Approximate controllability of the wave equation with mixed boundary conditions
We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cy...
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Pestov, L. Strelnikov, D. 2020-06-12T15:38:13Z 2020-06-12T15:38:13Z 2018 Approximate controllability of the wave equation with mixed boundary conditions / L. Pestov, D. Strelnikov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 251-263. — Бібліогр.: 14 назв. — англ. 1810-3200 2010 MSC. Primary 35R30; Secondary 35M33, 46E35 https://nasplib.isofts.kiev.ua/handle/123456789/169401 We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cylinder Ω × (0, T). The domain of observation is Σ × [0, 2T], and the pressure on another part (∂Ω\Σ) × [0, 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincar´e inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound. This work was supported by the Volkswagen Foundation project “Modeling, Analysis and Approximation Theory toward Applications in Tomography and Inverse Problem”. The authors would like also to thank M. Belishev, V. Derkach and T. Fastovska for useful discussions and valuable remarks. en Інститут прикладної математики і механіки НАН України Український математичний вісник Approximate controllability of the wave equation with mixed boundary conditions Article published earlier |
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Approximate controllability of the wave equation with mixed boundary conditions |
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Approximate controllability of the wave equation with mixed boundary conditions Pestov, L. Strelnikov, D. |
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Approximate controllability of the wave equation with mixed boundary conditions |
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Approximate controllability of the wave equation with mixed boundary conditions |
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Approximate controllability of the wave equation with mixed boundary conditions |
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Approximate controllability of the wave equation with mixed boundary conditions |
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approximate controllability of the wave equation with mixed boundary conditions |
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Pestov, L. Strelnikov, D. |
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Pestov, L. Strelnikov, D. |
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2018 |
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Український математичний вісник |
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Інститут прикладної математики і механіки НАН України |
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We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cylinder Ω × (0, T). The domain of observation is Σ × [0, 2T], and the pressure on another part (∂Ω\Σ) × [0, 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincar´e inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.
|
| issn |
1810-3200 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/169401 |
| citation_txt |
Approximate controllability of the wave equation with mixed boundary conditions / L. Pestov, D. Strelnikov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 251-263. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT pestovl approximatecontrollabilityofthewaveequationwithmixedboundaryconditions AT strelnikovd approximatecontrollabilityofthewaveequationwithmixedboundaryconditions |
| first_indexed |
2025-11-25T13:06:47Z |
| last_indexed |
2025-11-25T13:06:47Z |
| _version_ |
1850512743281459200 |
| fulltext |
Український математичний вiсник
Том 15 (2018), № 2, 251 – 263
Approximate controllability of the wave
equation with mixed boundary conditions
Leonid Pestov, Dmytro Strelnikov
(Presented by V. O. Derkach)
Abstract. We consider the initial boundary value problem for acoustic
equation in time space cylinder Ω× (0, 2T ) with unknown variable speed
of sound, zero initial data, and mixed boundary conditions. We assume
that (Neumann) controls are located at some part Σ× [0, T ], Σ ⊂ ∂Ω of
lateral surface of the cylinder Ω×(0, T ). The domain of observation is Σ×
[0, 2T ] and the pressure at another part (∂Ω\Σ)× [0, 2T ]) is assumed to
be zero for any control. We prove approximate boundary controllability
for functions from subspace V ⊂ H1(Ω) which traces have vanished on Σ
provided that the observation time is 2T more than two acoustical radii
of the domain Ω. We give an explicit procedure for solving Boundary
Control Problem (BCP) for smooth harmonic functions from V (i.e.
we are looking for a boundary control f which generates a wave uf
such that uf (., T ) approximates any prescribed harmonic function from
V ). Moreover using Friedrichs-Poincare inequality we obtain conditional
estimate for this BCP. Notice that for solving BCP for these harmonic
functions we do not need the knowledge of the speed of sound.
2010 MSC. Primary 35R30; Secondary 35M33, 46E35.
Key words and phrases. Acoustical tomography, inverse problem,
wave equation, Boundary Control method, generalized Friedrichs-Poin-
care inequality.
1. Introduction
Let Ω be a bounded domain in Rn (n ≥ 2) with a smooth boundary
∂Ω. Let Σ ⊂ ∂Ω be an open set with a smooth boundary. The problem,
Received 29.03.2018
This work was supported by the Volkswagen Foundation project “Modeling, Analysis
and Approximation Theory toward Applications in Tomography and Inverse Problem”.
