0
q(ξ)dξ − 2p(t)δ, t ∈ (0, T ).
Taking into account (3.9), we can see that the equation above is reduced to (2.2).
The theorem is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 313
K.S. Khalina
Theorem 3.4. Let V(x, t) be the solution of system (2.2), (2.3). Let also
V(·, t) = M−1V(·, t), t ∈ (0, T ), V0
j = M−1V0
j , j = 0, 1. Suppose that the function
p is connected with the control u of system (2.2), (2.3) by the following formula:
p(t) = u(t) +
∞∫
0
Nx(0, ξ)V(ξ, t)dξ +
1
2
V(0, t)
∞∫
0
q(ξ)dξ, t ∈ (0, T ). (3.11)
Then V(x, t) is the solution of system (3.1), (3.2) with the control p determined
by (3.11).
Pr o o f. Apply the operator M−1 to system (2.2), (2.3). Evidently, (2.3) is
reduced to (3.2). Equation (2.2) takes the form
M−1Vtt(·, t) = M−1Vxx(·, t)−M−1(QV)(·, t)−2u(t)M−1δ, t ∈ (0, T ). (3.12)
Since Mδ = δ, we have M−1δ = δ. Using Lemma 5.7, from (3.12) we get
d2
dt2
M−1V(·, t) =
d2
dx2
M−1V(·, t)− 2δ
∞∫
0
Nx(0, ξ)V(ξ, t)dξ
− δV(0, t)
∞∫
0
q(ξ)dξ − 2δu(t), t ∈ (0, T ). (3.13)
Taking into account (3.11), it is easy to see that (3.13) is reduced to (2.2). The
theorem is proved.
R e m a r k 3.1. Theorems 3.3 and 3.4 establish a one-to-one correspondence
between the solutions of systems (2.2), (2.3) and (3.1), (3.2) under the condition
that the controls are connected by the corresponding relations.
Lemma 3.2. Let V(x, t) be the solution of system (2.2), (2.3). Let also
V(·, t) = M−1V(·, t), t ∈ (0, T ), V0
j = M−1V0
j , j = 0, 1, and (3.11) holds. Then
Vx(+0, t) = u(t), t ∈ (0, T ).
Pr o o f. Applying the operator M−1 to equation (2.2), we get (3.13). Taking
into account expression (2.6) after di�erentiation and (5.17), we obtain
∞∫
0
Nx(0, ξ)V(ξ, t)dξ + V(0, t)
1
2
∞∫
0
q(ξ)dξ =
signx
∞∫
|x|
Nx(|x|, ξ)V(ξ, t)dξ
+ signxV(|x|, t)1
2
∞∫
|x|
q(ξ)dξ
∣∣∣∣∣∣∣
x=+0
=
[
d
dx
(M−1V)(x, t)−Vx(x, t)
]∣∣∣∣
x=+0
= Vx(+0, t)−Vx(+0, t), t ∈ (0, T ). (3.14)
314 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Substituting (3.14) in (3.13), we get
Vtt(x, t) = Vxx(x, t)− 2δ[Vx(+0, t)−Vx(+0, t)]− 2δu(t), x ∈ R, t ∈ (0, T ).
(3.15)
Since the conditions of Theorem 3.4 hold, V(x, t) is the solution of system (3.1),
(3.2). Hence, due to Lemma 3.1, Vx(+0, t) = p(t), t ∈ (0, T ). Thus equation
(3.15) takes the form
Vtt(x, t) = Vxx(x, t)− 2δ(x)p(t)+2δ(x)Vx(+0, t)− 2δ(x)u(t), x ∈ R, t ∈ (0, T ).
From the above, it is seen that V(x, t) is the solution of system (3.1), (3.2) when-
ever Vx(+0, t) = u(t), t ∈ (0, T ). The lemma is proved.
R e m a r k 3.2. Let V be the solution of system (2.2), (2.3). By Lemma 3.2,
it follows that the restriction of V(·, t) to [0,∞), t ∈ (0, T ), is the solution of
system (1.1), (1.2), (2.1). Thus we prove that control systems (1.1), (1.2), (2.1)
and (2.2), (2.3) are equivalent.
Lemma 3.3. Formulas (3.9) and (3.11) are equivalent.
Pr o o f. Let V(·, t) = MV(·, t), t ∈ (0, T ), and (3.11) be valid. We prove
that (3.9) is also valid. From (3.11), we have
u(t) = p(t)−
∞∫
0
Nx(0, ξ)V(ξ, t)dξ − 1
2
V(0, t)
∞∫
0
q(ξ)dξ, t ∈ (0, T ).
Using (2.5), (2.6), (5.10) and (5.17), we obtain
u(t) = p(t)−
signx
∞∫
|x|
Nx(|x|, ξ)V(ξ, t)dξ − signxV(|x|, t)N(|x|, |x|)
∣∣∣∣∣∣∣
x=+0
= p(t)− [
d/dx(M−1V)(x, t)−Vx(x, t)
]∣∣
x=+0
= p(t)− [Vx(x, t)− d/dx(MV)(x, t)]|x=+0
= p(t)−
− signx
∞∫
|x|
Mx(|x|, ξ)V(ξ, t)dξ + signxV(|x|, t)M(|x|, |x|)
∣∣∣∣∣∣∣
x=+0
= p(t) +
∞∫
0
Mx(0, ξ)V(ξ, t)dξ − 1
2
V(0, t)
∞∫
0
q(ξ)dξ, t ∈ (0, T ).
Analogously, it can be proved that (3.11) is valid whenever (3.9) holds. The
lemma is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 315
K.S. Khalina
Lemma 3.4. Let (3.9) hold for the controls u and p of systems (2.2), (2.3)
and (3.1), (3.2), respectively. Let p ∈ L∞(0, T ). Let also a state V0 of control
system (3.1), (3.2) be approximately null-controllable with respect to (3.1), (3.2).
Then u ∈ L∞(0, T ).
P r o o f. Let p ∈ L∞(0, T ). Taking into account (3.9), we have to prove
that V(ξ, ·) ∈ L∞(0, T ), ξ ∈ R. Due to (3.7), it remains to show that V0
0 (x± t),
Ṽ0
1 (x ± t), (∂−1ΩP t)(x ± t), (∂−1ΞP t)(x ± t) ∈ L∞(0, T ) when x is �xed. Since
p ∈ L∞(0, T ), then P t ∈ L∞(R). Therefore, ΩP t ∈ L∞(R) and ΞP t ∈ L∞(R).
