On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment
The control system wtt = wxx - q(x)w, wx(0; t) = u(t), wx(d, t) = 0, x is in (0; d), t is in (0; T), d > 0, 0 < T ≤ d is considered. Here q is in C¹[0, d], q > 0, q'₊(0) = q₋(d) = 0, u is a control, |u(t)| ≤ 1 on (0, T). The necessary and suffcient conditions of null-controllability an...
Gespeichert in:
| Datum: | 2011 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2011
|
| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/106688 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment / K.S. Khalina // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 4. — С. 333-351. — Бібліогр.: 23 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-106688 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1066882016-10-03T03:02:20Z On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment Khalina, K.S. The control system wtt = wxx - q(x)w, wx(0; t) = u(t), wx(d, t) = 0, x is in (0; d), t is in (0; T), d > 0, 0 < T ≤ d is considered. Here q is in C¹[0, d], q > 0, q'₊(0) = q₋(d) = 0, u is a control, |u(t)| ≤ 1 on (0, T). The necessary and suffcient conditions of null-controllability and approximate null-controllability are obtained for this system. The controllability problems are considered in the modified Sobolev spaces. The controls that solve these problems are found explicitly. It is proved that among the solutions of the Markov trigonometric moment problem there are bang-bang controls solving the approximate null-controllability problem. 2011 Article On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment / K.S. Khalina // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 4. — С. 333-351. — Бібліогр.: 23 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106688 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The control system wtt = wxx - q(x)w, wx(0; t) = u(t), wx(d, t) = 0, x is in (0; d), t is in (0; T), d > 0, 0 < T ≤ d is considered. Here q is in C¹[0, d], q > 0, q'₊(0) = q₋(d) = 0, u is a control, |u(t)| ≤ 1 on (0, T). The necessary and suffcient conditions of null-controllability and approximate null-controllability are obtained for this system. The controllability problems are considered in the modified Sobolev spaces. The controls that solve these problems are found explicitly. It is proved that among the solutions of the Markov trigonometric moment problem there are bang-bang controls solving the approximate null-controllability problem. |
| format |
Article |
| author |
Khalina, K.S. |
| spellingShingle |
Khalina, K.S. On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment 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 Segment |
| title_short |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment |
| title_full |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment |
| title_fullStr |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment |
| title_full_unstemmed |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment |
| title_sort |
on the neumann boundary controllability for the non-homogeneous string on a segment |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106688 |
| citation_txt |
On the Neumann Boundary Controllability for the Non-Homogeneous String on a Segment / K.S. Khalina // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 4. — С. 333-351. — Бібліогр.: 23 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khalinaks ontheneumannboundarycontrollabilityforthenonhomogeneousstringonasegment |
| first_indexed |
2025-07-07T18:52:09Z |
| last_indexed |
2025-07-07T18:52:09Z |
| _version_ |
1837015320415436800 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 4, pp. 333–351
On the Neumann Boundary Controllability for the
Non-Homogeneous String on a Segment
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 April 27, 2011
The control system wtt = wxx − q(x)w, wx(0, t) = u(t), wx(d, t) = 0,
x ∈ (0, d), t ∈ (0, T ), d > 0, 0 < T ≤ d is considered. Here q ∈ C1[0, d],
q > 0, q′+(0) = q′−(d) = 0, u is a control, |u(t)| ≤ 1 on (0, T ). The neces-
sary and sufficient conditions of null-controllability and approximate null-
controllability are obtained for this system. The controllability problems
are considered in the modified Sobolev spaces. The controls that solve these
problems are found explicitly. It is proved that among the solutions of the
Markov trigonometric moment problem there are bang-bang controls solving
the approximate null-controllability problem.
Key words: wave equation, controllability problem, Neumann control
bounded by a hard constant, modified Sobolev space, Sturm–Liouville prob-
lem, tranformation operator.
Mathematics Subject Classification 2000: 93B05, 35B37, 35L05, 34B24.
1. Introduction
In the paper, the wave equation for a non-homogeneous string on a segment
is considered. The velocity of the string is fixed at the right end point. At the
left end point a control of the Neumann type is applied. The control is bounded
by a hard constant. We assume that the potential q is not a constant, gene-
rally speaking. The time T is constrained (T ∈ (0, d]). The problems of null-
controllability and approximate null-controllability are studied in the spaces Hs
Q,
s ≤ 1 (the modified Sobolev spaces of even periodic functions under the operator
(1 + D2 + q(x))s/2 instead of (1 + D2)s/2, D = −id/dx).
c© K.S. Khalina, 2011
K.S. Khalina
Note that many results are available for controllability problems for hyperbolic
partial differential equations (see [1–16] and others). We describe the papers most
similar to ours.
The wave equation for a homogeneous string on a finite segment is well stud-
ied. The boundary Lp-controllability (2 ≤ p ≤ ∞) with the Dirichlet control
for the equation was considered in [1–7] and others. In [8], the problem of exact
null-controllability for the wave equation was considered in a bounded domain
Ω ⊂ Rm with the Neumann boundary control. The potential of the string was
equal to zero. The problem was considered in L2(Ω)× (
H1
0 (Ω)
)′. The controlla-
bility results were obtained for the controls from a special class larger than L2.
Note that the potential q cannot be equal to zero in the present work.
The problems of boundary controllability for a non-homogeneous string were
studied in a bounded domain Ω ⊂ Rn in [9, 10] and on a segment in [11, 12]. In
all these papers, the Dirichlet controls were extended on a part of a boundary and
were of the class L2 in [9, 10], of the class W 1
2 in [11], and of the class L∞ in [12].
Note that in [11] the potential q was equal to a constant, and in [12], the potential
was not equal to a constant, generally speaking. In [13], the problem of null- and
approximate null-controllability for the wave equation was considered on a half-
axis. The potential q was equal to a constant. The Neumann control at the point
x = 0 was applied. The control was bounded by a hard constant. The problems
were considered in the Sobolev spaces Hs
0 , s ≤ 1. The necessary and sufficient
conditions of null- and approximate null-controllability for the considered system
were obtained. An explicit formula for the control was also found. Notice that if
we put q = const > 0 in the present paper, then the results are close to those of
[13]. Note also that the condition q 6= const is essential distinction of the present
paper from those where q = const, and it makes the study more complicated.
To study the controllability problems for the given wave equation we apply the
method used in [12], namely, we apply the operators adjoint to the transformation
operators for the Sturm–Liouville problem on a considered segment. We extend
the transformation operators and their adjoints to Hs
Q, s ∈ R, and study them.