The authors would like also to thank M. Belishev, V. Derkach and T. Fastovska for
useful discussions and valuable remarks.
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
252 Approximate controllability of the wave equation...
which we refer to as a forward one is an initial boundary value problem
for the wave equation with boundary conditions
ρutt −∆u = 0 in Ω× (0, T ), (1.1)
uν |Σ×[0,T ] = f, (1.2)
u|(∂Ω\Σ)×[0,T ] = 0, (1.3)
u|t=0 = ut|t=0 = 0 in Ω. (1.4)
so f = 0 outside of Σ× [0, T ]. Here ρ(x) = 1/c2(x) is a smooth positive
function (c(x) is the speed of sound) and ν is the outward normal vector
to the boundary ∂Ω, uν is the normal derivative. We call function f
Neumann boundary control. Introduce the control space
FT = L2(Σ× (0, T ))
and the set of smooth controls
MT = C∞
0 (Σ× (0, T )).
Let uf be the solution to the forward problem (a wave). Notice, that
smooth controls generate classical waves. Due to the finiteness of the
wave propagation speed one has
suppuf (·, s) ⊂ Ωs, s > 0,
where
Ωs =
{
x ∈ Ω : distρ(x,Σ) ≤ s
}
, s ≥ 0
and the distance being understood in the sense of the Riemannian metric√
ρ(x) |dx|. The subdomain Ωs is the part of Ω filled with waves at the
moment t = s. In particular, for T > T ∗ := sup
x∈Σ
distρ(x, ∂Ω), the relation
ΩT = Ω holds. With the system (1.1)–(1.4) one associates the response
operator RT , which acts by the rule
RT f = uf |Σ×[0,T ].
The inverse problem consists of determining function ρ in Ω via the
response operator R2T provided T > T ∗. One of the natural ways to solve
this problem is the Boundary Control method (BC-method, Belishev,
1986, see e.g. [2–4] and works cited there, and the version of the BC-
method proposed in [12,13]). We do not give a BC-solution of the inverse
problem in this paper. We study the boudary controllability problem of
the dynamical system (1.1)–(1.4) only (see Remark 5.3 at the end of the
paper where we shortly comment on the inverse problem).
L. Pestov, D. Strelnikov 253
The boundary controllability is the principal question in the BC-
method. For scalar hyperbolic equations like (1.1), the system (1.1)–(1.4)
turns out to be approximately controllable. To formulate this property,
observe that the set of final states UT := {uf (·, T ) : f ∈ MT } is con-
tained in L2(Ω
T ). Then the approximate boundary controllability means
that this set is dense in L2(Ω
T ). However the set of final states UT is
not dense in the Sobolev space H1(ΩT ). The main result of this paper
(Theorem 4.3) states that the closure of UT in H1(ΩT ) coincides with the
subspace V = {u ∈ H1
ρ (Ω)|u∂Ω\Σ = 0} of functions u ∈ H1
ρ (Ω) vanishing
on ∂Ω \ Σ. The definition of the real Hilbert space H1
ρ (Ω) will be given
in Section 4.
In contrast to the works cited above, we use measurements (waves) at
the same part of the boundary as controls. It corresponds to the scheme
of using the boundary triple technique in [2]. The boundary triple used in
the present paper is associated with the Zaremba Laplacian with mixed
boundary conditions studied in [9]. In order to give an H1 estimate for
the difference φ− uf (·, T ), where φ in V is a harmonic function, we use
an analogue of the Friedrichs inequality with an estimate from a part of
the boundary.
2. Bilinear forms
Here we introduce one of the main tools of the BC method — the
symmetric energy forms defined on the set of smooth controls. In what
follows we fix T > T ∗ so that ΩT = Ω. We define two symmetric bilinear
forms on MT ×MT
[f, g]1 :=
∫
Ω
ρ(x)uf (x, T )ug(x, T ) dx, (2.1)
[f, g]2 :=
∫
Ω
(
∇uf (x, T ),∇ug(x, T )
)
dx. (2.2)
Both forms [ ·, · ]1 and [ ·, · ]2 are explicitly determined by the inverse data,
i.e. the response operator R2T . We derive these formulas here. For a
function u, which depends on time and, possibly, other variables, denote
u±(·, t) =
u(·, t)± u(·, 2T − t)
2
,
(Iu)(·, t) =
∫ t
0
u(·, s) ds, t ∈ [0, 2T ].