Hence, (∂−1ΩP t)(x± t) ∈ L∞(0, T ), (∂−1ΞP t)(x± t) ∈ L∞(0, T ).
Since the control p solves the approximate null-controllability problem for
system (3.1), (3.2), then conditions (3.5) and (3.6) hold. Thus, V0
1 ∈ L∞(R).
Hence, Ṽ0
1 (x ± t) = (∂−1V0
1 )(x ± t) ∈ L∞(0, T ). It follows from (3.6) that V0
0 =
∂−1(signxV0
1 ). Therefore, V0
0 (x± t) ∈ L∞(0, T ). The lemma is proved.
Theorem 3.5. Let (3.9) hold and V(·, t) = MV(·, t), t ∈ (0, T ), V0
j = MV0
j ,
j = 0, 1. Let a state V0 of control system (3.1), (3.2) be approximately null-
controllable with respect to (3.1), (3.2). Then a state V0 of control system (2.2),
(2.3) is approximately null-controllable with respect to (2.2), (2.3).
P r o o f. Let a state V0 be approximately null-controllable with respect to
(3.1), (3.2). Therefore, for each m ∈ N there exist Tm > 0 and pm ∈ L∞(0, Tm)
such that |||V(·, Tm)|||10 → 0 as m →∞. Here V is the solution of (3.1), (3.2) with
the control pm. Since MV(·, Tm) = V(·, Tm), m = 1,∞, and the operator M is
continuous in the spaces Hs
0,e, s = 1, 0, we obtain |||V(·, Tm)|||10 → 0 as m → ∞.
Thus, for each m ∈ N there exist Tm > 0 and um = pm +
∫∞
0 Mx(0, ξ)Vm(ξ, ·)dξ−
1
2Vm(0, ·) ∫∞
0 q(ξ)dξ such that um ∈ L∞(0, Tm) (due to Lemma 3.4), moreover,
|||V(·, Tm)|||10 → 0 as m → ∞. This implies that a state V0 is approximately
null-controllable with respect to (2.2), (2.3). The theorem is proved.
Due to Theorems 3.3, 3.5, Lemma 5.8 and Theorems 3.1, 3.2, we obtain the
controllability conditions for system (2.2), (2.3) and, consequently, for system
(1.1), (1.2), (2.1).
Theorem 3.6. Suppose the conditions below hold:
V0
1 ∈ L∞(R), (3.16)
V0
1 = M sign ξ(M−1V0
0)
′. (3.17)
Then a state V0 ∈ H1
0,e is approximately null-controllable with respect to (2.2),
(2.3). Under conditions (3.16), (3.17) there exists a sequence {Tn}∞n=1 such that
316 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Tn|M−1V0
0(Tn)|2 → 0 as n →∞, and for this sequence the controls
un(t) = pn(t) +
∞∫
0
Mx(0, ξ)V(ξ, t)dξ − 1
2
V(0, t)
∞∫
0
q(ξ)dξ
= (M−1V0
1)(t) +
∞∫
0
Mx(0, ξ)V(ξ, t)dξ − 1
2
V(0, t)
∞∫
0
q(ξ)dξ (3.18)
a.e. on (0, Tn), n ∈ N, solve the approximate null-controllability problem for
system (2.2), (2.3), where V(ξ, t) is de�ned by (3.7).
Theorem 3.7. Suppose conditions (3.16), (3.17) hold and there exists T > 0
such that suppM−1V0
1 ⊂ (−T, T ). Then a state V0 ∈ H1
0,e is null-controllable
with respect to (2.2), (2.3). In addition, the control solving the null-controllability
problem for system (2.2), (2.3) is of the form (3.18) a.e. on (0, T ).
R e m a r k 3.3. Unfortunately, we can not prove the necessity of conditions
(3.16), (3.17) for approximate null-controllability of the state V0 ∈ H1
0,e as it
is not proved that V(x, ·) ∈ L∞(0, T ) in general. Nevertheless, in the following
theorem we will prove the necessity of these conditions under some restrictions.
Theorem 3.8. Let conditions (1.3) hold. Let a time T > 0 be �xed and
suppV0
j ⊂ (−T, T ), j = 0, 1. Then conditions (3.16), (3.17) are not only su�-
cient, but also necessary for approximate null-controllability and null-controllability
of a state V0 ∈ H1
0,e at a �xed time.
P r o o f. In [20], the controllability problems at a �xed time for system
(1.1), (1.2), (2.1) were considered in the class of functions with bounded supports.
Let u ∈ L∞(0, T ) and a state V0 be approximately null-controllable at a time
T with respect to (2.2), (2.3). One can conclude from [20, Lemma 4.1] that
suppV(·, t) ⊂ (−2T, 2T ) and V(x, ·) ∈ L∞(0, T ) in this case. Due to these facts,
the proof of the following statement is trivial.
A) Let (3.11) hold for the controls u and p of systems (2.2), (2.3) and (3.1),
(3.2), respectively. Let u ∈ L∞(0, T ). Let also a state V0 of control system
(2.2), (2.3) be approximately null-controllable with respect to (2.2), (2.3). Then
p ∈ L∞(0, T ).
It is obvious that suppM−1V(·, t) ⊂ (−2T, 2T ), t ∈ (0, T ), and suppM−1V0
j ⊂
(−T, T ), j = 0, 1. The following statement is proved in a similar way as that to
Theorem 3.5 but for each m ∈ N the �xed time T is taken instead of Tm.
B) Let (3.11) hold and V(·, t) = M−1V(·, t), t ∈ (0, T ), V0
j = M−1V0
j , j = 0, 1.
Let a state V0 of control system (2.2), (2.3) be approximately null-controllable at
a time T with respect to (2.2), (2.3). Then a state V0 of control system (3.1),
(3.2) is approximately null-controllable at a time T with respect to (3.1), (3.2).
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 317
K.S. Khalina
Thus, using statements A), B) and Theorems 3.1, 3.2, we may conclude that
conditions (3.16), (3.17) hold. The theorem is proved.
R e m a r k 3.4. In practice, to �nd the controls un, n ∈ N, solving the
approximate null-controllability problem for system (2.2), (2.3), another formula
is more convenient than (3.18). Let us transform (3.18) using (5.10) and (2.5).