As the investigation of the mentioned operators in Hs
Q, s ∈ R, is fundamental
and very voluminous, it is carried out in Appendix. We also prove that Hs
Q is
equivalent to the standard Sobolev space of even periodic functions. For the
proof, the theorem (Hörmander, [14]) about a transformation of the Sobolev
space under pseudo-differential operator is used. The application of the adjoint
operators to the given control system allows to obtain an explicit formula for
the control (that belongs to L∞) and the necessary and sufficient conditions of
null- and approximate null-controllability. In the paper, it is also proved that the
control, the initial and the corresponding steering states of the control system are
connected. This connection is described by the mentioned operators. We also
prove that the system is (approximately) null-controllable from not an arbitrary
334 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
initial state. The initial state of the string depends on its initial velocity, and
this dependence is also described by the operators adjoint to the transformation
operators. The distinctions of [12] from the present paper are the following: in
[12], the control was of the Dirichlet type, therefore the control problem was
considered in the modified Sobolev spaces of odd periodic functions under the
operator (1 + D2 + q(x))s/2, s ≤ 0. That is why the transformation operators
considered in the present paper have different than in [12] kernels with different
properties.
We also construct the bang-bang controls that solve the approximate null-
controllability problem for s < 1/2. We prove that these controls are the so-
lutions of the Markov trigonometric moment problem on (0, T ). Note that the
construction of the bang-bang controls as the solutions of the Markov trigono-
metric moment problem on (0, T ) was applied in [15]. The authors studied the
wave equation with q = 0 on a half-axis in the Sobolev spaces Hs
0 , s ≤ 0. A
control of the Dirichlet type and of the class L∞ was considered. The authors
reduced the approximate null-controllability problem for s < −1/2 to the Markov
trigonometric moment problem on (0, T ). We use their method of reducing. We
should note that the distinctions of the present paper from [15] are the following:
the potential q is not equal to zero, we consider the wave equation on a finite
segment, and the control is of the Neumann type (therefore the bang-bang con-
trols solve the approximate null-controllability problem for s < 1/2). Further,
the Markov trigonometric moment problem can be solved by the algorithm given
in [16]. In [13], the approximate null-controllability problem was studied for the
wave equation with the potential q = const on a half-axis. The equation was
controlled by the control of the Neumann type. This problem was considered
in Hs
0 , s ≤ 1. The bang-bang controls were constructed as the solutions of the
Markov power moment problem on (0, T ). The bang-bang controls were proved
to solve the approximate null-controllability problem for s < 1/2.
2. Notation
Consider the wave equation on a finite segment
wtt(x, t) = wxx(x, t)− q(x)w(x, t), x ∈ (0, d), t ∈ (0, T ), (2.1)
controlled by the boundary conditions
wx(0, t) = u(t), wx(d, t) = 0, t ∈ (0, T ), (2.2)
where d > 0, 0 < T ≤ d, and u is a control. We suppose that
q ∈ E(0, d) =
{
r ∈ C1[0, d] : r(x) > 0, r′+(0) = r′−(d) = 0
}
,
u ∈ B(0, T ) = {v ∈ L∞(0, T ) : |v(t)| ≤ 1 a.e. on (0, T )} .
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 335
K.S. Khalina
Consider control system (2.1), (2.2) with the initial conditions
(
w
wt
)
(x, 0) =
(
w0
0
w0
1
)
(x) = w0(x). (2.3)
To introduce the spaces used in this work, we have to consider the Sturm–
Liouville problem on the segment (0, d)
Gv ≡ −v′′(x) + q(x)v(x) = λ2v(x), v′(0) = v′(d) = 0, x ∈ (0, d), (2.4)
where q ∈ E(0, d).
Let {µn = λ2
n}∞n=1 be a set of eigenvalues, and {yn(λn, x)}∞n=1 be a system of
the corresponding eigenfunctions of the operator G ≡ − (
d
dx
)2
+ q(x). As it is
known (see, e.g., [17]), the set of eigenvalues is countable, they are real, nonne-
gative and simple, and λn 6= 0, n = 1,∞. It is also known that {yn(λn, x)}∞n=1
are real and form an orthonormal basis in L2[0, d].
Let S be the Schwartz space [18]
S =
{
ϕ ∈ C∞ (R) : ∀m, l ∈ N ∪ 0∃Cml > 0∀x ∈ R
∣∣∣ϕ(m)(x)
(
1 + |x|2)l
∣∣∣ ≤ Cml
}
,
S′ be the dual space.
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.
Let Ω : S′ → S′ be the odd extension operator, Ξ : S′ → S′ be the even
extension operator. Thus (Ωf) (x) = f(x) − f(−x), (Ξf) (x) = f(x) + f(−x)
for f ∈ S′. Let Th be the translation operator: Thϕ(x) = ϕ(x + h), ϕ ∈ S and
(Thf, ϕ) = (f,T−hϕ), f ∈ S′, ϕ ∈ S.
We assume that q and yn(λn, ·), n = 1,∞, are defined on R and equal to
zero on R \ [0, d]. Denote Q =
∑
k∈Z T2dkΞq, Yn(λn, ·) =
∑
k∈Z T2dkΞyn(λn, ·),
n = 1,∞. Since q ∈ E(0, d), then Q ∈ C1(R). Introduce the operator D2
Q =
Q(x) + D2, where D = −id/dx. Thus D2
QYn(λn, x) = λ2
nYn(λn, x), n = 1,∞.
Consider the modified Sobolev spaces
Hs
Q =
{
f ∈ S′ : f is even and 2d-periodic,
(
1 + D2
Q
)s/2
f ∈ L2
loc(R)
}
, s ∈ R,
with the norm ‖f‖s
Q =
(
d∫
−d
∣∣∣∣
(
1 + D2
Q
)s/2
f(x)
∣∣∣∣
2
dx
)1/2
, and
Hs
0 =
{
f ∈ S′ : f(x) =
∞∑
n=1
fne−iλnx |
{
fn(1 + λ2
n)s/2
}∞
n=1
∈ l2
}
, s ∈ R,
336 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
with the norm ‖f‖s
0 =
(∑∞
n=1
∣∣∣
(
1 + λ2
n
)s/2
fn
∣∣∣
2
)1/2
, where λn are the arithmetic
roots of eigenvalues of the Sturm–Liouville problem (2.4). (For convenience in
further reasoning we will number {λn}∞n=1 in ascending order.)
