Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis
In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from thes...
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irk-123456789-1405462018-07-11T01:23:03Z Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis Fardigola, L.V. In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from these spaces depends on the growth of ρ and k. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system zₜₜ = zξξ − q²z, zξ(0, t) = v(t), ξ > 0, t belongs (0, T), and vise versa. Here q ≥ 0 is a constant and v belongs L∞(0, ∞) is a control. For the main system, necessary and sufficient conditions of the L∞-controllability and the approximate L∞-controllability are obtained from those known for the auxiliary system. 2016 Article Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis / L.V. Fardigola // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 17-47. — Бібліогр.: 40 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.01.017 Mathematics Subject Classification 2000: 93B05, 35B30, 35L05. http://dspace.nbuv.gov.ua/handle/123456789/140546 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from these spaces depends on the growth of ρ and k. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system zₜₜ = zξξ − q²z, zξ(0, t) = v(t), ξ > 0, t belongs (0, T), and vise versa. Here q ≥ 0 is a constant and v belongs L∞(0, ∞) is a control. For the main system, necessary and sufficient conditions of the L∞-controllability and the approximate L∞-controllability are obtained from those known for the auxiliary system. |
| format |
Article |
| author |
Fardigola, L.V. |
| spellingShingle |
Fardigola, L.V. Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis Журнал математической физики, анализа, геометрии |
| author_facet |
Fardigola, L.V. |
| author_sort |
Fardigola, L.V. |
| title |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis |
| title_short |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis |
| title_full |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis |
| title_fullStr |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis |
| title_full_unstemmed |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis |
| title_sort |
transformation operators and modified sobolev spaces in controllability problems on a half-axis |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140546 |
| citation_txt |
Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis / L.V. Fardigola // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 17-47. — Бібліогр.: 40 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT fardigolalv transformationoperatorsandmodifiedsobolevspacesincontrollabilityproblemsonahalfaxis |
| first_indexed |
2025-07-10T10:42:00Z |
| last_indexed |
2025-07-10T10:42:00Z |
| _version_ |
1837256276760854528 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 1, pp. 17–47
Transformation Operators and Modified Sobolev
Spaces in Controllability Problems on a Half-Axis
L.V. Fardigola
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Nauki Ave., Kharkiv, 61103, Ukraine
E-mail: fardigola@ukr.net, fardigola@ilt.kharkov.ua
Received September 20, 2015,
published online December 4, 2015
In the paper, the control system wtt = 1
ρ (kwx) x + γw, wx(0, t) = u(t),
x > 0, t ∈ (0, T ), is considered in special modified spaces of Sobolev type.
Here ρ, k, and γ are given functions on [0,+∞); u ∈ L∞(0,∞) is a control;
T > 0 is a constant. The growth of distributions from these spaces depends
on the growth of ρ and k. With the aid of some transformation operators,
it is proved that the control system replicates the controllability properties
of the auxiliary system ztt = zξξ − q2z, zξ(0, t) = v(t), ξ > 0, t ∈ (0, T ), and
vise versa. Here q ≥ 0 is a constant and v ∈ L∞(0,∞) is a control. For the
main system, necessary and sufficient conditions of the L∞-controllability
and the approximate L∞-controllability are obtained from those known for
the auxiliary system.
Key words: wave equation, half-axis, controllability problem, transfor-
mation operator, modified space of Sobolev type.
Mathematics Subject Classification 2010: 93B05, 35B30, 35L05.
1. Introduction
Controllability problems for hyperbolic equations with constant and variable
coefficients were studied in a number of recent papers, e.g., [2–4, 8–18, 22, 23,
27–29, 31, 33–38, 40] and many others. However, controllability problems for a
distributed parameter system on domains unbounded with respect to the space
variables have not been investigated fully. Nevertheless, these problems were
studied for the wave equation in R3 in [3], on a half-plane in [9, 16, 17], and on
a half-axis in [10–15, 27–29, 36].
c© L.V. Fardigola, 2016
L.V. Fardigola
In the present paper, we study the wave equation with variable coefficients
wtt =
1
ρ
(kwx) x + γw, x > 0, t ∈ (0, T ), (1.1)
controlled by the Neumann boundary condition
wx(0, ·) = u on (0, T ), (1.2)
under the initial conditions
w(·, 0) = w0
0, wt(·, 0) = w0
1 on (0,+∞), (1.3)
where T > 0 is a constant; ρ, k, γ, w0
0, and w0
1 are given functions; u ∈ L∞(0, T )
is a control. In addition, we assume ρ, k ∈ C1[0,+∞) are positive on [0,+∞),
(ρk) ∈ C2[0,+∞), (ρk)′(0) = 0, and
σ(x) =
x∫
0
√
ρ(µ)/k(µ) dµ→ +∞ as x→ +∞. (1.4)
Moreover, we assume
P (k, ρ)− γ ∈ L∞(0,+∞)
⋂
C1[0,+∞) (1.5)
and
∃q = const ≥ 0 σ
√
ρ
k
(
P (k, ρ)− γ − q2
)
∈ L1(0,+∞), (1.6)
where P (k, ρ) = 1
4
√
k
ρ
(√
k
ρ
(
k′
k + ρ′
ρ
))′
+
(
1
4
√
k
ρ
(
k′
k + ρ′
ρ
))2
. This control sys-
tem is considered in modified Sobolev spaces (see Sec. 2).
Note that (1.1) can be reduced to a wave equation with constant coefficients
and a constant potential by using transformation operators. Various transforma-
tion operators were used for studying different PDE’s in [14–17, 24–30, 32] and
others. Note that the application of transformation operators is a key point of
the paper.
Let us recall the results obtained in [10–14, 27–29, 36] for (1.1)–(1.3) with
ρ = k = const, i. e., for the wave equation
ytt = yλλ − (q2 + r)y, λ > 0, t ∈ (0, T ), (1.7)
controlled by the Dirichlet boundary condition
y(0, ·) = v on (0, T ) (1.8)
18 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
or the Neumann boundary condition
yλ(0, ·) = v on (0, T ). (1.9)
Here q ≥ 0 is a given constant, r ∈ C1[0,+∞) is a given function such that
xr ∈ L1(0,+∞), and v ∈ L∞(0, T ) is a control. This system is considered in
Sobolev spaces (see Sec. 2).
For control problem (1.7), (1.8), the approximate L∞-controllability and the
L∞-controllability were studied at a given time and at a free time for r = q = 0
in [36], and for r = 0 in [12]. In this case (r = 0), the solutions to (1.7),
(1.8) were found in explicit form, and necessary and sufficient conditions for
the (approximate) L∞-controllability were obtained. In [12], it was proved that
controllability properties of (1.7), (1.8) with r = 0 are similar at a given time
for the cases q = 0 and q > 0. Therein, it was also proved that the case q = 0
essentially differs from the case q > 0 at a free time. In particular, if q > 0,
then each initial state of control system (1.7), (1.8) with r = 0 is approximately
L∞-controllable at a free time. But, if q = 0, then the initial state of this system
is approximately L∞-controllable at a free time iff yt(·, 0) = yλ(·, 0). The results
of [12] were used in [14].
In [14], control system (1.7), (1.8) was studied under the addition restriction
|r(λ)| ≤ αe−λ, λ > 0, by using the well-known transformation operator (see, e.g.,
[32, Chap. 3]) of the Sturm–Liouville problem saving the asymptotics of solu-
tions at infinity. This operator and its inverse were extended from L2(0,+∞)
to the Sobolev spaces H̃m
0 , m = −2, 2, in [14]. Here H̃m
0 is the subspace of all
odd distributions in Hm
0 , m = −2, 2 (see Sec. 2). This extended transformation
operator was studied and its properties were obtained in [14]. In particular, this
operator is an automorphism of H̃m
0 , m = −2, 2. Using the extended transforma-
tion operator, it was shown that control system (1.7), (1.8) with exponentially
perturbed potential q2 replicated the controllability properties of its original con-
trol system of the same type with the constant potential q2 (r = 0) and vise versa
(see [14]). In particular, necessary and sufficient conditions for the approximate
L∞-controllability and the L∞-controllability of (1.7), (1.8) with the exponential
perturbed potential q2 were obtained at a given time and at a free time. A bit
later, in [29], sufficient conditions of the (approximate) L∞-controllability were
proved at a given time and at a free time for (1.7), (1.8) in the case q = 0 and
a general r. Therein, it was announced that the main properties obtained in
[14] for the transformation operator and its inverse are true for r without the
exponential restriction. Thus, all results of [14] are still valid, but without this
exponential restriction.
For control system (1.7), (1.9), the approximate L∞-controllability and the
L∞-controllability were studied at a given time (for q ≥ 0 and r = 0 in [10], for
q = 0 and r 6= 0 in [27]) and at a free time (for q = r = 0 in [11], for q ≥ 0 and
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 19
L.V. Fardigola
r = 0 in [13], for q = 0 and r 6= 0 in [28]). In [28], the transformation operator
of the Sturm–Liouville problem saving the asymptotics of solutions at infinity
and its inverse were extended from L2(0,+∞) to the Sobolev spaces Ĥm
0 , m =
−1, 1. Here Ĥm
0 is the subspace of all even distributions in Hm
0 , m = −1, 1 (see
Sec. 2). This extended transformation operator was studied and its properties
were obtained in [28]. In particular, this operator is an automorphism of Ĥm
0 ,
m = −1, 1. Using the properties of the extended transformation operator, only
sufficient conditions of the (approximate) L∞-controllability were proved at a
given time and at a free time for (1.7), (1.9) for the case q = 0 and r 6= 0 in [28].