Let t ∈ (0, Tn), n ∈ N. Then
un(t) = pn(t) +
signx
∞∫
|x|
Mx(|x|, ξ)V(ξ, t)dξ − signxM(|x|, |x|)V(|x|, t)
∣∣∣∣∣∣∣
x=+0
= pn(t) +
d
dx
∞∫
|x|
M(|x|, ξ)V(ξ, t)dξ
∣∣∣∣∣∣∣
x=+0
= pn(t) +
[
d
dx
(MV)(x, t)
]∣∣∣∣
x=+0
− Vx(+0, t).
Using Lemma 3.1, we get
un(t) =
[
d
dx
(MV)(x, t)
]∣∣∣∣
x=+0
, t ∈ (0, Tn), n ∈ N. (3.19)
R e m a r k 3.5. Let conditions (1.3) hold, a time T > 0 be �xed, and suppV0
j ⊂
(−T, T ), j = 0, 1. We have proved that conditions (3.16) and (3.17) are neces-
sary and su�cient for null-controllability and approximate null-controllability of
system (1.1), (1.2), (2.1) at a �xed time. On the other hand, in [20] it is proved
that the conditions
V0
1 ∈ L∞(R), (3.20)
V0
1 = K−1
e signx(KeV0
0)
′ (3.21)
are necessary and su�cient for null-controllability and approximate null-control-
lability of the system at a �xed time. Here Ke and K−1
e are other transformation
operators with other kernels.
One can see that conditions (3.16) and (3.20) coincide. Since under consid-
eration are necessary and su�cient conditions, we obtain that conditions (3.17)
and (3.21) are di�erent forms of the same relation between the initial functions.
4. Examples
Let q(x) = e−x, x > 0. It is obvious that conditions (1.3) are valid. In this
section, the kernels of the transformation operators will be found explicitly for the
318 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
given q, and thus relation (3.17) will be rewritten in a simpler form. The controls
solving the approximate null-controllability problem for system (1.1), (1.2), (2.1)
with given initial functions will also be found.
E x a m p l e 4.1. Find the kernel M(x, t) of the operator M. System (5.9)�
(5.11) takes the form
Mxx(x, t)− Mtt(x, t) = e−xM(x, t), 0 < x < t, (4.1)
M(x, x) =
1
2
e−x, x > 0, (4.2)
lim
x+t→∞ Mx(x, t) = lim
x+t→∞ Mt(x, t) = 0. (4.3)
Put ξ = e−
x+t
2 , η = e
t−x
2 − 1 and denote A(ξ, η) = M(x, t). It is easy to see that
system (4.1)�(4.3) is equivalent to the system
Aξη(ξ, η) = A(ξ, η), 0 < η < ξ−1 − 1,
A(ξ, 0) =
ξ
2
, 0 < ξ < 1,
Aη(0, η) = 0, η > 0.
Then A(ξ, η) = ξ
2
I1(2
√
ξη)√
ξη
is the unique solution of this system. Here I1(z) is the
modi�ed Bessel function of order one, I1(z) = 1
i J1(iz), where J1(y) is the Bessel
function of order one. Thus the kernel of the operator M is
M(x, t) =
e−
x+t
2
2
I1
(
2
√
e−x − e−
x+t
2
)
√
e−x − e−
x+t
2
, 0 < x < t. (4.4)
For the kernel N(x, t) of the operator M−1 we have the system
Nxx(x, t)− Ntt(x, t) = −e−tN(x, t), 0 < x < t,
N(x, x) = −1
2
e−x, x > 0,
lim
x+t→∞ Nx(x, t) = lim
x+t→∞ Nt(x, t) = 0.
Putting µ = e−
x+t
2 , ν = e
x−t
2 − 1 and denoting B(µ, ν) = N(x, t), we reduce this
system to the form
Bµν(µ, ν) = B(µ, ν), 0 < ν < µ−1 − 1,
B(µ, 0) = −µ
2
, 0 < µ < 1,
Bν(0, ν) = 0, ν > 0.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 319
K.S. Khalina
Then B(µ, ν) = −µ
2
I1(2
√
µν)√
µν is its unique solution. Hence, for 0 < x < t, we have
N(x, t) = −e−
x+t
2
2
I1
(
2
√
e−t − e−
x+t
2
)
√
e−t − e−
x+t
2
= −e−
x+t
2
2
J1
(
2
√
e−
x+t
2 − e−t
)
√
e−
x+t
2 − e−t
. (4.5)
Thus the kernels of the operators M and M−1 are of the forms (4.4) and (4.5),
respectively, when q(x) = e−x, x > 0.
E x a m p l e 4.2. Consider (3.17). Substituting (4.5) in (2.6), we obtain
(M−1V0
0)(ξ) = − sign ξ
d
dξ
∞∫
|ξ|
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
V0
0(y)dy = − sign ξG′(ξ),
where ξ ∈ R and
G(ξ) =
∞∫
|ξ|
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
V0
0(y)dy. (4.6)
For any ϕ ∈ H0
0,e, we have
(
sign ξ(M−1V0
0)
′(ξ), ϕ(ξ)
)
=
(
sign ξ(− sign ξG′(ξ))′, ϕ(ξ)
)
=
(
sign ξG′(ξ), 2δ(ξ)ϕ(ξ) + sign ξϕ′(ξ)
)
= 2ϕ(0)
(
sign ξG′(ξ), δ(ξ)
)
+
(
sign ξG′(ξ), sign ξϕ′(ξ)
)
=
(
2δ(ξ)G′(+0)−G′′(ξ), ϕ(ξ)
)
.
Thus, sign ξ(M−1V0
0)
′(ξ) = 2δ(ξ)G′(+0)−G′′(ξ), ξ ∈ R. Substituting this equality
in (3.17) and taking into account that Mδ = δ, we get
V0
1(x) = 2δ(x)G′(+0)− (MG′′)(x), x ∈ R.