We denote H̃s
Q = Hs
Q ×Hs−1
Q and use the norm
|||f |||sQ =
((
‖f1‖s
Q
)2
+
(
‖f2‖s−1
Q
)2
)1/2
for f =
(
f1
f2
)
∈ H̃s
Q.
R e m a r k 2.1. In Lemma A.1, it is proved that Hs
Q and Hs
0,per (the Sobolev
space of even 2d-periodic functions [19]), s ∈ R, coincide as sets, have equivalent
norms, and
Hs
Q =
{
f ∈ S′ : f(x) =
+∞∑
n=1
fnYn(λn, x) ∧
{
fn(1 + λ2
n)s/2
}+∞
n=1
∈ l2
}
, s ∈ R,
with the norm ‖f‖s
Q =
(∑∞
n=1
∣∣fn(1 + λ2
n)s/2
∣∣2
)1/2
.
R e m a r k 2.2. The series of exponentials in the definition of Hs
0 converges
with respect to the norm of the standard Sobolev space Hs
0 . In fact, it is proved
in Lemma A.2. If f ∈ Hs
0 is even, then f(x) =
∑∞
n=1 fn cosλnx. One can find
the properties of the functions from Hs
0 in Lemmas A.2–A.5.
Denote by Hs
Q(a, b) and Hs
0(a, b) the restrictions of the spaces Hs
Q and Hs
0 to
(a, b), respectively.
R e m a r k 2.3. It is proved in [12, Lemma A.1] that the system
{
eiλnx
}∞
n=1
is the Riesz basis in the space L2(−d, d). Hence H0
Q(−d, d) =
{
f ∈ H0
0(−d, d) :
f is even}.
Further, throughout the paper we will assume that s ≤ 1.
3. Definitions
Consider control system (2.1)–(2.3), where w0 ∈ H̃s
Q(0, d). We consider the
solution of (2.1)–(2.3) in the space Hs
Q.
Extend w(x, t) and w0(x) from the segment (0, d) on the whole axis. Consider
the even 2d-periodic extensions (with respect to x)
W (·, t) =
∑
k∈Z
T2dkΞw(·, t), W 0 =
∑
k∈Z
T2dkΞw0, t ∈ (0, T ).
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 337
K.S. Khalina
Obviously, W 0 ∈ H̃s
Q, W (·, t) ∈ Hs
Q (t ∈ (0, T )).
It is easy to see that control problem (2.1)–(2.3) is equivalent to the following
problem:
Wtt(x, t) = Wxx(x, t)−Q(x)W (x, t)− 2u(t)
∑
k∈Z
T2dkδ(x), x ∈ R, t ∈ (0, T ),
(3.1)
W (x, 0) = W 0
0 (x), Wt(x, 0) = W 0
1 (x), x ∈ R, (3.2)
where δ is the Dirac distribution. Consider this system with the steering condi-
tions
W (x, T ) = W T
0 (x), Wt(x, T ) = W T
1 (x), x ∈ R, (3.3)
where W T =
(
W T
0
W T
1
)
∈ H̃s
Q. We consider the solution of (3.1)–(3.3) in Hs
Q.
Let T > 0, w0 ∈ H̃s
Q(0, d). Denote by RT (w0) the set of the states W T ∈ H̃s
Q
for which there exists a control u ∈ B(0, T ) such that problem (3.1)–(3.3) has a
unique solution in Hs
Q.
Definition 3.1. A state w0 ∈ H̃s
Q(0, d) is called null-controllable at a given
time T > 0 if 0 belongs to RT (w0) and approximately null-controllable at a given
time T > 0 if 0 belongs to the closure of RT (w0) in H̃s
Q.
Definition 3.2. Denote by ST : H̃
p
Q → H̃
p
Q, p ∈ R, the operator
(ST f)(x) =
∞∑
n=1
(
cosλnT sin λnT
λn
−λn sinλnT cosλnT
)(
f1
n
f2
n
)
Yn(λn, x),
where f(x) =
(
f1(x)
f2(x)
)
=
∑∞
n=1
(
f1
n
f2
n
)
Yn(λn, x), D(ST ) = R(ST ) = H̃
p
Q.
The operator ST is linear and continuous (the proof of these facts is similar
to the proof of [12, Lemma A.3]). It was also proved there that
|||ST f |||sQ ≤ CS |||f |||sQ , (3.4)
where C2
S = 2 max{2, 2T 2 + 1}.
Definition 3.3. Denote by ∂−1 : Hp
0 → Hp+1
0 , p ∈ R, the operator
∂−1f = i
∞∑
n=1
fn
1
λn
e−iλnx,
where f =
∑∞
n=1 fne−iλnx, D(∂−1) = Hp
0, R(∂−1) = Hp+1
0 .
338 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
In Lemma A.6, ∂−1 is proved to be continuous.
Further we define the transformation operators for the Sturm–Liouville prob-
lem on a segment. As it is known from [20], the integral operator (Kf)(x) =
f(x) +
∫ x
0 K(x, t; 0)f(t) dt transfers the solution of the Cauchy problem y′′ +
λ2y = 0, y(0) = 1, y′(0) = 0 on [−d, d] to the solution of the Cauchy problem
y′′−q(x)y+λ2y = 0, y(0) = 1, y′(0) = 0 on [−d, d]. According to [20], the operator
K has the inverse one that we denote by L (see Appendix B for details). In the
present paper we determine these operators for p ∈ R in the following spaces:
K : H−p
0 (−d, d) −→ H
−p
Q (−d, d), L : H
−p
Q (−d, d) −→ H−p
0 (−d, d), where D(K) ={
f ∈ H−p
0 (−d, d) : f is even
}
, D(L) = H
−p
Q (−d, d), R(K) = D(L), R(L) = D(K)
(see Lemma B.1). We prove that these operators are linear and isometric in
Lemma B.2. In Definition B.1 (Appendix B), we also determine the adjoint ope-
rators K∗ : H
p
Q(−d, d) −→ Hp
0(−d, d), L∗ : Hp
0(−d, d) −→ H
p
Q(−d, d), where
D(K∗) = H
p
Q(−d, d), D(L∗) = {f ∈ Hp
0(−d, d) : f is even}, R(K∗) = D(L∗),
R(L∗) = D(K∗). They are also linear and isometric. The properties of the
operators K, L, K∗, L∗ are studied in Appendix B.
4. Controllability Conditions
The following two theorems are the main result of the paper. Theorem 4.1
gives the description of the set RT (w0), and Theorem 4.2 gives the necessary and
sufficient conditions of null-controllability and approximate null-controllability
for the initial state of (2.1)–(2.3).
