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IMA Journal of Mathematical Control and Information Advance Access published October 18, 2016
IMA Journal of Mathematical Control and Information (2016) 00, 1–17
doi: 10.1093/imamci/dnw055
Controllability of the wave equation via multiplicative controls
I. El Harraki and A. Boutoulout∗
Laboratory of Modeling Analysis and Control Systems, Department of Mathematics and Computer
Science, Faculty of Sciences Meknes, Moulay Ismail University, Meknes 50000, Morocco
∗
Corresponding author: [email protected]
[Received on 15 November 2015; revised on 15 June 2016; accepted on 12 September 2016]
Keywords: controllability; multiplicative controls; semi-linear damped wave; stabilization.
1. Introduction
In the last decades, many controllability problems for various types of non-linear dynamical systems have
been considered in several monographs and publications (see Lions, 1988; Coron, 2007). In most cases,
internal traditionally additive control is employed (see Haraux, 2004; Fu et al., 2007). Nevertheless,
these models are unfit to study other important problems, in which the control in the system acts in
a multiplicative way. The bilinear controls also play an important role in physical systems modelling,
when the linear representation is not significant enough. This explains the increasing interest in bilinear
controllability. In Ball et al. (1982), the authors considered the controllability of the abstract infinite
dimensional bilinear system, which seems to be one of the most important works in this field. They
treated the rod equation
utt + uxxxx + p(t)uxx = 0,
with hinged ends, and the wave equation
utt − uxx + p(t)u = 0,
with Dirichlet boundary conditions, where p is the control (the axial load). The global approximate
controllability was shown using the non-harmonic Fourier series approach under the assumption that all
the modes in the initial data are active. We cite also the paper by Kime (1995), on bilinear simultaneous
controllability for PDEs, dealing with the controllability of a simple Schrodinger equation and the rod
equation. Owing to the negative result for multiplicative controls given by Ball et al. (1982), bilinear
systems have been considered not controllable for a long time. However, in the last few years advances
have been made. In Beauchard (2011), the author have taken a linear wave equation on the interval
(0, 1), with bilinear control and Neumann boundary conditions. She studied the controllability locally
© The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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In this paper, we study the controllability for the multidimensional wave equation. The first part deals
with the linear wave equation with unbounded damping. Two multiplicative controls are constructed to
conduct the solution of the system to a neighbourhood of the desired equilibrium state. The second part
treats the controllability of the semi-linear wave equation. We show the approximate controllability for
this problem, and we prove the exact controllability for the one-dimensional case using bilinear controls.
2
I. EL HARRAKI AND A. BOUTOULOUT
around a constant reference trajectory in the space H 3 (0, 1) × H 2 (0, 1). In Khapalov (2010), the author
investigated the approximate controllability of the state equilibriums for the following control system:
⎧
ytt = yxx + v(x, t)y + γ (t)yt − F(y) x ∈ (0, 1), t > 0
⎪
⎪
⎨
y(x, 0) = y0 (x), yt (x, 0) = y1 (x)
x ∈ (0, 1)
(1.1)
⎪
⎪
⎩
y(0, t) = y(1, t) = 0,
2. Approximate controllability of the linear wave equation
2.1. Problem statement
Let Ω be an open bounded subset of Rn with regular boundary ∂Ω. We consider the space H = L 2 (Ω)
with its inner product and the corresponding norm denoted by H . We define the unbounded operator
A = −Δ with the domain D(A) = H 2 (Ω) ∩ H01 (Ω). Let us also consider the space V = H01 (Ω) with
1
the norm vV = A 2 vH , v ∈ V . For a time T1 > 0 large enough, we set Q = Ω × (0, T1 ) and
Σ = ∂Ω × (0, T1 ). We further consider a complex Hilbert space U endowed with its inner product
., .U , and the induced norm U . We denote by (D(A1/2 ) ) the dual space of (D(A1/2 )) with respect
to the inner product in H. Let us consider the following initial and boundary value problem for the
n-dimensional wave equation:
⎧
y = Δy + v(x, t)y − u(t)B̄yt
in Q
⎪
⎪
⎨ tt
(2.1)
y(x, 0) = y0 (x), yt (x, 0) = y1 (x) in Ω
⎪
⎪
⎩
y(ξ , t) = 0
on Σ,
where B̄ = BB∗ for B ∈ L(U, (D(A1/2 ) ) with adjoint operator B∗ .