Using Lemma 5.6, we have
V0
1(x) = 2δ(x)G′(+0)− (MG)′′(x) + e−|x|(MG)(x) + 2δ(x)
∞∫
0
Mx(0, ξ)G(ξ)dξ
+ δ(x)G(0)
∞∫
0
e−ydy = 2δ(x)G′(+0)− (MG)′′(x) + e−|x|(MG)(x)
+ 2δ(x)
d
dx
∞∫
|x|
M(|x|, ξ)G(ξ)dξ
∣∣∣∣∣∣∣
x=+0
= −(MG)′′(x) + e−|x|(MG)(x) + 2δ(x)(MG)′(+0). (4.7)
320 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Consider (MG)(x). Substituting (4.4) and (4.6) in (2.5) and changing the order
of integration, we obtain
(MG)(x) =
∞∫
|ξ|
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
V0
0(y)dy
+
∞∫
|x|
V0
0(y)
y∫
|x|
e−
x+t
2
2
I1
(
2
√
e−x − e−
x+t
2
)
√
e−x − e−
x+t
2
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
dt dy.
Consider the inner integral. Putting e−t/2 = z, e−|x|/2 = h, e−y/2 = g, we reduce
it to the form
y∫
|x|
e−
x+t
2
2
I1
(
2
√
e−x − e−
x+t
2
)
√
e−x − e−
x+t
2
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
dt
= h
∫ h
g
I1(2
√
h(h− z))√
h(h− z)
J0(2
√
g(z − g))dz.
After expanding I1(τ) and J0(τ) into series over τn, n = 0,∞, and integrating,
we obtain
y∫
|x|
e−
x+t
2
2
I1
(
2
√
e−x − e−
x+t
2
)
√
e−x − e−
x+t
2
J0
(
2
√
e−
|ξ|+y
2 − e−y
)
dt
= I0
(
2
(
e−
|x|
2 − e−
y
2
))
− J0
(
2
√
e−
|x|+y
2 − e−y
)
.
Thus,
(MG)(x) =
∞∫
|x|
I0
(
2
(
e−
|x|
2 − e−
y
2
))
V0
0(y)dy, x ∈ R. (4.8)
Di�erentiating (4.8), we get
(MG)′(+0) = −
∞∫
0
I1
(
2
(
1− e−
y
2
))
V0
0(y)dy −V0
0(+0); (4.9)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 321
K.S. Khalina
(MG)′′(x) = −2δ(x)
∞∫
0
I1
(
2
(
1− e−
y
2
))
V0
0(y)dy − 2δ(x)V0
0(+0)
− signx
(
V0
0
)′ (x)−
∞∫
|x|
e−
|x|+y
2
2
I1
(
2
(
e−
|x|
2 − e−
y
2
))
e−
|x|
2 − e−
y
2
V0
0(y)dy
+ e−|x|
∞∫
|x|
I0
(
2
(
e−
|x|
2 − e−
y
2
))
V0
0(y)dy. (4.10)
Substituting (4.8)�(4.10) in (4.7), we have
V0
1(x) = signx
(
V0
0
)′ (x) +
∞∫
|x|
e−
|x|+y
2
2
I1
(
2
(
e−
|x|
2 − e−
y
2
))
e−
|x|
2 − e−
y
2
V0
0(y)dy, x ∈ R.
(4.11)
Thus, condition (3.17) is of the form (4.11) when q(x) = e−x, x > 0.
E x a m p l e 4.3. Let q(x) = e−x, x > 0; V0
0(x) = I1(2e−|x|/2), V0
1(x) =
−1
2I1(2e−|x|/2), x ∈ R. Consider the approximate null-controllability problem for
system (1.1), (1.2), (2.1). Evidently, (3.16) is valid. One can see that (4.11) is
also valid. Therefore, due to Theorem 3.6, the initial state V0 is approximately
null-controllable with respect to system (1.1), (1.2), (2.1). To �nd the controls
un, n ∈ N, solving the approximate null-controllability problem for this system,
we reduce the given system to a system with Q = 0 applying the operator M−1.
Putting e−y/2 = z, e−|x|/2 = h, we have
M−1V0
0(x) = I1(2e−
|x|
2 )−
∞∫
|x|
e−
|x|+y
2
2
J1
(
2
√
e−
|x|+y
2 − e−y
)
2
√
e−
|x|+y
2 − e−y
I1
(
2e−
y
2
)
dy
= I1(2h)− h
h∫
0
I1(2z)
J1(2
√
z(h− z))√
z(h− z)
dz.
After expanding I1(τ) and J1(τ) into series over τn, n = 0,∞, and integrating,
we obtain
V0
0 (x) = M−1V0
0(x) = h
∞∑
k=0
∞∑
n=0
(−1)nh2k+2n(2k + n)!
k!(k + 1)!n!(2k + 2n)!
.
It is easy to check that V0
0 (x) = h = e−|x|/2. Hence, V0
1 (x) = −1/2e−|x|/2. It
is evident that conditions (3.5), (3.6) are valid. Therefore the state V0 is ap-
322 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
proximately null-controllable with respect to system (3.1), (3.2), and the con-
trols pn(t) = V0
1 (t) = −1/2e−t/2 a.e. on (0, n), n ∈ N, solve the approximate
null-controllability problem for system (3.1), (3.2). To �nd V(x, t), we substi-
tute the explicit expressions for V0
0 , V0
1 and pn, n ∈ N, in (3.8) and obtain
Vx(x, t) = V0
1 (x + t)H(x)− V0
1 (x− t)H(−x), x ∈ R, t ∈ (0, Tn). Hence,
V(x, t) =
x∫
−∞
[V0
1 (ξ + t)H(ξ)− V0
1 (ξ − t)H(−ξ)
]
dξ.
Obviously, Vx(x, t) is odd on x. Since
∫ x
−∞ f(ξ)dξ =
∫ −|x|
−∞ f(ξ)dξ for any odd
function f , we have
V(x, t) =
−|x|∫
−∞
Vξ(ξ, t)dξ = −
−|x|∫
−∞
V0
1 (ξ − t)dξ = −
−|x|−t∫
−∞
V0
1 (y)dy = e−
|x|−t
2 ,
where x ∈ R, t ∈ (0, Tn). To �nd the controls un, n ∈ N, we use (3.19). Thus,
(MV)(x, t) = e−t/2M(e−ξ/2)(x, t) = e−t/2I1(2e−|x|/2), x ∈ R, t ∈ (0, Tn).
Hence,
d
dx
(MV)(x, t) = signxe−
t
2
[
1
2
I1
(
2e−
|x|
2
)
− e−
|x|
2 I0
(
2e−
|x|
2
)]
, x ∈ R, t ∈ (0, Tn).