Theorem 4.1. Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s ≤ 1. Then
RT (w0) =
{
ST
[
W 0 −
∑
k∈Z
T2kdL∗
(−∂−1ΩU
ΞU
)]
: u ∈ B(0, T )
}
, (4.1)
where U = u on [0, T ] and U = 0 on R \ [0, T ].
P r o o f. The proof of the theorem is similar to that of [12, Theorem 3.1].
Therefore we only give its scheme. According to Lemma A.1, we have
W (x, t) =
∞∑
n=1
wn(t)Yn(λn, x),
∑
k∈Z
T2dkδ(x) =
∞∑
n=1
δnYn(λn, x), (4.2)
W γ
0 (x) =
∞∑
n=1
wγ
0nYn(λn, x), W γ
1 (x) =
∞∑
n=1
wγ
1nYn(λn, x), γ = 0, T, (4.3)
where wn(t) = (w(·, t), yn(λn, ·)), δn = 1
2 (δ, Yn(λn, ·)) = 1
2 , wγ
0n = (wγ
0 (·), yn(λn, ·)),
wγ
1n = (wγ
1 (·), yn(λn, ·)), γ = 0, T , n = 1,∞.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 339
K.S. Khalina
Substituting (4.2) and (4.3) into (3.1)–(3.3), we obtain
w′′n(t) + wn(t)λ2
n = −u(t), t ∈ [0, T ], n = 1,∞. (4.4)
wn(0) = w0
0n,
∂wn
∂t
(0) = w0
1n, wn(T ) = wT
0n,
∂wn
∂t
(T ) = wT
1n. (4.5)
It is easy to see that system (4.4), (4.5) is equivalent to the linear system (n =
1,∞)
v′n(t) = Anvn(t) + bn(t), t ∈ [0, T ],
vn(0) =
(
w0
0n
w0
1n
)
, vn(T ) =
(
wT
0n
wT
1n
)
,
where vn =
(
wn
w′n
)
, An =
(
0 1
−λ2
n 0
)
, bn(t) =
(
0
−u(t)
)
. Thus we have for
n = 1,∞
(
cosλnT sin λnT
λn
−λn sinλnT cosλnT
) (
w0
0n +
∫ T
0
sin λnt
λn
u(t) dt
w0
1n −
∫ T
0 cosλnt · u(t) dt
)
=
(
wT
0 n
wT
1 n
)
. (4.6)
Since u ∈ B(0, T ) ⊂ L2(0, d), then U ∈ L2(−d, d). Using Lemmas A.3, A.4
and A.6, we conclude that U ∈ H0
0(−d, d), ΩU ∈ H0
0(−d, d), ΞU ∈ H0
0(−d, d),
(ΩU)′ = ΞU ′ ∈ H−1
0 (−d, d). Hence we can apply the operator L∗ to ΞU and ΞU ′.
Reasoning similarly to [12], we get
T∫
0
sinλnt
λn
u(t) dt =
1
2λ2
n
(
(ΩU)′ , cosλnt
)
=
1
λ2
n
(L∗ΞU ′, yn
)
,
T∫
0
cosλntu(t) dt =
1
2
(ΞU, cosλnt) = (L∗ΞU, yn) .
Denote un = (L∗ΞU, yn), u′n = (L∗ΞU ′, yn). Thus (4.6) is equivalent to the
equality
(
cosλnT sin λnT
λn
−λn sinλnT cosλnT
) (
w0
0n + u′n
λ2
n
w0
1n − un
)
=
(
wT
0 n
wT
1 n
)
, n = 1,∞. (4.7)
Due to Lemma A.1, we get
∑
k∈Z
T2kd (L∗ΞU) (x) =
∞∑
n=1
unYn(λn, x),
∑
k∈Z
T2kd
(L∗ΞU ′) (x) =
∞∑
n=1
u′nYn(λn, x).
340 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Therefore, using Definition 3.3, from (4.7) we obtain
ST
[
W 0 −
∑
k∈Z
T2kdL∗
(−∂−2ΞU ′
ΞU
)]
= W T .
Since ΞU ′ = (ΩU)′ = ∂ΩU , we obtain (4.1). The theorem is proved.
Theorem 4.2. Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s ≤ 1. Then the following
statements are equivalent:
(i) the state w0 is null-controllable at the time T ;
(ii) the state w0 is approximately null-controllable at the time T ;
(iii) the conditions below hold
suppw0
1 ⊂ [0, T ], (4.8)
w0
1 ∈ L∞(0, d) and
∣∣(K∗Ξw0
1
)
(x)
∣∣ ≤ 1 a.e. on [−d, d], (4.9)
Ξw0
0 = −L∗∂−1
(
sign tK∗Ξw0
1
)
. (4.10)
In addition, the control u is given by the formula
u(t) = w0
1(t) +
T∫
t
K(x, t; 0)w0
1(x) dx, t ∈ [0, T ]. (4.11)
The proof of this theorem is rather similar to the proof of [12, Theorem 3.2],
but formula (4.1) is used instead of the corresponding formula in [12], as well as
Lemmas A.3, A.5, B.2, B.4, B.5, and the Riesz theorem.
R e m a r k 4.1. We have from (4.11) that there exists U > 0 such that
|u| ≤ U on (0, T ) iff there exists V > 0 such that
∣∣w0
1
∣∣ ≤ V on (0, d).
R e m a r k 4.2. Let q(x) ≡ q = const > 0 on (0, d). Hence, Q(x) ≡ q on
(−d, d). Find the kernels L(x, t; 0) and K(x, t; 0) of the transformation operators
on (−d, d)× (−d, d). We have K(x, t; 0) = K(x, t) + K(x,−t), L(x, t; 0) = L(x, t) +
L(x,−t), where K(x, t) and L(x, t) are the solutions of the following systems (see
Appendix B):
Kxx(x, t)− Ktt(x, t) = qK(x, t), Lxx(x, t)− Ltt(x, t) = −qL(x, t),
K(x, x) =
1
2
qx, K(x,−x) = 0, L(x, x) = −1
2
qx, L(x,−x) = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 341
K.S. Khalina
It is proved in [12] that
K(x, t; 0) = qx
I1
(√
q(x2 − t2)
)
√
q(x2 − t2)
, |t| < |x|, K(x, t; 0) = 0, |t| ≥ |x|;
L(x, t; 0) = −qx
J1
(√
q(x2 − t2)
)
√
q(x2 − t2)
, |t| < |x|, L(x, t; 0) = 0, |t| ≥ |x|,
where Jm(z) is the Bessel function, Im(z) = i−mJm(iz) is the modified Bessel
function, m ∈ Z. Thus in the case of q = const, we get
u(t) = w0
1(t) + q
T∫
t
xI1
(√
q(x2 − t2)
)
√
q(x2 − t2)
w0
1(x) dx, t ∈ (0, T ),
w0
1(t) = u(t)− q
T∫
t
xJ1
(√
q(x2 − t2)
)
√
q(x2 − t2)
u(x) dx, t ∈ (0, T ).