254 Approximate controllability of the wave equation...
Proposition 2.1. For any controls f, g ∈ MT the equalities∫
Ω
ρ(x)ug(x, T )uf (x, T ) dx =
∫
Σ×[0,T ]
[(
R2T g
)
+
If − g+IR
2T f
]
dt dσ
(2.3)∫
Ω
(
∇ug,∇uf
)
(x, T ) dx =
∫
Σ×[0,T ]
[
f
∂
∂t
(Rg)+ + g+
∂
∂t
(Rf)
]
dt dσ,
(2.4)
are valid, where dσ is the standard measure on the boundary ∂Ω.
Proof. For any smooth solution v to the wave equation (1.1), the equality
ρ
(
vuft − ufvt
)
t
= div
(
v∇uf − uf∇v
)
holds. Clearly functions ug± satisfy the wave equation. Substituting ug+
for v and integrating over Ω × [0, T ], we obtain (note, that ug+(·, T ) =
ug(·, T ), (ug+)t(·, T ) = 0)∫
Ω
ρ(x)ug(x, T )uft (x, T ) dx =
∫
∂Ω×[0,T ]
(
ug+
∂uf
∂ν
− uf
∂ug+
∂ν
)
dt dσ
(1.3)
=
∫
Σ×[0,T ]
(
ug+
∂uf
∂ν
− uf
∂ug+
∂ν
)
dt dσ
(1.2)
=
∫
Σ×[0,T ]
(
ug+f − ufg+
)
dt dσ.
Taking into account that uft = uft and uIf = Iuf , and denoting ft = f̃
we arrive at∫
Ω
ρ(x)ug(x, T )uf̃ (x, T ) dx =
∫
Σ×[0,T ]
(
ug+If̃ − Iuf̃g+
)
dt dσ. (2.5)
Since
(R2T f)(x, t) = (RT f)(x, t) for all x ∈ ∂Ω, t ∈ (0, T ) (2.6)
then
ug+(x, t) =
1
2
{(
RT g
)
(x, t) +
(
R2T g
)
(x, 2T − t)
}
=
(
R2T g
)
+
(x, t) for x ∈ ∂Ω, t ∈ (0, T ),
(2.7)
uf̃ (x, t) = (RT f̃)(x, t) = (R2T f̃)(x, t) for x ∈ ∂Ω, t ∈ (0, T ). (2.8)
By substituting (2.7) and (2.8) into (2.5) and replacing f̃ by f one obtains
(2.3).
L. Pestov, D. Strelnikov 255
Consider the form [ ·, · ]2 in (2.2). For any smooth solution v to the
wave equation the equality[
ρvtu
f
t + (∇v,∇uf )
]
t
= div
(
vt∇uf + uft∇v
)
holds. Substituting ug+ for v and integrating over Ω × [0, T ], we get
(using (1.2)) the equality∫
Ω
(
∇ug,∇uf
)
(x, T ) dx =
∫
Σ×[0,T ]
(
(ug+)tf + uft g+
)
dt dσ,
which coincides with (2.4).
Remark 2.2. In the case when Σ is the whole boundary ∂Ω the for-
mula (2.3) coincides with the formula of Blagoveshchenskii presented
in [1]. The formula (2.4) in the case when Σ = ∂Ω was obtained in [13].
3. Boundary triple and Zaremba operator
Different versions of dynamical systems with boundary controls are
related (see [2]) to different choices of boundary triples for the operator in
the space domain, see definitions in [6,8]. Here we introduce the boundary
triple corresponding to the system (1.1)–(1.4).