Note that if supp r is bounded, the transformation operator of the Sturm–
Liouville problem saving the initial data of solutions was used to study the (ap-
proximate) L∞-controllability of (1.7) with q = 0 only at a given time in [27].
In [15], the approximate L∞-controllability and the L∞-controllability for
equation (1.1) controlled by the Dirichlet boundary condition were studied. Note
that the Sobolev spaces Hs
0 , m ∈ R, are the natural “environment” for the
solutions to hyperbolic equations with constant coefficients, in particular, to Eq.
(1.7) (see, e.g., [21]). Evidently, the growth of solutions to equations with variable
coefficients depends on the properties of these coefficients at infinity. To study
equation (1.1) with the Dirichlet boundary control, new transformation operators
were introduced and investigated in [15]. They transform the solutions to (1.7),
(1.8) with r = 0 and some q into the solutions to (1.1) with the Dirichlet boundary
control. Together with one of these operators, special modified spaces of Sobolev
type were introduced where Eq. (1.7) was considered. Using these operators,
it was proved that Eq. (1.1) controlled by the Dirichlet boundary condition
replicated the controllability properties of control system (1.7), (1.8) with r = 0
and some q. In particular, necessary and sufficient conditions for the approximate
L∞-controllability and the L∞-controllability were obtained at a given time and
at a free time for (1.1) with the Dirichlet boundary control.
A similar approach is used in the present paper. To study control system
(1.1), (1.3), we apply the composition STr of two operators transforming the
solutions to (1.7), (1.9) with r = 0 and q defined by (1.6) to the solutions to
(1.1), (1.2). The first of them is the operator S transforming the solutions to
(1.7), (1.9) with q and r defined by (1.6) and (3.3) to the solutions to (1.1), (1.2).
This operator was introduced and studied in [15]. Together with the operator
S, there were introduced and studied special modified spaces Hm, m = −2, 2,
of Sobolev type where the space L2(R) is replaced by the space L2
ρ(R) with
the weight
√
ρ̂, and the differential operator d/dx is replaced by the “linearly
deformed” one,
√
k̂/ρ̂
(
d/dx+
(
ρ̂′/ρ̂+ k̂′/k̂
)
/4
)
, k̂ and ρ̂ are the even extensions
of k and ρ, respectively (see Sec. 2). The operator S is isometric from Hm
0
into Hm, in particular, S(Hm
0 ) = Hm, m = −2, 2. The operator S saves the
20 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
value of functions at the origin, but it does not save their asymptotics at infinity.
The growth of distribution from Hm is associated with the data ρ and k of
equation (1.1), m = −2, 2. The second of these two operators is the operator Tr
transforming the solutions to (1.7), (1.9) with r = 0 and q defined by (1.6) to the
solutions to (1.7), (1.9) with the same q and r defined by (3.3). This operator
was studied in [28]. The operator Tr and its inverse are bounded from Hm
0 to
Hm
0 and Tr(Ĥm
0 ) = Ĥm
0 , m = −1, 1. It saves the asymptotics of functions at
infinity, but it does not save their values at the origin. The operators S, Tr and
the spaces Hm, m = −1, 1, associated with the operator S, are the maim tools
of this paper. Using the operator STr, we conclude that control system (1.1),
(1.2) replicates the controllability properties of control system (1.7), (1.9) with
r = 0 and q determined by (1.6) (Corollaries 5.1, 5.6). In particular, we obtain
necessary and sufficient conditions of the approximate L∞-controllability and the
L∞-controllability at a given time and at a free time for (1.1), (1.2). If q > 0, then
each initial state of control system (1.1), (1.2) is approximately L∞-controllable
at a free time (Theorem 5.8). But, if q = 0, then an initial state of this system
is approximately L∞-controllable at a free time iff its coordinates (i. e., w(x, 0)
and wt(x, 0)) are related (Theorem 5.7). The similar relation is necessary for the
L∞-controllability and the approximate L∞-controllability at a given time for
both cases: q = 0 and q > 0 (Theorem 5.5), i. e., for each time T > 0 we have
the set of admissible initial states described by (5.4) and (5.5).
Note that the main results of [28] are particular cases of the results of the
present paper. Moreover, according to Corollary 5.7, sufficient conditions from
[28, Theorems 3.6 and 3.7] are also necessary for the approximate L∞-controlla-
bility of (1.7), (1.9) with q = 0 at a free time.
Thus the application of the transformation operators S and TR and the modi-
fied Sobolev spaces Hm, m = −1, 1, is the focal point in the study of the L∞-
controllability problems for equation (1.1) with variable coefficients.
In Secs. 2 and 3, the definitions of the spaces and the operators used in the
paper are given and their properties are studied. In addition, problem (1.1)–(1.3)
is reduced to (2.3), (2.4).
In Sec. 1, some transformation between the solutions to control systems (2.3),
(2.4) and (2.6), (2.7) are performed.
In Sec. 2, necessary and sufficient conditions for the approximate L∞-control-
lability and the L∞-controllability are obtained at a given time and at a free time.
In Sec. 6, some examples illustrating the results of Secs. 3–5 are considered.
In Sec. 7, some auxiliary assertion is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 21
L.V. Fardigola
2. Notation
Let us give the definitions of the spaces used in the paper. Let D be the
space of infinitely differentiable functions with compact supports, where ϕn → 0
as n → ∞ iff there exists a > 0 such that for each n = 1,∞ we have suppϕn ∈
[−a, a], and for each m = 1,∞ we have
∥∥∥ϕ(m)
n
∥∥∥
L∞[−a,a]
→ 0 as n→∞. Let D′ be
the dual space. Let also S be the Schwartz space of rapidly decreasing functions
on R, and S′ be the dual space of tempered distributions.
By Hs
l (s, l ∈ R), denote the Sobolev spaces:
Hs
l =
{
ϕ ∈ S′ |
(
1 + |D|2
)s/2 (1 + |x|2
)l/2
ϕ ∈ L2 (R)
}
,
‖ϕ‖sl =
∞∫
−∞
∣∣∣(1 + |D|2
)s/2 (1 + |x|2
)l/2
ϕ(x)
∣∣∣2 dx
1/2
,
whereD = −i∂/∂x, |·| is the Euclidean norm. In particular, we haveH0
0 = L2(R),
‖·‖00 = ‖·‖L2(R). For a positive integer p, we have Hp
0 = {ϕ ∈ L2
loc(R) | ∀m =
0, p ϕ(m) ∈ H0
0}, ‖ϕ‖
p
0 =
(∑p
m=0
(∥∥ϕ(m)
∥∥0
0
)2
)1/2
, and H−p0 = (Hp
0 )′. It is well
known [21, Chap. 1] that
‖ϕ‖sl ≤ ‖ϕ‖
s′
l′ , s ≤ s′, l ≤ l′, ϕ ∈ Hs′
l′ . (2.1)
Therefore, Hs
l ⊃ Hs′
l′ is a continuous embedding, s ≤ s′, l ≤ l′.
A distribution f ∈ D′ is said to be odd if 〈f, ϕ(ξ)〉 = −〈f, ϕ(−ξ)〉, ϕ ∈ D,
and it is said to be even if 〈f, ϕ(ξ)〉 = 〈f, ϕ(−ξ)〉, ϕ ∈ D. Here 〈f, ϕ〉 is the value
of the distribution f ∈ D′ on the test function ϕ ∈ D. By H̃s
l and Ĥs
l , denote
the subspaces of odd and even, respectively, distributions in Hs
l . Denote also
Ĥ1 = Ĥ1
0 × Ĥ0
0 with the norm |||·|||0.
By ρ̂, k̂, and γ̂, denote the even extension of ρ, k, and γ, respectively. Denote
η =
(
k̂ρ̂
)1/4
, θ =
(
k̂/ρ̂
)1/4
, and Dηθ = θ2 (d/dx+ η′/η) =
√
k̂/ρ̂
(
d/dx+
(
ρ̂′/ρ̂+
k̂′/k̂
)
/4
)
. We see that η ∈ C2(R) and θ ∈ C1(R).
Further throughout the section, we will assume p = 0, 1.
Introduce the space
Hp =
{
ϕ ∈ L2
loc(R) | ∀m = 0, p
(η
θ
(
Dmηθϕ
))
∈ H0
0
}
with the norm
[]ϕ[]p =
(
p∑
m=0
(∥∥∥η
θ
(
Dmηθϕ
)∥∥∥0
0
)2
)1/2
, ϕ ∈ Hp,
22 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
and the dual space H−p = (Hp)′ with the norm []f []−p = sup{|〈〈f, ϕ〉〉| / []ϕ[]p |
[]ϕ[]p 6= 0}, where 〈〈f, ϕ〉〉 is the value of the distribution f ∈ H−p on the test
function ϕ ∈ Hp. In particular, we have H0 =
(
H0
)′ and 〈〈f, ϕ〉〉 = 〈ηθ f,
η
θϕ〉 =∫∞
−∞
√
ρ̂(x)f(x)ϕ(x) dx, f, ϕ ∈ H0. Put, 〈〈Dηθf, ϕ〉〉 = −〈〈f,Dηθϕ〉〉, f ∈ H−p,
ϕ ∈ Hp+1, p 6= 2. The spaces Hp and H−p are studied in Section 3.. In particular,
it is proved that D ⊂ Hp ⊂ H−p ⊂ D′ are continuous embeddings [15].