One can see that if
∣∣w0
0
∣∣ ≤ Cw on (0, d), then |u| ≤ CwI0(
√
qT ) and if |u| ≤ Cu
on (0, T ), then
∣∣w0
0
∣∣ ≤ Cu
(
1 +
√
qT
)
. Note that similar formulas were obtained
for the semi-infinite string with q = const in [13].
E x a m p l e 4.1. Assume that d = 24, T = 20, q(x) ≡ q > 0, x ∈ (0, 24). Let
w0
1(x) = x2
3·202I0(20
√
q)
on (0, 20), w0
1(x) = 0 on (20, 24). Let w0
0(x) such that Ξw0
0 =
−L∗∂−1
(
signxK∗Ξw0
1
)
on (−24, 24). Thus,
∣∣w0
1(x)
∣∣ ≤ 202
3·202I0(20
√
q)
= 1
3I0(20
√
q) .
Due to Remark 4.2, we get the estimate
∣∣K∗Ξw0
1(x)
∣∣ ≤ 1
3I0(20
√
q)I0(20
√
q) = 1
3 .
Consequently, proposition (iii) of Theorem 4.2 holds. Hence the state w0 is null-
controllable at the time T = 20. From the formula for a control obtained in
Remark 4.2, we get
u(t) =
202qI0(
√
q(202 − t2))− 2
√
q(202 − t2)I1(
√
q(202 − t2))
3 · 202qI0(20
√
q)
, t ∈ (0, 20).
5. The Moment Problem
We consider the Markov trigonometric moment problem on (0, T ) in this
section. The main goal of the section is to prove that among the solutions of
the moment problem there are bang-bang solutions of the approximate null-
controllability problem for s < 1/2. As stated in Introduction, we use the meth-
ods from [15] and [16] to prove the results of this section. The proofs of theorems
have the same scheme as the proofs of the corresponding theorems in [12].
342 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Consider control system (2.1)–(2.3). Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s ≤ 1.
Assume that proposition (iii) of Theorem 4.2 holds. Put
ωm =
T∫
0
ei πmx
d
(K∗Ξw0
1
)
(x) dx, m = −∞,∞. (5.1)
The problem of determination of a function u ∈ B(0, T ) such that
T∫
0
ei πmx
d u(x) dx = ωm, m = −∞,∞, (5.2)
for a given {ωm}∞m=−∞ and T > 0 is called the Markov trigonometric moment
problem on (0, T ) for the infinite sequence {ωm}∞m=−∞.
Consider (5.2) for a finite set of m. The problem of determination of a function
u ∈ B(0, T ) such that
T∫
0
ei πmx
d u(x) dx = ωm, m = −M, M, M ∈ N, (5.3)
for a given {ωm}M
m=−M and T > 0 is called the Markov trigonometric moment
problem on (0, T ) for the finite sequence {ωm}M
m=−M .
The following theorem is proved similarly to [12, Theorem 5.1].
Theorem 5.1. Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s ≤ 1. Assume that proposition
(iii) of Theorem 4.2 holds. Define the sequence {ωm}∞m=−∞ by (5.1). Then the
state w0 is null-controllable at the time T iff the Markov trigonometric moment
problem (5.2) has a unique solution on (0, T ). Moreover, this solution is of the
form (4.11).
R e m a r k 5.1. Since the system {e−i πmx
d }∞m=−∞ forms an orthonormal basis
in L2(−d, d), then the moment problem (5.2) has a unique solution in L2(−d, d).
If T < d, then the system {e−i πmx
d }∞m=−∞ is complete in L2(−T, T ), and the
solution is unique, if exists. Therefore the solution of (5.2) is in B(0, T ), and
it is unique, if exists, and coincides with K∗Ξw0
1. Thus Theorem 5.1 is close to
Theorem 4.2.
R e m a r k 5.2. It is evident that u of the form (4.11) is a solution of problem
(5.3), but it is not unique.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 343
K.S. Khalina
Theorem 5.2. Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s < 1/2. Assume that
proposition (iii) of Theorem 4.2 holds. Define the sequence {ωm}∞m=−∞ by (5.1).
Let some M ∈ N. If uM ∈ B(0, T ) is the solution of the Markov trigonometric
moment problem (5.3), then the state w0 is approximately null-controllable at the
time T , and the following estimate is valid:
∣∣∣∣∣∣W T
∣∣∣∣∣∣s
Q
≤
2
5
2 πs−1CS
√
C2
∂−1 + 1PTM s− 1
2
ds
√−2s + 1
, (5.4)
where W is the corresponding solution of control system (3.1)–(3.3), CS > 0 is the
constant from estimate (3.4), C∂−1 > 0 is the constant from Lemma A.6, P > 0
is the constant from Remark A.1.
P r o o f. Since proposition (iii) of Theorem 4.2 holds, then there exists
the solution ũ ∈ B(0, T ) of the controllability problem of system (2.1)–(2.3).
Put Ũ = ũ on [0, T ], Ũ = 0 on R \ [0, T ]. Then Ξw0 = L∗
(
−∂−1ΩŨ
ΞŨ
)
. Let
uM ∈ B(0, T ) be the solution of the Markov trigonometric moment problem (5.3)
on (0, T ) for the finite sequence {ωm}M
m=−M . Put UM = uM on [0, T ], UM = 0
on R \ [0, T ]. From (4.1) we have
W T = ST
∑
k∈Z
T2kdL∗
−∂−1Ω
(
Ũ − UM
)
Ξ
(
Ũ − UM
)
,
where W is the solution of (3.1)–(3.3). Using (3.4) and Lemmas A.4, A.6 and B.3,
we obtain ∣∣∣∣∣∣W T
∣∣∣∣∣∣s
Q
≤ 2CS
√
C2
∂−1 + 1
∥∥∥
(
Ũ − UM
)∥∥∥
s−1
0
. (5.5)
Since Ũ and UM belong to L∞(−d, d), then we can consider the
following series expansions on (−d, d): Ũ(x) = 1
d
∑∞
m=−∞ ωme−i πmx
d , UM (x) =
1
d
∑∞
m=−∞ νme−i πmx
d , where
ωm =
d∫
−d
ei πmx
d Ũ(x) dx =
T∫
0
ei πmx
d
(K∗Ξw0
1
)
(x) dx,
νm =
d∫
−d
ei πmx
d UM (x) dx =
T∫
0
ei πmx
d u(x) dx.