Let the minimal operator−∆min (resp. the maximal operator−∆max)
be defined as the closure in L2(Ω) of the operator −∆ restricted to C∞
0 (Ω)
(resp. C∞(Ω)). It is known (see, for instance, [5, Theorem 4.8]) that
−∆max = (−∆min)
∗ and
dom(−∆min) = H2
0 (Ω), dom(−∆max) =
{
u ∈ L2(Ω)|∆u ∈ L2(Ω)
}
,
where ∆ is understood in the sense of distributions. Let γD and γN be
the Dirichlet and the Neumann traces
γD : u 7→ u|∂Ω , γN : u 7→ uν |∂Ω , f ∈ H2(Ω). (3.1)
It is known, e.g. from Lions and Magenes [11] that γD and γN , defined
originally on H2(Ω) admit continuations to surjective operators
γD : dom(−∆max) → H−1/2(∂Ω), γN : dom(−∆max) → H−3/2(∂Ω).
Dirichlet −∆D and Neumann −∆N realizations of −∆, defined as restric-
tions of the operator −∆max to the domains
dom(−∆D) =
{
u ∈ H2(Ω): γDu = 0
}
,
256 Approximate controllability of the wave equation...
dom(−∆N ) =
{
u ∈ H2(Ω): γNu = 0
}
,
are selfadjoint extensions of the operator −∆min. One more selfadjoint
realization of −∆, so-called Zaremba extension −∆Σ of −∆min, is defined
as the restriction of −∆max to the set
dom(−∆Σ) =
{
u ∈ dom(−∆max) : γDu|∂Ω\Σ = 0, γNu|Σ = 0
}
, (3.2)
see [9]. Its domain is not contained in H3/2(Ω), however for every ϵ > 0
the following inclusion holds dom(−∆Σ) ⊂ H3/2−ϵ(Ω). Notice, that the
operator −∆Σ in L2(Ω) has a discrete spectrum.
Let H3/2
∆ := {u ∈ H3/2(Ω): ∆u ∈ L2(Ω)}. According to [11]
γD(H
3/2
∆ (Ω)) = H1(∂Ω), γN (H
3/2
∆ (Ω)) = L2(∂Ω)
and for all u, v ∈ H
3/2
∆ the following Green formula holds∫
Ω
(u∆v − v∆u) dx =
∫
∂Ω
(γDuγNv − γNuγDv) dσ. (3.3)
Let us define the subspace D∗ of H3/2
∆ (Ω) by
D∗ :=
{
u ∈ H
3/2
∆ (Ω): (γDu)|∂Ω\Σ = 0
}
, (3.4)
and let the operators γΣN , γ
Σ
D be defined as restrictions of the operators
u 7→ γNu|Σ, u 7→ γDu|Σ to the domain D∗.
γΣNu := γNu|Σ, γΣDu := γDu|Σ, (u ∈ D∗). (3.5)
The triple {L2(Σ), γ
Σ
N , γ
Σ
D} is a boundary triple for the operator −∆max
in the sense of [6]. As was shown in [7], the operator −∆0,Σ defined as
the restriction of −∆max to the domain
dom(−∆0,Σ) =
{
u ∈ H
3/2
∆ (Ω): (γDu)|∂Ω\Σ = (γNu)|Σ = 0
}
(3.6)
is essentially selfadjoint in L2(Ω). Namely, the closure of −∆0,Σ coincides
with the Zaremba operator −∆Σ.
Remark 3.1. By the terminology used in [7, Definition 1.8] the triple
{L2(Σ), γ
Σ
N , γ
Σ
D} is called an ES-generalized boundary triple for −∆max,
with “ES” meaning that the operator −∆0,Σ in (3.6) is essentially self-
adjoint.
The Green formula corresponding to this boundary triple takes the
form ∫
Ω
(u∆v − v∆u) dx =
∫
Σ
(γDuγNv − γNuγDv) dσ (3.7)
with u, v ∈ D∗. This formula will be used in the next section.
L. Pestov, D. Strelnikov 257
4. Approximate controllability
In order to apply the general results of [11] to the system (1.1)–(1.4)
let us consider the Hilbert space H(Ω) := L2
ρ(Ω) with the norm
∥u∥H(Ω) :=
(∫
Ω
{
ρ(x)|u(x)|2
}
dx
)1/2
. (4.1)
If there are ρ1, ρ2 > 0 such that ρ satisfies the inequalitues
ρ1 ≤ ρ(x) ≤ ρ2 (x ∈ Ω) (4.2)
then the norm (4.1) is equivalent to the standard norm in L2(Ω). Simi-
larly, the Sobolev space H1
ρ (Ω) is defined as the standard Sobolev space
H1(Ω) endowed with the inner product
(u, v)H1
ρ(Ω) :=
∫
Ω
{ρuv + (∇u,∇v)} dx. (4.3)
Let V be a subspace of H1
ρ (Ω) specified by the equality
V = {u ∈ H1
ρ (Ω): u∂Ω\Σ = 0}. (4.4)
Next we recall (see [10, Lemma 2.3]) the following Friedrichs type in-
equality:
k1
∫
Ω
|u|2 dx ≤
∫
Ω
|∇u|2 dx+
∫
Σ0
|u|2 dσ, (4.5)
which is valid for every open set Σ0 ⊂ ∂Ω with σ(Σ0) > 0, for some
constant k1 > 0 and for all u ∈ H1
ρ (Ω).