Introduce also the spaces
H0 =
{
ϕ ∈ L2
loc(0,+∞) |
(η
θ
ϕ
)
∈ L2(0,+∞)
}
,
H1 =
{
ϕ ∈ H0 |
(η
θ
Dηθϕ
)
∈ H0
}
with the norm
][ϕ][p =
(
p∑
m=0
(∥∥∥η
θ
(
Dmηθϕ
)∥∥∥
L2(0,+∞)
)2
)1/2
, ϕ ∈ Hp,
and the dual space H−p = (Hp)′ with the norm of the conjoint space ][f ][−p =
sup{|〈[f, ϕ]〉| / ][ϕ][p | ][ϕ][p 6= 0}, where 〈[f, ϕ]〉 is the value of the distribution
f ∈ H−p on the test function ϕ ∈ Hp. In particular, we have H0 =
(
H0
)′ and
〈[f, ϕ]〉 =
∫∞
0
(
η(x)
θ(x)
)2
f(x)ϕ(x) dx =
∫∞
0
√
ρ(x)f(x)ϕ(x) dx, f, ϕ ∈ H0.
We can see that the restriction of an even function from Hp to [0,+∞) belongs
to Hp and vise versa: the even extension of a function from Hp belongs to Hp.
Therefore the restriction of an even distribution from H−p to Hp belongs to H−p
and vise versa: the even extension of a distribution from H−p belongs to H−p.
By Ĥm, denote the subspace of even distributions in Hm, m = −1, 1, and
denote IHI1 = Ĥ1 × Ĥ0 with the norm [][]·[][]0.
We treat (1.2) as the value of the distribution w at the point x = 0 in D′ (see
[1, Chap.1] or [14]). We say that a distribution f ∈ H−p has the value f0 ∈ R
at the point x = 0 (f(0) = f0) iff for any ϕ ∈ D, suppϕ ∈ [0,+∞), we have
〈fα, ϕ〉 → 〈f0, ϕ〉 as α → +0. Here fα(x) = f(αx), i.e., 〈fα, ϕ〉 = 〈f, 1
αϕ1/α〉,
ϕ1/α(x) = ϕ(x/α), x ∈ R.
We consider control system (1.1)–(1.3) in the spaces H−m, m = 0, 1, i.e.,(
d
dt
)m
w : [0, T ] → H−m+1, m = 0, 1, 2, w0
0 ∈ H1, and w0
1 ∈ H0. Evidently,
Eq. (1.1) can be rewritten in the form
wtt = D2
ηθw + (γ − ν)w, x > 0, t ∈ (0, T ), (2.2)
where ν = Dηθ
(
θ2η′/η
)
.
Let w ∈ H0 be a solution to control system (2.2), (1.2), (1.3). By W , W 0
0 , and
W 0
1 , denote the even extension of w, w0
0, and w0
1 with respect to x, respectively.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 23
L.V. Fardigola
Taking into account Lemma 7.1, we see that
Wtt = D2
ηθW + (γ̂ − ν)W − 2η2(0)uδ, x ∈ R, t ∈ (0, T ), (2.3)
W (·, 0) = W 0
0 , Wt(·, 0) = W 0
1 on R, (2.4)
where
(
d
dt
)m
W : [0, T ]→ Ĥ−m+1, m = 0, 1, 2, W 0
0 ∈ Ĥ1, W 0
1 ∈ Ĥ0, δ is the Dirac
distribution with respect to x. Let W be a solution to (2.3), (2.4) and W+ be its
restriction to [0,+∞)× [0, T ]. According to Corollary 4.3 (see below Section 4.),
we have
DηθW+(0, ·) = u on (0, T ). (2.5)
Therefore, W+ is a solution to (2.2), (1.2), (1.3).
Together with control system (2.3), (2.4) with a general wave operator, con-
sider the auxiliary control system with the simplest wave operator
Ztt = Zξξ − q2Z − 2vδ, ξ ∈ R, t ∈ (0, T ), (2.6)
Z(·, 0) = Z0
0 , Zt(·, 0) = Z0
1 on R, (2.7)
where
(
d
dt
)m
Z : [0, T ] → H−m+1
0 , m = 0, 1, 2, Z0
0 ∈ Ĥ1
0 , Z0
1 ∈ Ĥ0
0 , δ is the Dirac
distribution with respect to ξ, v ∈ L∞(0, T ) is a control, q is the constant from
condition (1.6). Let Z be a solution to (2.6), (2.7) and Z+ be its restriction to
[0,+∞)× [0, T ]. It is proved (see Theorem 4.1 below) that
(Z+)x(0, ·) = v on (0, T ). (2.8)
To formulate controllability properties of control system (2.6), (2.7), recall
some definitions and assertions from [12]. Let β > 0, Φβ : S′ → S′, D (Φβ) =
{g ∈ S′ | g is odd and supp g ⊂ [−β, β]},
Φβg = F−1
σ→ξ
(
−i√
σ2 + q2
(Fg)
(√
σ2 + q2
))
, g ∈ D(Φβ),
where F is the Fourier transform operator. It is evident if q = 0, then Φβ = Id
(where Id is the identity operator). In particular,
(Φβg) (ξ) = −
∞∫
|ξ|
J0
(
q
√
τ2 − ξ2
)
g(τ) dτ, ξ ∈ R, g ∈ D (Φβ) ∩H0
0 .
Here Jν is the Bessel function, ν = 0,∞. The operator Φβ is invertible, and
Φ−1
β : S′ → S′, D
(
Φ−1
β
)
= R (Φβ) = {g ∈ S′ | g is even and supp g ⊂ [−β, β]},
Φ−1
β f = F−1
σ→ξ
(
−iµ (Fg)
(√
µ2 − q2
))
, g ∈ D(Φ−1
β ).
24 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
In particular,
(
Φ−1
β f
)
(τ) =f ′(τ) + qτ
∞∫
|τ |
I1
(
q
√
ξ2 − τ2
)
√
ξ2 − τ2
f ′(ξ) dξ
=− d
dτ
∞∫
|τ |
I0
(
q
√
ξ2 − τ2
)
f ′(ξ) dξ, τ ∈ R, f ∈ D
(
Φ−1
β
)
∩H0
0 .
Here Iν is the modified Bessel function, Iν(τ) = i−νJν(iτ), ν = 0,∞. The
operators Φβ and Φ−1
β are bounded from H−s0 to H−s0 , s ≥ 0. Moreover,
Φ
(
H̃−s0 ∩D (Φβ)
)
= H̃−s0 ∩D (Φβ), s ≥ 0. If f = Φβg, then f ′ ∈ L∞(−β, β) iff
g ∈ L∞(−β, β).
Set W 0 =
(
W 0
0
W 0
1
)
and Z0 =
(
Z0
0
Z0
1
)
. Evidently, W 0 ∈ IHI1 and Z0 ∈ Ĥ1.
Throughout the paper the domain and the range of an operator A are denoted
by D(A) and R(A), respectively.
3. Spaces and Operators
In this section the spaces Hm, m = −1, 1, are investigated and some transfor-
mation operators are introduced and studied.
Further throughout the section, we will assume p = 0, 1.
According to (1.4), we have σ(x) =
∫ x
0
dµ
θ2(µ)
, x ∈ R. Moreover, σ is an odd
increasing invertible function, and σ(x)→ +∞ as x→ +∞.
Consider the operator S introduced and studied in [15]. Let Sp : Hp
0 → Hp
with the domain D(Sp) = Hp
0 , and
Spψ =
ψ ◦ σ
η
, ψ ∈ D(Sp),
where ψ ◦ σ is the composition of ψ and σ, i.e., (ψ ◦ σ)(x) = ψ(σ(x)), x ∈ R.
Evidently, Sp is the restriction of S0 to Hp
0 , p 6= 0.
By the construction, Sp is invertible, S−1
p : Hp → Hp
0 with the domain
D(S−1
p ) = R(Sp), and
S−1
p ϕ = (ηϕ) ◦ σ−1, ϕ ∈ D(S−1
p )
In [15], the following theorem is proved.
Theorem 3.1. The following assertions hold:
(i) Dmηθ (Spψ) = Sp
(
ψ(m)
)
, ψ ∈ Hp
0 , m = 0, p;
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 25
L.V. Fardigola
(ii) []Spψ[]p = ‖ψ‖p0, ψ ∈ Hp
0 , m = 0, p;
(iii) R(Sp) = Hp;
(iv) Sp and S−1
p are bounded.
Let us continue S0 to H−p0 . Introduce the operator S−p : H−p0 → H−p with
the domain D(S−p) = H−p0 such that
〈〈S−pg, ϕ〉〉 = 〈g,S−1
p ϕ〉, g ∈ D(S−p), ϕ ∈ D(S−1
p ).
This extension S−p of S0 is also invertible, S−1
−p : H−p → H−p0 with the domain
D(S−1
−p), and
〈S−1
−pf, ψ〉 = 〈〈f,Spψ〉〉 , f ∈ D(S−1
−p), ψ ∈ D(Sp).