Hence,
Ũ(x)− UM (x) =
1
d
∞∑
m=−∞
(ωm − νm) e−i πmx
d . (5.6)
344 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Using Remark A.1 and reasoning similarly to the proof of [12, Theorem 5.2], we
obtain the estimate
∥∥∥Ũ − UM
∥∥∥
s−1
0
≤ 2
√
2πs−1PTM s− 1
2
ds
√−2s + 1
, s < 1/2, P > 0. (5.7)
Substituting (5.7) into (5.5), we obtain (5.4).
It is clear from (5.4) that
∣∣∣∣∣∣W T
∣∣∣∣∣∣s
Q
→ 0 as M → ∞. Thus the state w0 is
approximately null-controllable at the given time T . The theorem is proved.
Consider the set of bang-bang controls. Denote
BM (0, T ) = {uM ∈ B(0, T )|∃T∗ ∈ (0, T ) : (|uM | = 1 a.e. on (0, T∗)) ,
(uM = 0 a.e. on (T∗, T ))
(uM has not more than M discontinuities on (0, T∗))} .
Theorem 5.3. Let 0 < T ≤ d, w0 ∈ H̃s
Q(0, d), s < 1/2. Assume that
assertion (iii) of Theorem 4.2 holds. Define the sequence {ωm}∞m=−∞ by (5.1).
Then for all ε > 0 there exists M ∈ N such that for this M there exists a solution
uM ∈ BM (0, T ) of the Markov trigonometric moment problem (5.3). The number
M is defined from the condition
2
5
2 πs−1CS
√
C2
∂−1 + 1PTM s− 1
2
ds
√−2s + 1
< ε.
Moreover, control system (2.1)–(2.3) is approximate null-controllable at the time
T (the estimate
∣∣∣∣∣∣W T
∣∣∣∣∣∣s
Q
≤ ε is valid).
The proof of this theorem is absolutely similar to the proof of [12, Theorem
5.3].
A. The Spaces Hs
Q and Hs
0(−d, d), s ∈ R
Lemma A.1. The spaces Hs
Q and Hs
0,per (the Sobolev space of even 2d-periodic
functions, [19]), s ∈ R, coincide as sets, have equivalent norms, and
Hs
Q =
{
f ∈ S′ : f(x) =
+∞∑
n=1
fnYn(λn, x) and
{
fn(1 + λ2
n)s/2
}+∞
n=1
∈ l2
}
, s ∈ R,
(A.1)
where fn = 1
2 (f, Yn(λn, ·)) on (−d, d), with the norm
‖f‖s
Q =
( ∞∑
n=1
∣∣∣
(
1 + λ2
n
)s/2
fn
∣∣∣
2
)1/2
. (A.2)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 345
K.S. Khalina
P r o o f. We have to prove that
(
1 + |D|2)s/2
f ∈ L2(R) iff
(
1 + |D|2 + Q(x)
)s/2
f ∈ L2(R).
We use the following theorem from [14, Chap. 18]:
Theorem A.1 ((Hörmander, [14])). If a(x, ξ) ∈ Sm, then a(x,D) is a con-
tinuous operator from Hs
0 to Hs−m
0 for all s, where for m ∈ R
Sm = Sm(Rn × Rn)
=
{
a ∈ C∞(Rn × Rn) : ∀α, β |∂α
β a(x, ξ)| ≤ Cα
β (1 + |ξ|)m−|α|
}
,
∂α
β a(x, ξ) = ∂α
ξ ∂β
xa(x, ξ), x, ξ ∈ Rn.
Let us denote for x, ξ ∈ R
S̃m = S̃m(R× R) =
{
a ∈ Cx(R) ∩ C∞
ξ (R) : ∀α |∂α
ξ a(x, ξ)| ≤ Cα(1 + |ξ|)m−α
}
.
Analyzing the proof of Theorem A.1 we conclude that it remains true for a(x, ξ) ∈
S̃m. Thus, if we prove that a(x, ξ) =
(
1 + ξ2 + Q(x)
)s/2 ∈ S̃m, then we will have
that a(x, D) is a continuous operator from Hs
0 to Hs−m
0 for all s.
Since q ∈ E(0, d), then Q ∈ C1(R), and there exist mq > 0, Mq < ∞ such
that mq ≤ Q ≤ Mq on R. It is easy to prove that
√
m̂
2
(1 + |ξ|) ≤
√
1 + ξ2 + Q ≤
√
2M̂(1 + |ξ|),
where m̂ = min{mq, 1}, M̂ = max{Mq, 1}.
Using these two inequalities, one can easily prove that
(
1 + ξ2 + Q(x)
)s/2 ∈
S̃s. Therefore,
(
1 + |D|2 + Q(x)
)s/2 is the continuous operator from Hs
0 to H0
0 =
L2(R). This implies that f ∈ Hs
0 iff
(
1 + |D|2 + Q(x)
)s/2
f ∈ L2(R) as required.
We prove representation (A.1) with norm (A.2) for the space Hs
Q in the same
way as in the proof of [12, Lemma A.2].
It is easy to prove that
Hs
0,per =
{
f ∈ S′ : f(x) =
+∞∑
n=1
f1
n cos
πnx
d
and
{
f1
n(1 + n2)s/2
}+∞
n=1
∈ l2
}
, s ∈ R,
where f1
n = 1
d
(
f, cos πnx
d
)
on (−d, d) with the norm
‖f‖s
0 =
( ∞∑
n=1
∣∣∣
(
1 + n2
)s/2
f1
n
∣∣∣
2
)1/2
. (A.3)
346 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
It is known [21, chap. V] that λn = nπ
d + εn, where εn = O
(
1
n
)
, n = 1,∞.
It follows easily from here that norms (A.2) and (A.3) are equivalent. The lemma
is proved.
The proofs of the following four lemmas are absolutely similar to the proofs
of the corresponding lemmas in [12, Appendix A].
Lemma A.2. f ∈ Hs
0, s ∈ R iff
(
1 + D2
)s/2
f ∈ H0
0.
Lemma A.3. f ∈ H0
0(−d, d) iff f ∈ L2(−d, d).