Proposition 4.1. Let Σ0 be an open subset of ∂Ω with σ(Σ0) > 0, and
let W 1
2,2(Ω,Σ0) be the completion of the set of functions C∞(Ω) ∩ C(Ω)
with respect to the norm
∥u∥W 1
2,2(Ω,Σ0) =
(∫
Ω
|∇u|2 dx
)1/2
+
(∫
Σ0
|u|2 dσ
)1/2
, u ∈ H1(Ω),
(4.6)
and let ρ satisfy the inequalities (4.2). Then:
(1) W 1
2,2(Ω,Σ0) = H1
ρ (Ω) and there exist constants C1, C2 > 0 such
that
C1∥u∥W 1
2,2(Ω,Σ) ≤ ∥u∥H1(Ω) ≤ C2∥u∥W 1
2,2(Ω,Σ) (4.7)
for all u ∈ H1
ρ (Ω).
258 Approximate controllability of the wave equation...
(2) If Σ is an open subset of ∂Ω with σ(∂Ω \ Σ) > 0, and ρ satis-
fies (4.2) then the norm ∥ · ∥H1
ρ(Ω) on V is equivalent to the norm(∫
Ω |∇u|2 dx
)1/2
.
Proof. (1) The proof of (1) follows from the Friedrichs type inequal-
ity (4.5) and from the estimate (see [11])
∥u∥L2(∂Ω) ≤ k2∥u∥H1(Ω) (u ∈ H1(Ω)),
which is valid for some k2 > 0.
(2) The statement (2) is immediate from (1) applied for Σ0 = ∂Ω\Σ,
since then the second integral in (4.6) vanishes for all u ∈ V .
Proposition 4.2. Let Σ be an open subset of ∂Ω with σ(∂Ω \ Σ) > 0,
let ρ satisfies the inequalities (4.2), let the form a(u, v) be defined on V
by
a(u, v) :=
∫
Ω
(∇u,∇v)dx, u, v ∈ V.
and let the operator A in L2
ρ(Ω) be given by A := ρ−1(−∆Σ). Then:
(1) The operator A is selfadjoint and nonnegative in H = L2
ρ(Ω). The
spectrum of A is discrete.
(2) The form a(u, v) admits the representation
a(u, v) = (Au, v)H , u, v ∈ dom(−∆Σ).
Proof. The statement (1) follows from the properties of the Zaremba
operator −∆Σ mentioned in Section 3. The discretness of the spectrum
of A is implied by the Courant minimax principle and the corresponding
statement for the operator −∆Σ in L2(Ω). The nonnegativity of A is
postponed until the next paragraph.
Substituting in the 1-st Green formula
−(∆u, v)L2(Ω) =
∫
Ω
(∇u(x),∇v) dx−
∫
∂Ω
u(x)vν(x) dσ,
u, v ∈ dom(−∆Σ) one obtains
a(u, v) = −(∆u, v)L2(Ω) = −(ρ−1∆u, v)L2
ρ(Ω) = (Au, v)H .
This formula proves (2) and the nonnegativity of A.
L. Pestov, D. Strelnikov 259
Recall that the set of reachable states of the wave at the instant of
time t = T is defined by the equality
UT :=
{
uf (·, T ) : f ∈ M
}
. (4.8)
It is clear, that UT ⊂ V .
Theorem 4.3. Let Σ be an open subset of ∂Ω with σ(∂Ω \ Σ) > 0 and
let ρ satisfy the inequalities (4.2). Then the variety UT is dense in V .