Finally, by S, denote the operator S−1 with the domain D(S) = D(S−1) =
H−1
0 . Evidently, S is invertible, and S−1 = S−1
−1 : H−1 → H−1
0 with the domain
D(S−1) = R(S) = H−1 (see Theorem 3.1). Taking into account the construction
of S and Theorem 3.1, we obtain
Theorem 3.2. For m = −1, 1, the following assertions hold:
(i) Dηθ (Sg) = S (g′), g ∈ Hm
0 , m 6= −1;
(ii) []Sg[]m = ‖g‖m0 , g ∈ Hm
0 ;
(iii) SHm
0 = Hm;
(iv) SĤm
0 = Ĥm;
(v) 〈〈f, ϕ〉〉 = 〈S−1f,S−1ϕ〉, f ∈ H−m, ϕ ∈ Hm.
To study control system (2.3), (2.4), we need
Theorem 3.3. S (δ) = η(0)Dηθδ.
P r o o f. Let ϕ ∈ H1. According to Theorem 3.2, we get 〈〈Sδ, ϕ〉〉 =
〈δ,S−1ϕ〉 = η(0) (ϕ) (0) = 〈〈η(0)δ, ϕ〉〉, which was to be proved.
Due to [15], the following theorem holds.
Theorem 3.4. We have:
(i) Hm ⊂ Hn is a continuous embedding, −1 ≤ n ≤ m ≤ 1;
(ii) D ⊂ Hm ⊂ D′ are continuous embeddings, −1 ≤ m ≤ 1;
(iii) D is dense in Hm, Hm is dense in Hn, Hn is dense in D′, −1 ≤ n ≤ m ≤ 1.
26 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
In [15], it is also shown that the relation between the Schwartz space S and
the space Hm depends on η and θ, i.e., on k and ρ, m = −2, 2.
To study control system (2.3), (2.4), we need the operator transforming each
L2(0,+∞)-solution to
−z′′ = µ2z, λ > 0, (3.1)
into an L2(0,+∞)-solution to
−y′′ + ry = µ2y, λ > 0, (3.2)
under the boundary condition y(λ,µ)
z(λ,µ) → 1 as λ → +∞, µ ∈ C, <µ ≥ 0. Here z′′
and y′′ are the derivatives of z and y with respect to λ, µ is a parameter,
r =
(
Dηθ
(
θ2η′/η
)
− γ̂ − q2
)
◦ σ−1
=
1
4
√
k
ρ
(√
k
ρ
(
k′
k
+
ρ′
ρ
))′
+
(
1
4
√
k
ρ
(
k′
k
+
ρ′
ρ
))2
− γ − q2
◦ σ−1.
(3.3)
Taking into account (1.5) and (1.6), we get
r ∈ L∞(0,+∞)
⋂
C1[0,+∞) and λr ∈ L1(0,+∞). (3.4)
It was proved [32, Chap. 3] that this operator is invertible, i.e., there exists
an operator transforming each L2(0,+∞)-solution to (3.2) into an L2(0,+∞)-
solution to (3.1) under the boundary condition saving the asymptotics of the
solutions at infinity.
In [28], this transformation operator and its inverse were extended to H−1
0 .
Let us recall the definition for these extensions. We assume (3.4) holds. Let
Ω = {y = (y1, y2) ∈ R2 | y2 > y1 > 0}. Let K be a solution to the system
Ky1y1 −Ky2y2 = r(y1)K, y = (y1, y2) ∈ Ω,
K(y1, y1) =
1
2
∞∫
y1
r(ξ) dξ, y1 > 0,
lim
y1+y2→∞
Ky1(y) = lim
y1+y2→∞
Ky2(y) = 0, y ∈ Ω.
(3.5)
Due to [32, Chap. 3], system (3.5) has the unique solution K, and K ∈ C2(Ω).
Definition 3.1. Denote T0 : H0
0 → H0
0 with the domain D (T0) = Ĥ0
0 ,
(T0g) (λ) = g(λ) + λ
∞∫
|λ|
K(|λ|, ξ)g(ξ) dξ, λ ∈ R, g ∈ D(T0).
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 27
L.V. Fardigola
From [32, Chap. 3], it follows that the operator T0 is invertible, and T−1
0 :
H0
0 → H0
0 , D
(
T−1
0
)
= Ĥ0
0 ,
(
T−1
0 f
)
(ξ) = f(ξ) + ξ
∞∫
|ξ|
L(|ξ|, λ)f(λ) dλ, ξ ∈ R, f ∈ D(T−1
0 ),
where L ∈ C2(Ω) is determined by
L(y) +K(y) +
y2∫
y1
L(y1, ξ)K(ξ, y2) dξ = 0, y ∈ Ω, (3.6)
or
L(y) +K(y) +
y2∫
y1
K(y1, ξ)L(ξ, y2) dξ = 0, y ∈ Ω. (3.7)
For the adjoint operators T∗0 and
(
T−1
0
)∗ = (T∗0)−1 we have T∗0 : H0
0 → H0
0 ,
D (T∗0) = Ĥ0
0 = R
(
(T∗0)−1
)
,
(T∗0ϕ) (ξ) = ϕ(ξ) + ξ
|ξ|∫
0
K(λ, |ξ|)ϕ(λ) dλ, ξ ∈ R, ϕ ∈ D (T∗0) ,
and (T∗0)−1 : H0
0 → H0
0 , D
(
(T∗0)−1
)
= Ĥ0
0 = R (T∗0),
(
(T∗0)−1 ψ
)
(λ) = ψ(λ) + sgnλ
|λ|∫
0
L(ξ, |λ|)ψ(ξ) dξ, λ ∈ R, ψ ∈ D
(
(T∗0)−1)
)
.
Due to [28], the following theorem is valid.
Theorem 3.5. We have:
(i) T∗0 is bounded from Hp
0 to Hp
0 ;
(ii) (T∗0)−1 is bounded from Hp
0 to Hp
0 ;
(iii) T∗0(Ĥp
0 ) = Ĥp
0 and (T∗0)−1 (Ĥp
0 ) = Ĥp
0 .
Definition 3.2. Denote by Tr the operator
(
T∗0|H1
0
)∗
. We have Tr : H−1
0 →
H−1
0 , D (Tr) = Ĥ−1
0 , 〈Trg, ϕ〉 = 〈g,T∗0ϕ〉, g ∈ D (Tr) = Ĥ−1
0 , ϕ ∈ Ĥ1
0 .
28 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
Then T−1
r =
(
(T∗0)−1 |H1
0
)∗
and T−1
r : H−1
0 → H−1
0 , D
(
T−1
r
)
= Ĥ−1
0 ,
〈T−1
r f, ψ〉 = 〈g, (T∗0)−1 ψ〉, f ∈ D
(
T−1
r
)
= Ĥ−1
0 , ψ ∈ H1
0 .
The following four theorems were proved in [28].
Theorem 3.6. We have
(i) Tr is bounded from H−p0 to H−p0 ;
(ii) T−1
r is bounded from H−p0 to H−p0 ;
(iii) D (Tr) = R
(
T−1
r
)
= Ĥ−1
0 = D
(
T−1
r
)
= R (Tr);
(iv) Tr
(
Ĥ−p0
)
= Ĥ−p0 and T−1
r
(
Ĥ−p0
)
= Ĥ−p0 .
Theorem 3.7. We have:
(
(d/dλ)2 − r(|λ|)
)
(Trg) =Tr
(
(d/dξ)2g
)
+ 2δ(λ)
∞∫
0
Ky1(0, ξ)g(ξ) dξ
− δ(λ)
∞∫
0
r(ξ) dξg(+0), g ∈ Ĥ1
0 .
Moreover, if g ∈ Ĥ1
0 and g′(+0) exists, then(
(d/dλ)2 − r(|λ|)
)
(Trg) = Tr
(
(d/dξ)2g
)
+ 2δ(λ)
(
(Trg)′(+0)− g′(+0)
)
.
Theorem 3.8. We have
(d/dξ)2
(
T−1
r f
)
=T−1
r
((
(d/dλ)2 − r(|λ|)
)
f
)
+ 2δ(ξ)
∞∫
0
Ly1(0, λ)f(λ) dλ
− δ(ξ)
∞∫
0
r(λ) dλf ′(+0) f ∈ Ĥ0
0 .
Moreover, if f ∈ Ĥ1
0 and f ′(+0) exists, then
(d/dξ)2
(
T−1
r f
)
= T−1
r
((
(d/dλ)2 − r(|λ|)
)
f
)
+ 2δ(ξ)
(
(T−1
r f)r′(+0)− f ′(+0)
)
.
Theorem 3.9. Tr (δ) = δ.
The kernels K and L of the integral operators Tr and T−1
r were found ex-
plicitly in [14, Example 5.1] for r(λ) = βe−λ, λ > 0, where β > 0 is a constant.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 29
L.V. Fardigola
4. Transformations between Solutions to the Main and the
Auxiliary Control Systems
In this section, it is shown that, using the transformation operator STr and
its inverse, we can transform each solution to auxiliary control system (2.6), (2.7)
into a solution to main control system (2.3), (2.4) and vise versa (Theorems 4.2
and 4.4). Note that Theorems 4.2, 4.4 and Corollary 4.3 are analogs of some
assertions from [28] for the case q = 0.
Theorem 4.1. Let Z be a solution to (2.6), (2.7) for some v ∈ L∞(0, T ) and
Z0 ∈ Ĥ1. Then (2.8) holds.
P r o o f. Put J = 1
2J0
(
q
√
t2 − ξ2
)
(H(ξ + t) − H(ξ − t)), ξ ∈ R, t ≥ 0.