Thus the controls
un(t) = e−
t
2 [I1(2)/2− I0(2)] a.e. on (0, n), n ∈ N,
solve the approximate null-controllability problem for system (1.1), (1.2), (2.1)
with the given initial state.
5. The Transformation Operators for the Sturm�Liouville
Equation that do not Change a Solution Asymptotic
at In�nity
At the beginning of the section we recall de�nitions and some properties of the
transformation operators from [23, Chap. 3]. Further, we will extend these oper-
ators to the Sobolev spaces and prove their continuity. Consider two di�erential
equations
− y′′(x) = λ2y(x), x ∈ (0,+∞), λ ∈ C, (5.1)
− y′′(x) + q(x)y(x) = λ2y(x), x ∈ (0,+∞), λ ∈ C. (5.2)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 323
K.S. Khalina
As it is known [23, Chap. 3], the integral operator (I + K)f = f(x)+∫∞
x M(x, t)f(t)dt transfers the solution of (5.1) to the solution of (5.2), and it is the
transformation operator that does not change a solution asymptotic at in�nity.
Due to [23, Chap. 3], this operator is a bijection of L2[0,∞) onto L2[0,∞),
and the inverse operator (I + K)−1 = I + L is of the same form: (I + L)f =
f(x) +
∫∞
x N(x, t)f(t)dt.
For the operators kernels M(x, t) and N(x, t), the following estimates were ob-
tained in [23, Chap. 3]:
|M(x, t)| ≤ 1
2
σ
(
x + t
2
)
eσ1(x)−σ1((x+t)/2), (x, t) ∈ (0,∞)× (0,∞), (5.3)
|N(x, t)| ≤ 1
2
σ
(
x + t
2
)
eσ1((x+t)/2)−σ1(t), (x, t) ∈ (0,∞)× (0,∞), (5.4)
where σ(x) =
∫∞
x |q(ξ)|dξ, σ1(x) =
∫∞
x σ(ξ)dξ. It is also known that
M(x, t) = 0 when 0 < t < x. (5.5)
R e m a r k 5.1. The method of �nding the kernel M(x, t) is obtained in [23,
Chap. 3]. The function M(x, t) is the kernel of the operator I+K i� the function
M̃(α, β) is the solution of the following problem:
M̃αβ(α, β) = −q(α− β)M̃(α, β), 0 < β < α, (5.6)
M̃(α, 0) =
1
2
∞∫
α
q(ξ)dξ, α > 0, (5.7)
lim
α→∞ M̃α(α, β) = lim
α→∞ M̃β(α, β) = 0. (5.8)
Hence, M(x, t) = M̃(x+t
2 , t−x
2 ) when 0 < x < t.
R e m a r k 5.2. Problem (5.6)�(5.8) is equivalent to the problem
Mxx(x, t)− Mtt(x, t) = q(x)M(x, t), 0 < x < t, (5.9)
M(x, x) =
1
2
∞∫
x
q(ξ)dξ, x > 0, (5.10)
lim
x+t→∞ Mx(x, t) = lim
x+t→∞ Mt(x, t) = 0. (5.11)
R e m a r k 5.3. Using the properties of the kernel M(x, t), from the obvious
equation N(x, t)+M(x, t)+
∫ t
x N(x, ξ)M(ξ, t)dξ = 0 one can easily obtain the follow-
ing statements:
a) N(x, t) = 0, when 0 < t < x. (5.12)
324 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
b) The function N(x, t) is the kernel of the operator I+L i� the function Ñ(α, β)
is the solution of the following problem:
Ñαβ(α, β) = q(α + β)Ñ(α, β), 0 < β < α, (5.13)
Ñ(α, 0) = −1
2
∞∫
α
q(ξ)dξ, α > 0, (5.14)
lim
α→∞ Ñα(α, β) = lim
α→∞ Ñβ(α, β) = 0. (5.15)
Hence, N(x, t) = Ñ(x+t
2 , t−x
2 ) when 0 < x < t.
c) Problem (5.13)�(5.15) is equivalent to the problem
Nxx(x, t)− Ntt(x, t) = −q(t)N(x, t), 0 < x < t, (5.16)
N(x, x) = −1
2
∞∫
x
q(ξ)dξ, x > 0, (5.17)
lim
x+t→∞ Nx(x, t) = lim
x+t→∞ Nt(x, t) = 0. (5.18)
Passing to the integral equations
M̃(α, β) =
1
2
∞∫
α
q(ξ)dξ +
∞∫
α
β∫
0
q(y − z)M̃(y, z)dz dy, 0 < β < α,
Ñ(α, β) = −1
2
∞∫
α
q(ξ)dξ −
∞∫
α
β∫
0
q(y + z)Ñ(y, z)dz dy, 0 < β < α,
that are equivalent to boundary problems (5.6)�(5.8) and (5.13)�(5.15), respec-
tively, we can �nd the estimates for M̃α, M̃β , Ñα, Ñβ . Returning to variables x and
t, we obtain the following estimates for 0 < x < t:
|Mt(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
eσ1(x)−σ1(x+t
2
), (5.19)
|Mx(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
eσ1(x)−σ1(x+t
2
), (5.20)
|Nt(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
4
eσ1(x+t
2 )−σ1(t)σ
(
x + t
2
)[
σ
(
x + t
2
)
+ σ(t)
]
,
(5.21)
|Nx(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
4
eσ1(x+t
2 )−σ1(t)σ
(
x + t
2
)[
σ
(
x + t
2
)
+ σ(t)
]
.
(5.22)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 325
K.S. Khalina
In the following lemma, we obtain some properties of the functions σ(x) and σ1(x)
due to which estimates (5.3), (5.4), (5.19)�(5.22) will be somewhat simpli�ed.
Lemma 5.1. Let σ(x) =
∫∞
x |q(ξ)|dξ, σ1(x) =
∫∞
x σ(ξ)dξ, x ∈ [0,∞), where
conditions (1.3) hold for q. Then
(a) σ and σ1 are decreasing functions;
(b) σ ≤ σ(0) < ∞, σ1 ≤ σ1(0) < ∞ on [0,∞).