Proof. Let φ ∈ V ⊖ UT , i.e. for every control f ∈ MT the following
equality holds (
uf (·, T ), φ
)
= 0. (4.9)
Let us show that φ = 0. By [11, Theorem 3.8.1] the following system
ρ(x)vtt −∆v = 0 in Ω× (0, T ), (4.10)
v|t=T = 0, vt|t=T = φ in Ω, (4.11)
v|∂Ω\Σ = vν |Σ = 0 (4.12)
has a weak solution v ∈ L2(0, T ;V ) such that vt ∈ L2(0, T ;H). The
latter means, that
a(v, u) +
∫
Ω
ρuvtt dx = 0 (4.13)
Let the wave uf (x, t) be the solution of the problem (1.1)–(1.4), and let
v be the solution of the system (4.10)–(4.12). Then by the 1-st Green
formula
a(v, u)−
∫
Σ
vuνdσ +
∫
Ω
ρuttv dx = 0.
Subtracting the last equation from (4.13) and substituting ufν |Σ×[0,T ] by
f one obtains∫
Ω
ρ(ufvt − vuft )t dx = −
∫
Σ
ufνv dσ = −
∫
Σ
fv dσ. (4.14)
Integrating this identity on [0, T ] yields∫
Ω
ρuf (x, T )φ(x) dx = −
∫
Σ×[0,T ]
fv dσ dt. (4.15)
Let now w be the weak solution of the system (see [11, Theorem 3.8.1])
ρ(x)wtt −∆w = 0 in Ω× (0, T ), (4.16)
260 Approximate controllability of the wave equation...
w|t=T = φ, wt|t=T = 0 in Ω, (4.17)
w|∂Ω\Σ = wν |Σ = 0, (4.18)
such that w ∈ L2(0, T ;V ) such that wt ∈ L2(0, T ;H). Notice that
div
(
wt∇uf + uft∇w
)
=
(
ρuft wt + (∇uf ,∇w)
)
t
(4.19)
and by the Gauss–Ostrogradskii formula∫
Ω
(
ρuft wt + (∇uf ,∇w)
)
t
dx =
∫
∂Ω
(wtuν + utwν) dσ. (4.20)
Integrating this identity on t ∈ [0, T ] one obtains∫
Ω
(
∇uf ,∇φ
)
(x, T ) dx =
∫
Σ×[0,T ]
fwt dσ dt. (4.21)
Combining the equalities (4.15) and (4.21) and taking into account (4.9)
one obtains
0 = (uf , φ)H1
ρ(Ω) = −
∫
Σ×[0,T ]
f(v − wt) dσ dt. (4.22)
Since MT is dense in L2(Σ× [0, T ]) one obtains from (4.22)
(v − wt)|Σ×[0,T ] = 0. (4.23)
By virtue of (4.12), (4.18) one obtains from (4.23)
(v − wt)|∂Ω×[0,T ] = (v − wt)ν |Σ×[0,T ] = 0. (4.24)
Therefore, the function u = v − wt is a solution of the system
ρ(x)utt −∆u = 0 in Ω× (0, T ), (4.25)
u(x, T ) = 0 in Ω, (4.26)
u|∂Ω×[0,T ] = uν |Σ×[0,T ] = 0. (4.27)
Let us consider the odd extension of u to Ω× [T, 2T ]:
ũ(·, t) :=
{
u(·, t), t ∈ [0, T ),
−u(·, 2T − t), t ∈ [T, 2T ].
(4.28)
Then the function ũ satisfies the system
ρũtt −∆ũ = 0, in Ω× (0, 2T ), (4.29)
L. Pestov, D. Strelnikov 261
ũ|∂Ω×[0,2T ] = ũν |Σ×[0,2T ] = 0. (4.30)
The reasonings of [3] show that by the Holmgren–John–Tataru theo-
rem [14] ũ = 0 in the space-time domain
{(x, t) ∈ Ω× (0, 2T ) : distρ(x,Σ) < t < 2T − distρ(x,Σ)} . (4.31)
Therefore, v = wt in the domain {(x, t) ∈ Ω × (0, T ) : distρ(x,Σ) < t ≤
T} and hence
w(·, t) = φ−
∫ T
t
v(·, s) ds. (4.32)
By the last equality one gets φν |Σ = 0, φ|∂Ω\Σ = 0 and
0 = ρwtt −∆w = −∆φ−
∫ T
t
[ρvss −∆v](·, s)ds+ ρvt|t=T
= ρφ−∆φ = ρ(I +A)φ in Ω.