According to [10], we have
Z(ξ, t) =Jt(ξ, t) ∗ Z0
0 (ξ) + J(ξ, t) ∗ Z0
1 (ξ)
+ 2
t∫
0
J(ξ, t− τ)v(τ) dτ, ξ ∈ R, t ≥ 0, (4.1)
where ∗ is the convolution with respect to ξ. Therefore,
Zξ(ξ, t) =Jt(ξ, t) ∗ Z0
0
′(ξ) + J(ξ, t) ∗ Z0
1
′(ξ)
+ 2
t∫
0
Jξ(ξ, t− τ)v(τ) dτ + v(t), ξ ∈ R, t ≥ 0. (4.2)
Hence (2.8) holds.
Theorem 4.2. Let Z be a solution to (2.6), (2.7) for some v ∈ L∞(0, T ) and
Z0 ∈ Ĥ1. Let W (·, t) = STrZ(·, t), t ∈ [0, T ]. Then W is a solution to (2.3),
(2.4) with W 0 = STrZ
0 and
η(0)θ2(0)u(t) = v(t)−RZ+(0, t) +
∞∫
0
Ky1(0, ξ)Z(ξ, t) dξ, t ∈ [0, T ], (4.3)
and (2.5) is valid. Here R = 1
2
∫∞
0 r(ξ) dξ, and (·)+ means the restriction to
[0,+∞)× [0, T ]. Moreover,[][](
W (·, t)
Wt(·, t)
)[][]
≤ B0
∣∣∣∣∣∣∣∣∣∣∣∣(Z(·, t)
Zt(·, t)
)∣∣∣∣∣∣∣∣∣∣∣∣ , t ∈ [0, T ], (4.4)
‖u‖L∞(0,T ) ≤ B1
(
‖v‖L∞(0,T ) + (1 + T )
∣∣∣∣∣∣Z0
∣∣∣∣∣∣) , (4.5)
where B0 > 0 and B1 > 0 are some constants independent of T .
30 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
P r o o f. Applying Theorem 3.7, we see that W is a solution to (2.3),
(2.4) with W 0 = STRZ
0 and U determined by (4.3). Let us prove (2.5). Due to
Lemma 4.1, (2.8) holds. Taking into account (4.3), we get
W+
x (0, t) = (STRZ)+x (0, t) =
1
η(0)θ2(0)
Z+
x (0, t)
+
∞∫
0
Ky1(0, ξ)Z(ξ, t) dξ −RZ+(0, t)
= u(t), t ∈ [0, T ],
i.e., (2.5) holds. According to Theorem 3.6, there exists a constant B0 > 0 such
that (4.4) holds. To prove (4.5), we rewrite (4.3). Taking into account (4.1) and
setting G(ξ, t) =
(
RJ0
(
q
√
t2 − ξ2
)
−
t∫
ξ
Ky1(ξ, µ)J0
(
q
√
t2 − µ2
)
dµ
)
(H(ξ+ t)−
H(ξ − t)), ξ ∈ R, t ≥ 0, we obtain
η(0)θ2(0)u(t) =v(t)− ∂
∂t
t∫
0
G(ξ, t)Z0
0 (ξ) dξ
−
t∫
0
G(ξ, t)Z0
1 (ξ) dξ −
t∫
0
G(0, t− τ)v(τ) dτ
=v(t)− Z0
0 (t)−
∫ t
0
Gt(ξ, t)Z0
0 (ξ) dξ
−
t∫
0
G(ξ, t)Z0
1 (ξ) dξ −
t∫
0
G(0, t− τ)v(τ) dτ, t ∈ [0, T ]. (4.6)
Since Z0
0 ∈ Ĥ1
0 , we have
∣∣Z0
0 (t)
∣∣ ≤ ∞∫
−∞
1√
1 + σ2
∣∣∣√1 + σ2
(
FZ0
0
)
(σ)
∣∣∣2 dσ ≤ π ∥∥Z0
0
∥∥1
0
, (4.7)
where F is the Fourier transform operator. Then, using the estimates given in
[32, Chap. 3] for the kernel K and its derivatives, we get
|G(ξ, t)| ≤ A0 and |Gy(ξ, t)| ≤ A1t, t ≥ ξ ≥ 0, (4.8)
where A0 > 0 and A1 > 0 are constants. Summarizing (4.6)–(4.8), we conclude
that (4.5) is true. The theorem is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 31
L.V. Fardigola
Corollary 4.3. If W is a solution to (2.3), (2.4) for some u ∈ L∞(0, T ) and
W 0 ∈ IHI1, then (2.5) holds.
P r o o f. Put Z(·, t) = T−1
r S−1W (·, t), t ∈ [0, T ]. Applying Theorem 3.8,
we see that Z is a solution to (2.6), (2.7) with Z0 = T−1
r S−1W 0 and
v(t) =η(0)θ2(0)u(t) +Rη(0)W+(0, t)
+
∞∫
0
Ly1(0, λ)
(
S−1W (·, t)
)
(λ) dλ, t ∈ [0, T ]. (4.9)
Here R = 1
2
∫∞
0 r(ξ) dξ, and (·)+ means the restriction to [0,+∞) × [0, T ]. Sub-
stituting TrZ(·, t) for W (·, t) and taking into account (4.1), we get
v(t) =η(0)θ2(0)u(t) +
∂
∂t
t∫
0
G(ξ, t)Z0
0 (ξ) dξ
+
t∫
0
G(ξ, t)Z0
1 (ξ) dξ +
t∫
0
G(0, t− τ)v(τ) dτ, t ∈ [0, T ]. (4.10)
Theorem 4.2 and (4.6) imply W+
x (0, t) = ũ(t), t ∈ [0, T ], for
η(0)θ2(0)ũ(t) =v(t)− ∂
∂t
t∫
0
G(ξ, t)Z0
0 (ξ) dξ
−
t∫
0
G(ξ, t)Z0
1 (ξ) dξ −
t∫
0
G(0, t− τ)v(τ) dτ, t ∈ [0, T ].
Therefore, ũ(t) = u and that was to be proved.
By analogy with Theorem 4.2, we obtain
Theorem 4.4. Let W be a solution to (2.3), (2.4) for some u ∈ L∞(0, T )
and W 0 ∈ IHI1. Let Z(·, t) = T−1
r S−1W (·, t), t ∈ [0, T ]. Then Z is a solution to
(2.6), (2.7) with Z0 = T−1
r S−1W 0 and
v(t) =η(0)θ2(0)u(t) +Rη(0)W+(0, t)
+
∞∫
0
Ly1(0, x)S−1W (x, t) dx, t ∈ [0, T ]. (4.11)
32 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
Here R = 1
2
∫∞
0 r(ξ) dξ, and (·)+ means the restriction to [0,+∞)× [0, T ]. More-
over, ∣∣∣∣∣∣∣∣∣∣∣∣(Z(·, t)
Zt(·, t)
)∣∣∣∣∣∣∣∣∣∣∣∣ ≤ C0
[][](
W (·, t)
Wt(·, t)
)[][]
, t ∈ [0, T ], (4.12)
‖v‖L∞(0,T ) ≤ C1e
TC2
(
‖u‖L∞(0,T ) + (1 + T )
[][]
W 0
[][])
, (4.13)
where C0 > 0, C1 > 0, and C2 > 0 are constants independent of T .
P r o o f. By analogy with (4.3) and (4.4), using Theorem 3.8 instead
of Theorem 3.6, we obtain (4.11) and (4.12). Let us prove (4.13). Taking into
account (4.10), we get
v(t) = F1(t) +
t∫
0
F2(t− τ)v(τ) dτ, t ∈ [0, T ], (4.14)
where
F1(t) = η(0)θ2(0)u(t) +
∂
∂t
t∫
0
G(ξ, t)Z0
0 (ξ) dξ +
t∫
0
G(ξ, t)Z0
1 (ξ) dξ
= η(0)θ2(0)u(t) + Z0
0 (t) +
t∫
0
Gt(ξ, t)Z0
0 (ξ) dξ
+
t∫
0
G(ξ, t)Z0
1 (ξ) dξ, t ∈ [0, T ],
F2(t) = G(0, t), t ∈ [0, T ].
According to (4.7) and (4.8), we obtain
‖F1‖L∞(0,T ) ≤ C1
(
‖u‖L∞(0,T ) + (1 + T )
∣∣∣∣∣∣Z0
∣∣∣∣∣∣) , (4.15)
‖F2‖L∞(0,T ) ≤ C2. (4.16)
Taking into account (4.16) and the generalized Gronwall theorem, we can come
to the conclusion that the integral equation
v(t) =
t∫
0
F2(t− τ)v(τ) dτ, t ∈ [0, T ], (4.17)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 33
L.V. Fardigola
has only the trivial solution in L2(0, T ). Due to the Fredholm alternative,
Eq. (4.14) has the unique solution in L2(0, T ). Using again the generalized Gron-
wall theorem, we obtain
|v(t)| ≤ ‖F1‖L∞(0,T ) e
t‖F2‖L∞(0,+∞) , t ∈ [0, T ].
Taking into account (4.16), (4.15), and (4.12), we obtain (4.13). The theorem is
proved.
R e m a r k 4.1. Due to [10, Theorem 3.2], system (2.6), (2.7) has the unique
solution. Therefore Theorems 4.2, 4.4 yield the uniqueness of the solution to
system (2.3), (2.4).