P r o o f. Assertion (a) is evident. Prove (b). Using (1.3), we get
σ(0) =
∞∫
0
|q(ξ)|dξ =
1∫
0
|q(ξ)|dξ +
∞∫
1
|q(ξ)|dξ ≤ Cq +
∫ ∞
1
x|q(ξ)|dξ < ∞,
where Cq > 0 such that |q| ≤ Cq a.e. on (0,∞). Consider σ1(0). Integrating the
outer integral by parts, we get
σ1(0) =
∞∫
0
∞∫
ξ
|q(y)|dydξ =
ξ
∞∫
ξ
|q(y)|dy
∣∣∣∣∣∣∣
ξ=∞
ξ=0
+
∞∫
0
ξ|q(ξ)|dξ
≤
∞∫
ξ
y|q(y)|dy
∣∣∣∣∣∣∣
ξ=∞
+
∞∫
0
ξ|q(ξ)|dξ < ∞
due to (1.3). The lemma is proved.
Using Lemma 5.1, one can make the following conclusions for t > x > 0:
σ
(
x+t
2
) ≤ σ(x), σ(t) ≤ σ(x), eσ1(x)−σ1(x+t
2
) ≤ e2σ1(0), eσ1(x+t
2 )−σ1(t) ≤ e2σ1(0).
Therefore estimates (5.3), (5.4), (5.19)�(5.22) can be rewritten in the form
|M(x, t)| ≤ 1
2
σ
(
x + t
2
)
e2σ1(0), 0 < x < t, (5.23)
|N(x, t)| ≤ 1
2
σ
(
x + t
2
)
e2σ1(0), 0 < x < t, (5.24)
|Mt(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
e2σ1(0), 0 < x < t, (5.25)
|Mx(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
e2σ1(0), 0 < x < t, (5.26)
|Nt(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
e2σ1(0), 0 < x < t, (5.27)
|Nx(x, t)| ≤ 1
4
∣∣∣∣q
(
x + t
2
)∣∣∣∣ +
1
2
σ(x)σ
(
x + t
2
)
e2σ1(0), 0 < x < t. (5.28)
326 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Further, consider the extensions of the operators I+K and I+L, denoted by
M and M−1, respectively, extended to H0
0,e by formulas (2.5), (2.6).
Lemma 5.2. The operators M,M−1 : H0
0,e → H0
0,e, D(M) = D(M−1) = H0
0,e
de�ned by (2.5), (2.6) are continuous from H0
0,e to H0
0,e. In addition, R(M) =
R(M−1) = H0
0,e.
P r o o f. Let f ∈ H0
0,e. The evenness of Mf is evident. Since I + K is
continuous from L2[0,∞) to L2[0,∞), we can see that M is continuous from H0
0,e
to H0
0,e. The assertion on the operator M−1 is proved in a similar way. From the
continuity of the operators it follows that R(M) = R(M−1) = H0
0,e. The lemma
is proved.
Lemma 5.3. Let ϕ,ψ ∈ H0
0,e. Then the adjoint operators M∗, (M−1)∗ :
H0
0,e → H0
0,e, D(M∗) = D((M−1)∗) = H0
0,e are continuous from H0
0,e to H0
0,e
and can be de�ned by the formulas
(M∗ϕ)(t) = ϕ(t) +
|t|∫
0
M(x, |t|)ϕ(x)dx, t ∈ R, (5.29)
((M−1)∗ψ)(t) = ψ(t) +
|t|∫
0
N(x, |t|)ψ(x)dx, t ∈ R. (5.30)
In addition, R(M∗) = R((M−1)∗) = H0
0,e.
P r o o f. Let f ∈ H0
0,e. Substituting (2.5) into the known de�nition (Mf, ϕ) =
(f, M∗ϕ) and changing the order of integration, we obtain (5.29). In the same
way, we get (5.30). The continuity of the operators (M)∗ and (M−1)∗ from H0
0,e
to H0
0,e follows from the continuity of the operators M and M−1 from H0
0,e to
H0
0,e. The fact that R(M∗) = R((M−1)∗) = H0
0,e follows from the continuity of
the operators. The lemma is proved.
Lemma 5.4. The operators M,M−1 : H1
0,e → H1
0,e, D(M) = D(M−1) = H1
0,e
de�ned by (2.5), (2.6) are continuous from H1
0,e to H1
0,e, and R(M) = R(M−1)
= H1
0,e.
P r o o f. Let f ∈ H1
0,e. Taking into account that ‖y‖1
0 ≤ ‖y‖0
0 + ‖y′‖0
0 for
any y ∈ H1
0 , we have
‖Mf‖1
0 ≤ ‖f‖1
0 +
∥∥∥∥∥∥∥
∞∫
|x|
M(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
+
∥∥∥∥∥∥∥
d
dx
∞∫
|x|
M(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
. (5.31)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 327
K.S. Khalina
Taking into account (5.5), (5.23), using the Cauchy�Bunyakovsky�Schwartz in-
equality and the inequality ‖f‖0
0 ≤ ‖f‖1
0, we obtain the estimate for the second
summand in (5.31),
∥∥∥∥∥∥∥
∞∫
|x|
M(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
=
√
2
∞∫
0
∣∣∣∣∣∣
∞∫
0
M(x, t)f(t)dt
∣∣∣∣∣∣
2
dx
1/2
≤ ‖f‖0
0
∞∫
0
∞∫
x
|M(x, t)|2dtdx
1/2
≤ e2σ1(0)
2
‖f‖1
0
∞∫
0
∞∫
x
∣∣∣∣σ
(
x + t
2
)∣∣∣∣
2
dtdx
1/2
≤ 1
2
e2σ1(0) ‖f‖1
0
∞∫
0
σ(x)
∞∫
x
σ
(
x + t
2
)
dt dx
1/2
≤ σ1(0)√
2
e2σ1(0) ‖f‖1
0 . (5.32)
Using (5.10) and the evenness of f , we get the estimate for the third summand
in (5.31),
∥∥∥∥∥∥∥
d
dx
∞∫
|x|
M(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
≤
∥∥∥∥∥∥∥
signx
∞∫
|x|
Mx(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
+
1
2
∥∥∥∥∥∥∥
signxf(x)
∞∫
|x|
q(ξ)dξ
∥∥∥∥∥∥∥
0
0
.