Lemma A.4. Let g ∈ Hs
0, s ∈ R. Then we have Ωg ∈ Hs
0, Ξg ∈ Hs
0 and
‖Ωg‖s
0 = ‖Ξg‖s
0 = 2 ‖g‖s
0.
Lemma A.5. Hs
0 is dense in H0
0, s ≥ 0.
R e m a r k A.1. It follows from Lemma A.3 that there exist P, P1 > 0 such
that ‖f‖0
0 ≤ P ‖f‖L2 and ‖f‖L2 ≤ P1 ‖f‖0
0 for f ∈ H0
0(−d, d).
Lemma A.6. The operator ∂−1 : Hs
0 → Hs+1
0 , s ∈ R (see Definition 3.3) is
linear and continuous. If g is odd, then ∂−1g is even and if g is even, then ∂−1g
is odd.
P r o o f. Let s ∈ R. Let us denote by ∂ = d
dx : Hs+1
0 → Hs
0 the operator of dif-
ferentiation, D(∂) = Hs+1
0 , R(∂) = Hs
0. We have (∂g)(x) = −i
∑∞
n=1 λngne−iλnx
for g(x) =
∑∞
n=1 gne−iλnx. Using the trivial inequality λn <
√
1 + λ2
n, we get
‖∂g‖s
0 < ‖g‖s+1
0 for g ∈ Hs+1
0 . Thus the operator ∂ is linear and continuous. It is
also obvious that if g is odd, then ∂g is even and if g is even, then ∂g is odd.
Denote by ∂̂ : Hs
0 → Hs+1
0 the operator
(
∂̂g
)
(x) = i
∑∞
n=1
gn
λn
e−iλnx, where
g ∈ Hs
0, g(x) =
∑∞
n=1 gne−iλnx, D
(
∂̂
)
= Hs
0, R
(
∂̂
)
= Hs+1
0 . It also transfers
odd functions to even and vice versa. We have ∂̂∂f = f and ∂∂̂g = g for
f ∈ Hs+1
0 , g ∈ Hs
0. Therefore, ∂̂ = ∂−1, D(∂−1) = Hs
0, R(∂−1) = Hs+1
0 . Due
to the inverse operator theorem, we have that there exists C∂−1 > 0 such that∥∥∂−1g
∥∥s+1
0
≤ C∂−1 ‖g‖s
0. The lemma is proved.
B. The Transformation Operators for the Sturm–Liouville
Problem on a Segment and Their Adjoints
We make a quotation of definitions and properties of the transformation ope-
rators from [20]. Denote ỹn(λn, ·) = Ξyn(λn, ·), n = 1,∞. It is evident that
ỹn(λn, x) satisfies the following Cauchy problem for n = 1,∞:
−ỹ ′′n(λn, x) + Q(x)ỹn(λn, x) = λ2
nỹn(λn, x), x ∈ (−d, d),
ỹn(λn, 0) = 1, ỹ ′n(λn, 0) = 0.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 347
K.S. Khalina
Due to [20], we have
ỹn(λn, x) = K (cosλnt) (x) = cos λnx +
x∫
0
K(x, t; 0) cosλnt dt, n = 1,∞,
cosλnx = L (ỹn(λn, t)) (x) = ỹn(λn, x) +
x∫
0
L(x, t; 0)ỹn(λn, t) dt, n = 1,∞,
where K(x, t; 0) = K(x, t)+ K(x,−t), L(x, t; 0) = L(x, t)+ L(x,−t). Under the con-
dition Q ∈ C1[−d, d] the continuous functions K(x, t) and L(x, t) are the solutions
of the following systems on [−d, d]× [−d, d]:
Kxx(x, t)− Ktt(x, t) = Q(x)K(x, t), Lxx(x, t)− Ltt(x, t) = −Q(x)L(x, t),
K(x, x) =
1
2
x∫
0
Q(ξ) dξ, L(x, x) = −1
2
x∫
0
Q(ξ) dξ,
K(x,−x) = 0, L(x,−x) = 0.
It is also proved in [20] that the kernels K(x, t; 0), L(x, t; 0) are bounded functions
with respect to the both arguments on [−d, d]× [−d, d] and K(x, t) = L(x, t) = 0
for |t| ≥ |x|.
We determine the transformation operators in the spaces H−s
0 (−d, d) and
H−s
Q (−d, d) via series in the present paper.
Lemma B.1.
K : H−s
0 (−d, d) −→ H−s
Q (−d, d), L : H−s
Q (−d, d) −→ H−s
0 (−d, d), s ∈ R,
where D(K) =
{
f ∈ H−s
0 (−d, d) : f is even
}
, D(L) = H−s
Q (−d, d), R(K) = D(L),
R(L) = D(K).
P r o o f. Let s ∈ R. Let ψ ∈ H−s
Q (−d, d), ϕ ∈ H−s
0 (−d, d), ϕ be
even. Hence, ψ(x) =
∑∞
n=1 ψnΞyn(λn, x) and
{(
1 + λ2
n
)−s
2 ψn
}∞
n=1
∈ l2, ϕ(x) =
∑∞
n=1 ϕn cosλnx and
{(
1 + λ2
n
)−s
2 ϕn
}∞
n=1
∈ l2. Applying the transformation
operators, we obtain
(Kϕ) (x) =
∞∑
n=1
ϕnK (cosλnt) (x) =
∞∑
n=1
ϕnỹn(λn, x) = ψ̃(x),
(Lψ) (x) =
∞∑
n=1
ψnL (Ξyn(λn, t)) (x) =
∞∑
n=1
ψn cosλnt = ϕ̃(x).
Evidently, ϕ̃ ∈ H−s
0 (−d, d) and ψ̃ ∈ H−s
Q (−d, d). The lemma is proved.
348 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
Lemma B.2. The operators K and L are linear and isometric on their do-
mains.
P r o o f. Let s ∈ R and ϕ ∈ H−s
0 (−d, d) be even. Hence, ϕ(x) =
∑∞
n=1 ϕn cosλnx and
{(
1 + λ2
n
)−s
2 ϕn
}∞
n=1
∈ l2. According to Lemma B.1, Kϕ ∈
H−s
Q (−d, d) and (Kϕ) (x) =
∑∞
n=1 ϕnỹn(λn, x). Write down the norms:
(‖ϕ‖−s
0
)2
=
∑∞
n=1
∣∣∣ϕn
(
1 + λ2
n
)−s/2
∣∣∣
2
,
(
‖Kϕ‖−s
Q
)2
=
∑∞
n=1
∣∣∣ϕn
(
1 + λ2
n
)−s/2
∣∣∣
2
=
(‖ϕ‖−s
0
)2
.