Thus main control system (2.3), (2.4) is the transformation of auxiliary control
system (2.6), (2.7) by STr and vise versa: control system (2.6), (2.7) is the
transformation of control system (2.3), (2.4) by T−1
r S−1.
5. Conditions for Controllability
For given T > 0 and W 0 ∈ IHI1 (Z0 ∈ Ĥ1), denote by RgT (W 0) (RsT (Z0),
respectively) the set of the states W T ∈ IHI1 (ZT ∈ Ĥ1, respectively) for which
there exists a control u ∈ L∞(0, T ) (v ∈ L∞(0, T ), respectively) such that prob-
lem (2.3)–(2.4) ((2.6)–(2.7), respectively) has the unique solution W (Z, respec-
tively) and
(
W (·, T )
Wt(·, T )
)
= W T
((
Z(·, T )
Zt(·, T )
)
= ZT , respectively
)
.
Definition 5.1. A state W 0 ∈ IHI1 (Z0 ∈ Ĥ1) is called L∞-controllable with
respect to control system (2.3), (2.4) ( (2.6), (2.7), respectively) at a given time
T > 0 if 0 belongs to RgT (W 0) (RsT (Z0), respectively) and approximately L∞-
controllable with respect to control system (2.3), (2.4) ( (2.6), (2.7), respectively)
at a given time T > 0 if 0 belongs to the closure of RgT (W 0) in IHI1 (RsT (Z0) in
Ĥ1, respectively).
For a givenW 0 ∈ IHI0 (Z0 ∈ Ĥ0), denoteRg(W 0) =
⋃
T>0R
g
T (W 0) (Rs(Z0) =⋃
T>0RsT (Z0), respectively).
Definition 5.2. A state W 0 ∈ IHI1 (Z0 ∈ Ĥ1) is called approximately L∞-
controllable with respect to control system (2.3), (2.4) ( (2.6), (2.7), respectively)
at a free time if 0 belongs to the closure of Rg(W 0) in IHI1 (Rs(Z0) in Ĥ1,
respectively).
Theorems 4.2 and 4.4 imply
Corollary 5.1. Let W 0 ∈ IHI1, Z0 = STRW
0, and a time T > 0 be given.
Then
34 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
(i) W 0 is approximately L∞-controllable with respect to system (2.3), (2.4) at
the time T iff Z0 is approximately L∞-controllable with respect to system
(2.6), (2.7) at the same time;
(ii) W 0 is L∞-controllable with respect to system (2.3), (2.4) at the time T iff
Z0 is L∞-controllable with respect to system (2.6), (2.7) at the same time.
Due to [11, 12, 14], we have the following three theorems.
Theorem 5.2. Let Z0 ∈ Ĥ1 and a time T > 0 be given. Then
(i) Z0 is approximately L∞-controllable with respect to control system (2.6), (2.7)
at the time T iff
suppZ0
0 ⊂ [−T, T ], (5.1)
Z0
1 −ΦT
(
sgn ξ
(
Φ−1
T Z0
0
)′) = 0; (5.2)
(ii) Z0 is L∞-controllable with respect to control system (2.6), (2.7) at the time
T iff Z0
1 ∈ L∞(0, T ) and (5.1), (5.2) hold.
Theorem 5.3. Let q = 0. A state Z0 ∈ Ĥ1 is approximately L∞-controllable
with respect to control system (2.6), (2.7) at a free time iff
Z0
1 − sgn ξ Z0
0
′ = 0. (5.3)
Theorem 5.4. Let q > 0. Each state Z0 ∈ Ĥ1 is approximately L∞-
controllable with respect to control system (2.6), (2.7) at a free time.
Note that if q = 0, then (5.2) is equivalent to (5.3).
Corollary 5.1 and Theorem 5.2 imply
Theorem 5.5. Let W 0 ∈ IHI1 and a time T > 0 be given. Then
(i) W 0 is approximately L∞-controllable with respect to system (2.3), (2.4) at
the time T iff
suppW 0
0 ⊂ [−σ−1(T ), σ−1(T )], (5.4)
W 0
1 − (STrΦT )
(
sgnx
(
(STrΦT )−1W 0
0
)′)
= 0; (5.5)
(ii) W 0 is L∞-controllable with respect to system (2.3), (2.4) at the time T iff
W 0
1 ∈ L∞(0, T ) and (5.4), (5.5) hold.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 35
L.V. Fardigola
P r o o f. (i) Put Z0 = T−1
r S−1W 0. According to Corollary 5.1 and Theorem
5.2, W 0 is approximately L∞-controllable with respect to system (2.3), (2.4) at
the time T iff conditions (5.1), (5.2) are valid. Evidently, (5.2) is equivalent to
(5.5). Taking into account the construction of STr and (STr)−1, we see that
(5.1) is equivalent to (5.4).
(ii) Taking into account Corollary 5.1, Theorem 5.2, and (i), it is sufficient to
show that
W 0
1 ∈ L∞(0, T ) iff Z0
1 ∈ L∞(0, T ) (5.6)
for proving (ii). Using the estimate given in [32, Chap. 3] for the kernel K, we
get
∥∥W 0
1
∥∥
L∞(0,T )
≤
∥∥Z0
1
∥∥
L∞(0,T )
max
λ∈[0,T ]
∣∣∣∣∣∣ 1
η(λ)
1 +
∞∫
0
|K(λ, ξ)|2 dξ
∣∣∣∣∣∣
≤ C
∥∥Z0
1
∥∥
L∞(0,T )
,
i.e., if Z0
1 ∈ L∞(0, T ), then W 0
1 ∈ L∞(0, T ). The converse assertion can be
obtained analogously by using the estimate given in for the kernel L instead of
that given for the kernel K [32, Chap. 3]. Hence (5.6) holds, which was to be
proved.
Note that if q = 0, then (5.5) is equivalent to
W 0
1 − STr
(
sgn ξ
(
T−1
r S−1W 0
0
)′) = 0. (5.7)
Theorems 4.2 and 4.4 imply
Corollary 5.6. Let W 0 ∈ IHI1 and Z0 = TRW
0. Then W 0 is approximately
L∞-controllable with respect to control system (2.3), (2.4) at a free time iff Z0
is approximately L∞-controllable with respect to control system (2.6), (2.7) at a
free time.
Theorems 5.3, 5.4 and Corollary 5.6 yield the following two theorems
Theorem 5.7. Let q = 0. A state W 0 ∈ IHI1 is approximately L∞-controllable
with respect to control system (2.3), (2.4) at a free time iff (5.7) holds.
Theorem 5.8. Let q > 0. Each state W 0 ∈ IHI1 is approximately L∞-
controllable with respect to control system (2.3), (2.4) at a free time.
Taking into account [12], we give the following remark.
R e m a r k 5.2. Let W 0 ∈ IHI1 and suppW 0 ⊂ [−β, β], where β > 0. Thus,
36 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
(i) If q = 0, then the state W 0 is approximately L∞-controllable with respect to
control system (2.3), (2.4) at a free time iff condition (5.5) holds, and under
this condition the state W 0 is approximately L∞-controllable with respect
to control system (2.3), (2.4) at a given time T ≥ β;
(ii) If q > 0, then the state W 0 is always approximately L∞-controllable with
respect to control system (2.3), (2.4) at a free time, but the state W 0 is
approximately L∞-controllable with respect to control system (2.3), (2.4) at
a given time T ≥ β iff condition (5.5) holds.
Thus, transformed control system (2.3), (2.4) replicates controllability pro-
perties of its original control system (2.6), (2.7) and vise versa.
6. Examples
In this section, we give the examples illustrating the results of Secs. 2–5.
E x a m p l e 6.1. Let α > 0, k̂(x) = (1 + |x|)
(
1− tanh2 x
)
, ρ̂(x) =
1
(1+|x|) cosh2 x
, γ̂(x) = (1 + |x|)2
(
2 tanh2 x− 1
)
− (1 + |x|) tanh |x| − 1
4(1+|x|) ,
W 0
0 (x) =
H
(
α2 − x2
)
ln(1 + α)
coshx
α∫
|x|
I0
(√√
1 + µ−
√
1 + |x|
√
1 + µ+
√
1 + |x|
)
dµ
1 + µ
,
W 0
1 (x) =
H
(
α2 − x2
)
ln(1 + α)
coshxI0
(√√
1 + α−
√
1 + |x|
√
1 + α+
√
1 + |x|
)
, x ∈ R.
Evidently, η(x) = 1/ coshx, θ =
√
1 + |x|, σ(x) = sgnx ln(1 + |x|), x ∈ R,
σ−1(λ) = sgnλ
(
e|λ| − 1
)
, λ ∈ R. Therefore, ϕ ∈ H0 iff ϕ√
1+|x| coshx
∈ H0
0 ;
ϕ ∈ H1 iff
√
1 + |x|
( ϕ
coshx
)′ ∈ H0
0 . Hence, W 0
0 ∈ H1. We have
ν(x) =
(
θ2
(
θ2 η
′
η
)′)
(x) +
(
θ2 η
′
η
)2
(x)
= (1 + |x|)2
(
2 tanh2 x− 1
)
− (1 + |x|) tanh |x| = γ̂(x) +
1
4(1 + |x|)
, x ∈ R.
Hence conditions (1.5) and (1.6) hold for q = 0, r(λ) = e−|λ|/4, λ ∈ R (see (3.4)).