Taking into account Lemma 5.1 and the inequality ‖f‖0
0 ≤ ‖f‖1
0, we obtain
1
2
∥∥∥∥∥∥∥
signxf(x)
∞∫
|x|
q(ξ)dξ
∥∥∥∥∥∥∥
0
0
≤ 1
2
∞∫
−∞
∣∣∣∣∣∣∣
f(x)
∞∫
|x|
q(ξ)dξ
∣∣∣∣∣∣∣
2
dx
1/2
≤ 1
2
σ(0) ‖f‖1
0 .
Taking into account (5.5), and using the Cauchy�Bunyakovsky�Schwartz inequal-
ity and (5.26), we get
∥∥∥∥∥∥∥
signx
∞∫
|x|
Mx(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
=
√
2
∞∫
0
∣∣∣∣∣∣
∞∫
0
Mx(x, t)f(t)dt
∣∣∣∣∣∣
2
dx
1/2
≤
√
2
∞∫
0
∞∫
0
|Mx(x, t)|2dt
∞∫
0
|f(t)|2dtdx
1/2
= ‖f‖0
0
∞∫
0
∞∫
x
|Mx(x, t)|2dtdx
1/2
328 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
≤ ‖f‖1
0
∞∫
0
1
16
∞∫
x
∣∣∣∣q
(
x + t
2
)∣∣∣∣
2
dt +
1
4
e2σ1(0)σ(x)
∞∫
x
∣∣∣∣q
(
x + t
2
)∣∣∣∣σ
(
x + t
2
)
dt
+
1
4
e4σ1(0)(σ(x))2
∞∫
x
(
σ
(
x + t
2
))2
dt
dx
1/2
. (5.33)
Let us estimate the last three summands. From (1.3) it follows that there exists
Cq > 0 such that |q| ≤ Cq a.e. on [0,∞). Using Lemma 5.1, we obtain
1
16
∞∫
0
∞∫
x
∣∣∣∣q
(
x + t
2
)∣∣∣∣
2
dtdx =
1
8
∞∫
0
∞∫
x
|q(y)|2dydx ≤ Cq
8
∞∫
0
∞∫
x
|q(y)|dydx =
Cq
8
σ1(0).
Then we use Lemma 5.1 again to obtain
1
4
e2σ1(0)
∞∫
0
σ(x)
∞∫
x
∣∣∣∣q
(
x + t
2
)∣∣∣∣σ
(
x + t
2
)
dt dx
=
1
2
e2σ1(0)
∞∫
0
σ(x)
∞∫
x
|q(y)|σ(y)dy dx ≤ 1
2
e2σ1(0)σ(0)
∞∫
0
σ(x)σ(x)dx
≤ 1
2
e2σ1(0)(σ(0))2σ1(0);
1
4
e4σ1(0)
∞∫
0
(σ(x))2
∞∫
x
(
σ
(
x + t
2
))2
dt dx
=
1
2
e4σ1(0)
∞∫
0
(σ(x))2
∞∫
x
(σ(y))2dy dx ≤ 1
2
e4σ1(0)(σ(0))2
∞∫
0
σ(x)σ1(x)dx
≤ 1
2
e4σ1(0)(σ(0))2σ1(0)
∞∫
0
σ(x)dx =
1
2
e4σ1(0)(σ(0))2(σ1(0))2.
Continuing estimate (5.33), we get
∥∥∥∥∥∥∥
signx
∞∫
|x|
Mx(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
≤ P ‖f‖1
0 ,
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 329
K.S. Khalina
where P =
(
Cq
8 σ1(0) + e2σ1(0)
2 (σ(0))2σ1(0)
[
1 + e2σ1(0)σ1(0)
])1/2
> 0. Thus,
∥∥∥∥∥∥∥
d
dx
∞∫
|x|
M(|x|, t)f(t)dt
∥∥∥∥∥∥∥
0
0
≤ ‖f‖1
0
(
1
2
σ(0) + P
)
. (5.34)
Substituting (5.32) and (5.34) in (5.31), we obtain that the operator M is continu-
ous. Analogously, M−1 is continuous from H1
0,e to H1
0,e. From the continuity of the
operators it follows that R(M) = R(M−1) = H1
0,e when D(M) = D(M−1) = H1
0,e.
The lemma is proved.
Lemma 5.5. The operators M, M−1 : H−1
0,e → H−1
0,e , D(M) = D(M−1) = H−1
0,e ,
de�ned by (2.7), (2.8), are continuous from H−1
0,e to H−1
0,e , and R(M) = R(M−1) =
H−1
0,e .
P r o o f. Let us prove that the restrictions of the adjoint operators to H1
0,e
are continuous from H1
0,e to H1
0,e and their range is the space H1
0,e if the domain
is H1
0,e. The proof of the continuity of the operators M∗ and (M−1)∗ is similar
to the proof of the previous lemma. Here the adjoint operator is considered
instead of the original one. From the continuity of the operators it follows that
R(M∗) = R((M−1)∗) = H1
0,e. Thereby, the operators M and M−1 are well de�ned
by formulas (2.7), (2.8). Since the adjoint operators are continuous, then M and
M−1 are continuous from H−1
0,e to H−1
0,e , and thus R(M) = R(M−1) = H−1
0,e . The
lemma is proved.
Lemma 5.6. Let f ∈ H1
0,e. Then
M(f ′′) = (Mf)′′ −QMf − 2δ
∞∫
0
Mx(0, ξ)f(ξ)dξ + δf(0)
∞∫
0
q(ξ)dξ.
P r o o f. Let ϕ ∈ H1
0,e. Let us transform the expressions (M(f ′′), ϕ) and
((Mf)′′, ϕ) using (2.5), (5.29), (5.10) and the evenness of f and ϕ. Further, we
will consider the di�erence (M(f ′′), ϕ)− ((Mf)′′, ϕ). Thus,
(M(f ′′), ϕ) = −(f ′, (M∗ϕ)′) =
−
f ′(x), ϕ′(x) + signx
|x|∫
0
Mx(y, |x|)ϕ(y)dy + signxϕ(x)
1
2
∞∫
|x|
q(ξ)dξ
330 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
= (f ′′, ϕ) +
f(x), 2δ(x)
|x|∫
0
Mx(y, |x|)ϕ(y)dy
+
(
f(x), ϕ(x) Mx(y, |x|)|y=|x|
)
+
f(x),
|x|∫
0
Mxx(y, |x|)ϕ(y)dy
−
f ′(x) signx
1
2
∞∫
|x|
q(ξ)dξ, ϕ(x)
= (f ′′, ϕ) +
(
f(x) Mx(t, |x|)|t=|x| , ϕ(x)
)
+
∞∫
|x|
Mtt(|x|, t)f(t)dt, ϕ(x)
−
f ′(x) signx
1
2
∞∫
|x|
q(ξ)dξ, ϕ(x)
.