Hence K is isometric from H−s
0 (−d, d) to H−s
Q (−d, d). One can see that L is also
isometric from H−s
Q (−d, d) to H−s
0 (−d, d).
The linearity of the operators is obvious. The lemma is proved.
Definition B.1. Define by K∗ and L∗ the adjoint operators for K and L:
(K∗f, ϕ) = (f,Kϕ), (L∗g, ψ) = (g,Lψ), where f ∈ D(K∗) = Hs
Q(−d, d), ϕ ∈
D(K), g ∈ D(L∗) = {f ∈ Hs
0(−d, d) : f is even}, ψ ∈ D(L), s ∈ R.
Thus, K∗ : Hs
Q(−d, d) → Hs
0(−d, d), L∗ : Hs
0(−d, d) → Hs
Q(−d, d), and they
are linear and isometric. Moreover, (K∗f) (x) =
∑∞
n=1 fn cosλnx, (L∗g) (x) =∑∞
n=1 gnỹn(λn, x). Evidently, R(K∗) = D(L∗), R(L∗) = D(K∗). It is also obvious
that K∗f and L∗g are even if f and g are even.
Lemma B.3. Let f ∈ Hs
0(−d, d) × Hs−1
0 (−d, d), s ∈ R, f be even. Then
|||L∗f |||sQ = |||f |||s0.
The proof is trivial, taking into account that L∗ is isometric.
The following two lemmas are proved similarly to Lemmas B.4 and B.5 of
[12, Appendix B] by changing odd functions into even functions and taking into
account that L(x, t; 0) and K(x, t; 0) are even on t and odd on x.
Lemma B.4. Let f ∈ L2(−d, d) be even and supp f ⊂ [−T, T ]. Then
supp (L∗f) ⊂ [−T, T ] and (L∗f) (t) = f(t) +
∫ T
|t| L(x, t; 0)f(x) dx.
Lemma B.5. Let f ∈ L2(−d, d) be even and supp f ⊂ [−T, T ]. Then
supp (K∗f) ⊂ [−T, T ] and (K∗f) (t) = f(t) +
∫ T
|t| K(x, t; 0)f(x) dx.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 349
K.S. Khalina
References
[1] M. Gugat and G. Leugering, L∞-Norm Minimal Control of the Wave Equation: on
the Weakness of the Bang-Bang Principle. — ESAIM: COCV 14 (2008), 254–283.
[2] M. Gugat and G. Leugering, Solutions of Lp-Norm-Minimal Control Problems for
the Wave Equation. — Comput. Appl. Math. 21 (2002), 227–244.
[3] M. Gugat, G. Leugering and G. Sklyar, Lp-Optimal Boundary Control for the Wave
Equation. — SIAM J. Control Optim. 44 (2005), 49–74.
[4] M. Gugat, Optimal Boundary Control of a String to Rest in Finite Time with
Continuous State. — Z. Angew. Math. Mech. 86 (2006), 134–150.
[5] L.V. Fardigola and K.S. Khalina, Controllability Problems for the String Equation.
— Ukr. Math. J. 59 (2007), 1040–1058.
[6] W. Krabs and G. Leugering, On Boundary Controllability of One-Dimension Vib-
rating Systems by W 1,p
0 -Controls for p ∈ [0,∞). — Math. Methods Appl. Sci. 17
(1994), 77–93.
[7] M. Negreanu and E. Zuazua, Convergence of a Multigrid Method for the Con-
trollability of a 1-d Wave Equation. — C. R. Math. Acad. Sci. Paris 338 (2004),
413–418.
[8] I. Lasiecka and R. Triggiani, Exact Controllability of the Wave Equation with
Neumann Boundary Control. — Appl. Math. Optimization 19 (1989), 243–290.
[9] D.L. Russell, Controllability and Stabilizability Theory for Linear Partial Differen-
tial Equations: Recent Progress and Open Questions. — SIAM Review 20 (1978),
639–739.
[10] O.Y. Emanuilov, Boundary Controllability of Hyperbolic Equations. — Sib. Math.
J. 41 (2000), 785–799.
[11] V.A. Il’in and E.I. Moiseev, On a Boundary Control at One End of a Process
Described by the Telegraph Equation. — Dokl. Math. 66 (2002), 407–410.
[12] K.S. Khalina, Controllability Problems for the Non-Homogeneous String that is
Fixed at the Right End Point and has the Dirichlet Boundary Control at the Left
End Point. — J. Math. Phys. Anal. Geom. 7 (2011), 34–58.
[13] L.V. Fardigola, Controllability Problems for the String Equation on a Half-Axis with
a Boundary Control Bounded by a Hard Constant. — SIAM J. Control Optim. 47
(2008), 2179–2199.
[14] L. Hörmander, The Analysis of Linear Partial Differential Operators, III: Pseudo–
Differential Operators. Berlin, Heidelberg, New Tokyo, Springer–Verlag, 1958.
[15] G.M. Sklyar and L.V. Fardigola, The Markov Trigonometric Moment Problem in
Controllability Problems for the Wave Equation on a Half–Axis. — Mat. Fiz., Anal.,
Geom. 9 (2002), 233–242.
350 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
On the Neumann Boundary Controllability for the Non-Homogeneous String
[16] V.I. Korobov and G.M. Sklyar, Time-Optimal Control and the Trigonometric Mo-
ment Problem. — Izv. AN SSSR, Ser. Mat. 53 (1989), 868–885. (Russian) (Engl.
transl.: Math. USSR Izv. 35 (1990), 203–220.)
[17] V.S. Vladimirov, Equations of Mathematical Physics. Imported Pubn., 1985.
[18] L. Schwartz, Théorie Des Distributions, I, II. Paris, Hermann, 1950–1951.
[19] S.G. Gindikin and L.R. Volevich, Distributions and Convolution Equations.
Philadelphia, Gordon and Breach Sci. Publ., 1992.
[20] V.A. Marchenko, Sturm–Liouville Operators and Applications. Basel–Boston–
Stuttgart, Birkhauser Verlag, 1986.
[21] R. Courant and D. Gilbert, Methods of Mathematical Physics, Vol. 1. Wiley-
Interscience, 1989.
[22] M.G. Krein and A.A. Nudel’man, The Markov Moment Problem and Extremal
Problems. Moscow, Nauka, 1973. (Russian) (Engl. transl.: AMS Providence, R.I.,
1977.)
[23] I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 3: Theory of Differential
Equations. New York, Academic Press, 1964.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 351
|