Let us transform control system (2.3), (2.4) with the given k̂, ρ̂, γ̂, W 0 and
some T > 0 into control system (2.6), (2.7). Put Z0 = T−1
r S−1W 0, Z(·, t) =
T−1
r S−1W 0(·, t), t ∈ [0, T ]. For this r, the kernels K and L of the operators Tr
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 37
L.V. Fardigola
and T−1
r were calculated in [14, Example 5.1]:
K(y) = K̃y2(y) and L(y) = −L̃y1(y), y2 ≥ y1 ≥ 0, (6.1)
K̃(y) = I0
(√
e−
y1
2
(
e−
y1
2 − e−
y2
2
))
, y2 ≥ y1 ≥ 0, (6.2)
L̃(y) = J0
(√
e−
y2
2
(
e−
y1
2 − e−
y2
2
))
, y2 ≥ y1 ≥ 0. (6.3)
Setting A = ln(1 + α), we obtain
(
S−1W 0
0
)
(λ) =
1
A
H
(
A2 − λ2
) A∫
|λ|
K̃(|λ|, µ) dµ, λ ∈ R.
Therefore, for ξ ∈ R, we have
Z0
0 (ξ) =
(
T−1
r S−1W 0
0
)
(ξ)
=
1
A
H
(
A2 − λ2
) A∫
|ξ|
K̃(|ξ|, µ) dµ+
A∫
|ξ|
L(|ξ|, λ)
∫ A
λ
K̃(λ, µ) dµ dλ
=
1
A
H
(
A2 − λ2
) A∫
|ξ|
K̃(|ξ|, µ) +
µ∫
|ξ|
L(|ξ|, λ)K̃(λ, µ) dλ
dµ. (6.4)
Let us evaluate
B(ξ, µ) = K̃(ξ, µ) +
µ∫
ξ
L(ξ, λ)K̃(λ, µ) dλ, µ ≥ ξ ≥ 0.
Since K and L are the kernels of Tr and T−1
r , respectively, we get
K(ξ, µ) + L(ξ, µ) +
µ∫
ξ
L(ξ, λ)K(λ, µ) dλ = 0, µ ≥ ξ ≥ 0.
From here, taking into account (6.1), we obtain Bµ(ξ, µ) = 0, µ ≥ ξ ≥ 0. Since
B(ξ, ξ) = K̃(ξ, ξ) = 1, ξ ≥ 0, we have B(ξ, µ) = 1, µ ≥ ξ ≥ 0. Then, according
to (6.4), we obtain
Z0
0 (ξ) =
1
A
H
(
A2 − λ2
) A∫
|ξ|
dµ =
A− |ξ|
A
H
(
A2 − λ2
)
, ξ ∈ R.
38 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
We also have(
S−1W 0
0
)
(λ) =
1
A
H
(
A2 − λ2
)
K̃(|λ|, A), λ ∈ R,
Z0
0 (ξ) =
(
T−1
r S−1W 0
0
)
(ξ) =
1
A
H
(
A2 − λ2
)
B(|ξ|, A) =
1
A
H
(
A2 − λ2
)
, ξ ∈ R.
Since condition (5.3) holds for Z0, applying Theorem 5.2, we conclude that
Z0 is L∞-controllable with respect to system (2.6), (2.7) at any time T ≥ A. Let
T ≥ A. Put v(t) = 1
AH
(
A2 − λ2
)
, t ∈ [0, T ], then
Z(ξ, t) =
(A− t)− |ξ|
A
H
(
(A− t)2 − λ2
)
, ξ ∈ R, t ∈ [0, T ],
is the unique solution to (2.6), (2.7). Hence, Z(ξ, T ) = Zt(ξ, T ) = 0. Therefore
the control v solves the L∞-controllability problem with respect to system (2.6),
(2.7) at the time T for Z0.
Now we can find a control solving the L∞-controllability problem with respect
to system (2.3), (2.4) at the time T and the solution to this system for W 0. For
t ∈ [0, T ], put
u(t) = v(t)− 1
2
Z+(0, t)
∞∫
0
r(ξ) dξ +
∞∫
0
Ky1(0, ξ)Z(ξ, t) dξ
=
1
A
H
(
A2 − λ2
)1− 1
8
(A− t) +
A−t∫
0
K̃y1y2(0, ξ) ((A− t)− ξ) dξ
=
1
A
H
(
A2 − λ2
)1− 1
8
(A− t)− K̃y1(0, 0)(A− t) +
A−t∫
0
K̃y1(0, ξ) dξ
=
1
A
H
(
A2 − λ2
)1 +
A−t∫
0
I1
(√
1− e−
ξ
2
)
2
√
1− e−
ξ
2
(
1
2
e−
ξ
2 − 1
) dξ
=
1
A
H
(
A2 − λ2
)I0
√1− e
t
2
√
1 + α
− Ĩ
√1− e
t
2
√
1 + α
,
where Ĩ(p) = 2
∫ p
0
I1(µ)
1−µ2 dµ. According to Theorem 4.2, the control u solves the
L∞-controllability problem with respect to system (2.3), (2.4) at the time T
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 39
L.V. Fardigola
for W 0. Moreover, we can find the solution to this system explicitly. We have
(Trz(·, t)) (λ) = Z(λ, t) +
∞∫
|λ|
K̃y2(|λ|, ξ)Z(ξ, t) dξ
=
1
A
H
(
(A− t)2 − λ2
) ∫
|λ|A−tK̃(|λ|, ξ) dξ.
Therefore,
W (x, t) = (STrZ(·, t)) (x) =
H
((
eA−t − 1
)2 − x2
)
A
A−t∫
ln(1+|x|)
K̃(ln(1 + |x|), ξ) dξ
=H
((
(1 + α)e−t − 1
)2 − x2
) coshx
ln(1 + α)
×
(1+α)e−t−1∫
|x|
I0
(√√
1 + µ−
√
1 + |x|
√
1 + µ+
√
1 + |x|
)
dµ
1 + µ
, x ∈ R, t ∈ [0, T ],
is the unique solution to system (2.3), (2.4).
E x a m p l e 6.2. Let k̂(x) = ρ̂(x) = 1
1+x2 , γ̂(x) = − 2+x4
(1+x2)2
, W 0
0 (x) = 0, and
W 0
1 (x) = e−|x|
√
1 + x2, x ∈ R. Evidently, η(x) =
√
ρ̂(x), θ(x) = 1, σ(x) = x,
x ∈ R, σ−1(λ) = λ, λ ∈ R. Then we have η
θD
m
ηθ = (1 + x2)−1/2
(
d
dx
)m
, m = 1, 2.
Therefore, ϕ ∈ Hm iff ϕ ∈ Hm
−1, m = −1, 0, 1. Hence, W 0
0 ∈ H1. We have
ν(x) =
(
θ2
(
θ2 η
′
η
)′)
(x) +
(
θ2 η
′
η
)2
(x)
=
(
x
1 + x2
)′
+
(
x
1 + x2
)2
=
2x2 − 1
(1 + x2)2
= 1 + γ̂(x), x ∈ R.
Thus conditions (1.5) and (1.6) hold for q = 1, r = 0 (see (3.4)). Due to Theo-
rem 5.5, the state W 0 is not approximately L∞-controllable with respect to sys-
tem (2.3), (2.4) at any given time T > 0 because condition (5.4) is not valid for
this state. Nevertheless, according to Theorem 5.8, the state W 0 is approximately
L∞-controllable with respect to system (2.3), (2.4) at a free time.
Let us find controls solving the approximate L∞-controllability problem with
respect to system (2.3), (2.4) at a free time. Put Z0 = T−1
r S−1W 0 = ηW 0 =
(1 + x2)−1/2. We have Z0
0 (ξ) = 0 and Z0
1 (ξ) = e−|ξ|, ξ ∈ R. Put Tn = n18,
n = 1,∞. In [13, Example 6.1], it was proved that the controls
vn(t) = −8n8
t2
sin
t
2n
sin
t
2n6
sin
(
sin
t
2n
− t
2n6
)
, t ∈ [0, Tn], n = 1,∞,
40 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
solve the approximate L∞-controllability problem with respect to system (2.6),
(2.7) at a free time. From Theorem 4.2, it follows that the controls un(t) = vn(t),
t ∈ [0, Tn], n = 1,∞, solve the approximate L∞-controllability problem with
respect to system (2.3), (2.4) at a free time.
Note that for each t ∈ R, we have Un(t) → U(t) ≡ t as n → ∞, where
Un(t) = un(t)H
(
T 2
n − t2
)
, n = 1,∞. Therefore the sequence {Un}∞n=1 converges
neither in L2(R), nor in L∞(R).
E x a m p l e 6.3. Let k̂(x) = (1+|x|) cosh2 x, ρ̂(x) = cosh2 x
1+|x| , γ̂(x) = (1+|x|)2+
(1+|x|) tanh |x|− 2+|x|
4(1+|x|) coshx , W 0
0 (x) = 8
coshxI2
(
1√
1+|x|
)
, W 0
1 (x) = −
√
3
2 W
0
0 (x),
x ∈ R. Evidently, η(x) = coshx, θ =
√
1 + |x|, σ(x) = sgnx ln(1 + |x|), x ∈ R,
σ−1(λ) = sgnλ
(
e|λ| − 1
)
, λ ∈ R. Therefore, ϕ ∈ H0 iff coshx√
1+|x|
ϕ ∈ H0
0 ; ϕ ∈ H1 iff√
1 + |x| (ϕ coshx)′ ∈ H0
0 . Hence, W 0
0 ∈ H1. We have
ν(x) =
(
θ2
(
θ2 η
′
η
)′)
(x) +
(
θ2 η
′
η
)2
(x)
= (1 + |x|)2 − (1 + |x|) tanh |x| = γ̂(x) +
1
4
+
1
4(1 + |x|)
, x ∈ R.