Taking into account that q(|x|) = Q(x), x ∈ R, we have
((Mf)′′, ϕ) = −((Mf)′, ϕ′)
= −
f ′(x) + signx
∞∫
|x|
Mx(|x|, t)f(t)dt− signxf(x)
1
2
∞∫
|x|
q(ξ)dξ, ϕ′(x)
= (f ′′, ϕ) +
2δ(x)
∞∫
0
Mx(0, t)f(t)dt, ϕ(x)
+
∞∫
|x|
Mxx(|x|, t)f(t)dt, ϕ(x)
−
(
f(x) Mx(|x|, t)|t=|x| , ϕ(x)
)
−
δ(x)f(0)
∞∫
0
q(ξ)dξ, ϕ(x)
−
f ′(x) signx
1
2
∞∫
|x|
q(ξ)dξ, ϕ(x)
+
1
2
(f(x)Q(x), ϕ(x)).
Thus,
(M(f ′′), ϕ)− ((Mf)′′, ϕ) =
∞∫
|x|
[Mtt(|x|, |t|)− Mxx(|x|, |t|)]f(t)dt, ϕ(x)
+
(
f(x)
[
Mx(t, |x|)|t=|x| + Mx(|x|, t)|t=|x|
]
, ϕ(x)
)
− 1
2
(f(x)Q(x), ϕ(x))
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 331
K.S. Khalina
−
2δ(x)
∞∫
0
Mx(0, t)f(t)dt, ϕ(x)
+
δ(x)f(0)
∞∫
0
q(ξ)dξ, ϕ(x)
.
From (5.9), (5.10) it follows that Mtt(|x|, |t|)− Mxx(|x|, |t|) = −q(|x|)M(|x|, t) when
|x| < |t| and Mx(t, |x|)|t=|x| + Mx(|x|, t)|t=|x| = M′(|x|, |x|) = −1/2q(|x|). Conse-
quently,
(M(f ′′)− (Mf)′′, ϕ)
= −
Q(x)
∞∫
|x|
M(|x|, t)f(t)dt, ϕ(x)
− (f(x)Q(x), ϕ(x))
−
2δ(x)
∞∫
0
Mx(0, t)f(t)dt, ϕ(x)
+
δ(x)f(0)
∞∫
0
q(ξ)dξ, ϕ(x)
=
−Q(x)(Mf)(x)− 2δ(x)
∞∫
0
Mx(0, t)f(t)dt + δ(x)f(0)
∞∫
0
q(ξ)dξ, ϕ(x)
from which the assertion of the lemma follows. The lemma is proved.
Lemma 5.7. Let g ∈ H1
0,e. Then
M−1(g′′) = (M−1f)′′ + M−1(Qf)− 2δ
∞∫
0
Nx(0, ξ)f(ξ)dξ − δf(0)
∞∫
0
q(ξ)dξ.
P r o o f. Consider any ψ ∈ H1
0,e. As in the previous lemma, using (2.6),
(5.30), (5.17) and the evenness of g and ψ, we obtain
(M−1(g′′), ψ) = (g′′, ψ) +
∞∫
|x|
Ntt(|x|, t)g(t)dt, ψ(x)
+
(
g(x) Nx(t, |x|)|t=|x| , ψ(x)
)
+
g′(x) signx
1
2
∞∫
|x|
q(ξ)dξ, ψ(x)
,
((M−1g)′′, ψ) = (g′′, ψ) +
2δ(x)
∞∫
0
Nx(0, t)g(t)dt, ψ(x)
− 1
2
(g(x)Q(x), ψ(x))
332 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
+
∞∫
|x|
Nxx(|x|, t)g(t)dt, ψ(x)
−
(
g(x) Nx(|x|, t)|t=|x| , ψ(x)
)
+
δ(x)g(0)
∞∫
0
q(ξ)dξ, ψ(x)
+
g′(x) signx
1
2
∞∫
|x|
q(ξ)dξ, ψ(x)
.
Using (5.16), (5.17), we obtain
(M−1(g′′)− (M−1g)′′, ψ) =
∞∫
|x|
[Ntt(|x|, |t|)− Nxx(|x|, |t|)]g(t)dt, ψ(x)
+
(
g(x)
[
Nx(t, |x|)|t=|x| + Nx(|x|, t)|t=|x|
]
, ψ(x)
)
+
1
2
(g(x)Q(x), ψ(x))
−
2δ(x)
∞∫
0
Nx(0, t)g(t)dt, ψ(x)
−
δ(x)g(0)
∞∫
0
q(ξ)dξ, ψ(x)
=
∞∫
|x|
Q(t)N(|x|, t)g(t)dt, ψ(x)
+ (g(x)Q(x), ψ(x))
−
2δ(x)
∞∫
0
Nx(0, t)g(t)dt, ψ(x)
−
δ(x)g(0)
∞∫
0
q(ξ)dξ, ψ(x)
=
(M−1(Qg))(x)− 2δ(x)
∞∫
0
Nx(0, t)g(t)dt− δ(x)g(0)
∞∫
0
q(ξ)dξ, ψ(x)
.
The lemma is proved.
Lemma 5.8. Let f ∈ H0
0,e such that f ∈ L∞(R). Then Mf, M−1f ∈ L∞(R).
P r o o f. Using (2.5) and (5.23), we get
|Mf | ≤ |f |
1 +
∣∣∣∣∣∣∣
∞∫
|x|
|M(|x|, t)|dt
∣∣∣∣∣∣∣
≤ |f |
1 +
e2σ1(0)
2
∣∣∣∣∣∣∣
∞∫
|x|
σ
( |x|+ t
2
)
dt
∣∣∣∣∣∣∣
≤ |f |
(
1 + e2σ1(0)σ1(0)
)
< ∞.
The assertion about the operator M−1 is proved in a similar way. The lemma is
proved.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 333
K.S. Khalina
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