(4.33)
Since A > 0 this eigenvalue problem has no non-trivial solutions, there-
fore φ = 0.
5. Estimates for the error of approximation
Now we are going to estimate the error ∥uf (·, T ) − φ∥L2(Σ) of the
approximation of a harmonic function φ by the wave uf (·, T ) (f ∈ MT )
in terms of the response operator.
For every φ ∈ H1
ρ (Ω) let us set
Φ(f) :=
∫
Ω
∣∣∣∇uf (·, T )−∇φ
∣∣∣2 dx. (5.1)
Proposition 5.1. Let φ ∈ V be a harmonic function in Ω and let f ∈
MT . Then the functional Φ(f) takes the form
Φ(f) = [f, f ]2 − 2
∫
Σ
(R2T f)(·, T )φν dσ +
∫
Σ
φφν dσ. (5.2)
Proof. Indeed, it follows from (5.1) that
Φ(f) =
∫
Ω
|∇uf (·, T )|2dx− 2
∫
Ω
(∇uf (·, T ),∇φ) dx+
∫
Ω
|∇φ|2dx.
In view of the 1-st Green formula∫
Ω
(∇uf (x, T ),∇φ) dx =
∫
∂Ω
uf (x, T )φν(x) dσ,∫
Ω
|∇φ|2dx =
∫
∂Ω
φ(x)φν(x) dσ
and the assumptions uf |Ω\Σ = φ|Ω\Σ = 0 this implies (5.2).
262 Approximate controllability of the wave equation...
Proposition 5.2. Let Σ be an open subset of ∂Ω with σ(∂Ω \ Σ) > 0,
let ρ satisfy the inequalities (4.2) and let ε > 0. Then:
(1) For every harmonic function φ ∈ V there exists a control f ∈ MT
such that
Φ(f) ≤ ε2. (5.3)
(2) There exists a constant C > 0 such that if (5.3) holds for some
φ ∈ V and f ∈ MT then∥∥∥uf (·, T )− φ(·)
∥∥∥
H1
ρ(Ω)
≤ Cε. (5.4)
Proof. (1) Since UT is dense in V (⊂ H1
ρ (Ω)) then for every ε > 0 there
is f ∈ MT such that∥∥∥(uf )(·, T )− φ
∥∥∥2
L2(Ω)
+
∥∥∥(∇uf )(·, T )−∇φ
∥∥∥2
L2(Ω)
≤ ε2. (5.5)
In view of (5.1) this yields (5.3).
(2) It follows from (4.3), the Friedrichs type inequality (4.6) and (5.3)
that ∥∥∥uf (·, T )− φ(·)
∥∥∥2
H1
ρ(Ω)
≤ max
x∈Ω
ρ(x)1/2
∥∥∥(uf )(·, T )− φ
∥∥∥2
L2(Ω)
+
∥∥∥(∇uf )(·, T )−∇φ
∥∥∥2
L2(Ω)
≤
(
1 +
ρ2
k1
)∥∥∥(∇uf )(·, T )−∇φ
∥∥∥2
L2(Ω)
≤
(
1 +
ρ2
k1
)
ε2.
This proves (5.4).
Remark 5.3. In the case when Σ coincides with ∂Ω one has the equality
V = H1(Ω) and hence the system is approximately controllable inH1(Ω).
As is known the set of products of harmonic functions φ,ψ ∈ H1(Ω) is
dense in L2(Ω) and this allows to solve the inverse problem for the speed
of sound ρ. In the case when Σ ̸= ∂Ω and m(∂Ω\Σ) > 0 we do not know
how big the set {φψ : φ,ψ ∈ V } is. If it is dense in L2(Ω) then one can
use the strategy of [13] in order to give the procedure of reconstruction
of ρ.
L. Pestov, D. Strelnikov 263
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Contact information
Leonid Pestov Immanuel Kant Baltic Federal University,
Kaliningrad, Russia
E-Mail: lpestov@kantiana.ru
Dmytro Strelnikov Vasyl’ Stus Donetsk National University,
Vinnytsya, Ukraine
E-Mail: d.strelnikov@donnu.edu.ua
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