Thus conditions (1.5) and (1.6) hold for q = 1/2, r(λ) = e−|λ|/4, λ ∈ R (see
(3.4)). Due to Theorem 5.5, the state W 0 is not approximately L∞-controllable
with respect to system (2.3), (2.4) at any given time T > 0 because condition
(5.4) is not valid for this state. Nevertheless, according to Theorem 5.8, the state
W 0 is approximately L∞-controllable with respect to system (2.3), (2.4) at a free
time.
Let us find controls solving the approximate L∞-controllability problem with
respect to system (2.3), (2.4) at a free time. Put Z0 = T−1
r S−1W 0. For this r, the
kernelsK and L of the operators Tr and T−1
r were calculated in [14, Example 5.1].
In our paper they are given by (6.1)–(6.3). We obtain
(
S−1W 0
0
)
(λ) = 8I2
(
e−
|λ|
2
)
,
λ ∈ R. Setting a = e−
|ξ|
2 and b = e−
λ
2 , we get(
T−1
r S−1W 0
0
)
(ξ)
= 8I2
(
e−
|ξ|
2
)
− 2
∞∫
|ξ|
e−
|ξ|+λ
2
J1
(√
e−
λ
2
(
e−
|ξ|
2 − e−
λ
2
))
√
e−
λ
2
(
e−
|ξ|
2 − e−
λ
2
) I2
(
e−
λ
2
)
dλ
= 8I2(a)− 4a
a∫
0
J1
(√
b(a− b)
)
√
b(a− b)
I2(b) db (6.5)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 41
L.V. Fardigola
Setting n = k −m, we get
4a
a∫
0
J1
(√
b(a− b)
)
√
b(a− b)
I2(b) db
= 4a
∞∑
m=0
∞∑
m=0
(−1)n
n!(n+ 1)!m!(m+ 2)!22m+2n+3
a∫
0
(a− b)nb2m+n+2 db
= 4a
∞∑
m=0
∞∑
n=0
(−1)na2m+2n+3(2m+ n+ 2)!
(n+ 1)!m!(m+ 2)!(2m+ 2n+ 3)!22m+2n+3
=
∞∑
m=0
∞∑
k=m
(−1)k+ma2k+4(k +m+ 2)!
(k −m+ 1)!m!(m+ 2)!(2k + 3)!22k+1
=
∞∑
k=0
(−1)ka2k+4
(k + 1)!(2k + 3)!22k+1
k∑
m=0
(
k + 1
m
)
(−1)m
(k +m+ 2)!
(m+ 2)!
=
∞∑
k=0
a2k+4
(k + 1)!(k + 3)!22k+1
because
k∑
m=0
(
k + 1
m
)
(−1)m
(k +m+ 2)!
(m+ 2)!
=
(
xk+2(1− x)k+1
)∣∣∣
x=1
+ (−1)k
(2k + 3)!
(k + 3)!
= (−1)k
(2k + 3)!
(k + 3)!
.
Taking into account (6.5), we obtain
(
T−1
r S−1W 0
0
)
(ξ) = 8I2
(
e−
|ξ|
2
)
−
∞∑
k=0
a2k+4
(k + 1)!(k + 3)!22k+1
= a2 = e−|ξ|.
Therefore, Z0
0 (ξ) = e−|ξ| and Z0
1 (ξ) = −
√
3
2 e
−|ξ|, ξ ∈ R. Hence, Z(ξ, t) =
e−
√
3
2
te−|ξ|, ξ ∈ R, t ∈ [0, Tn], is the unique solution to (2.6), (2.7) with v = vn
and T = Tn. Here vn(t) = e−
√
3
2
t, t ∈ [0, Tn], {Tn}∞n=1 ⊂ (0,∞) is an increasing
sequence, and Tn →∞ as n→∞. When we have∣∣∣∣∣∣∣∣∣∣∣∣(Z(·, Tn)
Zt(·, Tn)
)∣∣∣∣∣∣∣∣∣∣∣∣ ≤ e−√3
2
Tn → 0 as n→∞.
Applying Theorem 4.2, we conclude that W (·, t) = STrZ(·, t), t ∈ [0, Tn], is the
42 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
unique solution to (2.3), (2.4) with u = un and T = Tn where
un(t) = e−
√
3
2
t
−9
8
+
∞∫
0
Ky1(0, ξ)e−ξ dξ
, t ∈ [0, Tn], (6.6)
and [][](
W (·, Tn)
Wt(·, Tn)
)[][]
→ 0 as n→∞.
Thus the controls un, n = 1,∞, solve the approximate L∞-controllability problem
with respect to system (2.6), (2.7) at a free time.
Finally, let us evaluate un, n = 1,∞. Taking into account (6.1)–(6.3), we get
Ky1(0, ξ) =
1
8
e−
ξ
2
(
I1(µ)
µ
+
I2(µ)
2
+
I2(µ)
µ2
)∣∣∣∣
µ=
√
1−e−
ξ
2
.
Then, setting a = e−
ξ
2 , we obtain
∞∫
0
Ky1(0, ξ)e−ξ dξ = −1
4
1∫
0
(
I1(µ)
µ
+
I2(µ)
2
+
I2(µ)
µ2
)∣∣∣∣
µ=
√
1−a
a2 da
= −
∞∑
m=0
4(m+ 1)(m+ 2) + 1
m!(m+ 2)!22m+5
1∫
0
(1− a)ma2 da
= −
∞∑
m=0
4(m+ 1)(m+ 2) + 1
(m+ 2)!(m+ 3)!22m+4
1∫
0
(1− a)ma2 da
= −2 (I3(1) + I1(1)) +
9
8
.
According to (6.6), we have
un(t) = −2 (I3(1) + I1(1)) e−
√
3
2
t, t ∈ [0, Tn], n = 1,∞.
7. Appendix
Lemma 7.1. Let f ∈ H0, ϕ ∈ H1. If there exists f(0), then
〈[Dηθf, ϕ]〉 = −〈[f,Dηθϕ]〉 − (f(0), ϕ(0)). (7.1)
P r o o f. Put F (x) = f(x) if x ≥ 0 and F (x) = 0 otherwise on R. Then
F ∈ H0. For each l = 1,∞, set Fl(x) = f(x) if 0 ≤ x ≤ l and Fl(x) = 0 otherwise
on R. Therefore, Fl ∈ H0
⋂
H0
0 , l = 1,∞, and
[]F − Fl[]00 → 0 as l→∞. (7.2)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 43
L.V. Fardigola
Let ψ ∈ D, ψ(x) ≥ 0 for x ∈ R, suppµ ⊂ [−1, 0],
∫∞
−∞ ψ(x) dx = 1. Put
F kl = Fl ∗ ψk, where ψk(x) = kψ(kx), x ∈ R, l = 1,∞, ∗ is the convolution sign.
Hence, F kl ∈ H0
⋂
C∞(R), suppF kl ∈ [−1/k, l], l, k = 1,∞, and for each l = 1,∞
we have ∥∥∥Fl − F kl ∥∥∥0
0
→ 0 as k →∞.
Therefore, for each l = 1,∞, we get[]
Fl − F kl
[]0
0
→ 0 as k →∞. (7.3)
Summarizing (7.2) and (7.3), we conclude that there exist increasing sequences
{lp}∞p=1 and {kp}∞p=1 such that[]
F − F̂p
[]0
0
→ 0 as p→∞, (7.4)
where F̂p = F
kp
lp
, p = 1,∞. According to (7.4), we obtain
[]
Dηθ
(
F − F̂p
)[]−1
≤
∥∥∥∥(S−1
(
F − F̂p
))′∥∥∥∥−1
0
≤
∥∥∥S−1
(
F − F̂p
)∥∥∥0
0
=
[]
F − F̂p
[]0
→ 0 as p→∞.
By fp, denote the restriction of F̂p to [0,+∞), p = 1,∞. Then fp ∈ H0
⋂
C∞(R),
supp fp ∈ [0, lp], p = 1,∞, and〈[
Dmηθfp, ϕ
]〉
→
〈[
Dmηθf, ϕ
]〉
as p→∞, ϕ ∈ Hm, m = 0, 1. (7.5)
Taking into account the definition of the value of a distribution at the point x = 0,
we have
fp(0) = F̂p(0) =
lp∫
0
f(x)kpψ(kpx) dx
=
〈
f
(
·
kp
)
, ψ−
〉
→ 〈f(0), ψ−〉 = f(0) as p→∞, (7.6)
where ψ−(x) = ψ(−x), x ∈ R.
44 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Transformation Operators and Modified Sobolev Spaces
Let ϕ ∈ H1. Integrating by parts, we obtain
〈[Dηθfp, ϕ]〉 =
∞∫
0
((
S−1fp
)′ (x),
(
S−1ϕ
)
(x)
)
dx
=
((
S−1fp
)
(x),
(
S−1ϕ
)
(x)
)∣∣∞
0
−
∞∫
0
((
S−1fp
)
(x),
(
S−1ϕ
)′ (x)
)
dx
= −(f(0), ϕ(0))− 〈[f,Dηθϕ]〉 .
Taking into account (7.5) and (7.6), we get (7.1), which was to be proved.
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