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in which the functions v ∈ L ∞ ((0, +∞) × (0, 1)) and γ ∈ L ∞ (0, ∞) are regarded as bilinear or
multiplicative controls. If the system (1.1) models the oscillations of vibrating string with clapped ends,
then v(x, t) can be viewed as an axial load at point x and time t, and γ (t) as the gain of viscous (motionactivated) damping, acting upon the string at time t. Such controllability problems may arise in the
context of smart materials, whose properties can be altered by applying various factors (temperature,
electric current, magnetic field). In Ouzahra (2014), the author considered the linear case of (1.1) with
bounded damping, and proved the approximate and the exact controllability for the equilibrium states.
The problem of optimal control of (1.1) without damping has been considered in Liang (1999), where a
bilinear control has been used to make the state solution close to a desired profile minimizing a quadratic
cost of control.
Our purpose in this paper is to study two cases for the multidimensional version of problem (1.1).
The first one deals with the linear wave with unbounded damping. The second one concerns the semilinear wave with bounded damping. The paper is organized as follows. The second section presents the
considered system and establishes an approximate controllability result of a set of target states in the
state space H01 (Ω) × L 2 (Ω). In the third section, we provide bilinear controls ensuring the approximate
controllability of the multi-dimensional semi-linear wave equation, and we give an extension result. The
rest of the section is devoted to exact controllability of the one-dimensional semi-linear wave equation
using the approximate controllability result, and the exact controllability of an equivalent system with
additive control.
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
3
α1 for t ∈ (0, T1 )
with α1 a real positive number. In order to reach
0
for t > T1 ,
a given target state (yd , 0) with yd ∈ D(A), we will consider the following assumptions:
∞
Here v ∈ L (Q), and u(t) =
(H1 ) A − v1 I is a strictly positive operator where v1 =
v1 ∈ L ∞ (Ω).
−Δyd
1G , with G = {x ∈ Ω : yd (x) = 0} and
yd
(H2 ) For a fixed positive constant β, the function H defined by
is a bounded function on Cβ := {λ ∈ C such that Re(λ) = β}.
(H3 ) There exists a time T > 0 and a constant C > 0 such that
T
0
(B∗ φ(t))t 2U dt ≥ C (φ0 , φ1 ) 2H×V ∀(φ0 , φ1 ) ∈ D(A) × V ,
where φ satisfies
⎧
φtt = Δφ + v1 (x)φ
in Q
⎪
⎪
⎨
φ(x, 0) = φ0 (x), φt (x, 0) = φ1 (x) in Ω
⎪
⎪
⎩
φ(ξ , t) = 0
on Σ.
(2.2)
Remark 2.1 Note that the assumption (H2 ) is less restrictive than the boundedness of B which was
considered in Ouzahra (2014) and Khapalov (2004, 2010), and covers a class of unbounded operators.
2.2. Well posedness
It is well known that for u(t) = 0 the system (2.1) is well posed (see Lions and Magenes, 1968). Further,
when u(t) = α1 , we have the following result:
Lemma 2.1 For all Z0 ∈ V × H, there exists a unique solution Z(t) ∈ V × H to system (2.1). Moreover,
if Z0 ∈ D(Ã1 ) then Z ∈ C(0, ∞; D(Ã1 )) ∩ C 1 (0, ∞; V × H), where Ã1 is the unbounded linear operator
given by
0
Δ
0
,
−α1 B̄
(2.3)
with the domain
D(Ã1 ) = {(x, y) ∈ V × V , Δx − α1 B̄y ∈ H}.
(2.4)
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H : λ ∈ C+ = {λ ∈ C such that Re(λ) > 0} −→ λB∗ (λ2 I + A − v1 I)−1 B ∈ L(H)
4
I. EL HARRAKI AND A. BOUTOULOUT
Proof. We write the system (2.1) in the form
Zt = A˜1 Z + P˜1 Z, Z0 =
y0
,
y1
(2.5)
with
y
y
Z=
, Zt =
yt
yt
and P̃1 Z =
t
0
.
vI
Remark 2.2 For v(x, t) = v1 (x) and u(t) = 1 the system (2.1) can be also seen as a classical control
problem
Zt = A˜2 Z + Bu1 , Z0 =
y0
,
y1
(2.6)
where
0
u1 = B yt , Ã2 =
Δ + v1 I
∗
0
0
and B =
0
.
B
It is clear that the operator Ã2 generates a strongly continuous group of operator on V × H, denoted by
(S2 (t))t∈R and has a natural extension to the Hilbert space H × V (see Tucsnak and Weiss, 2009). If
Z0 ∈ V × H, then Z ∈ C([0, T1 ], H × V ), ∀T1 > 0 and we have
T1
Z(t) = S2 (t)Z0 +
S2 (t − s)Bu1 (s)ds.
0
Definition 2.1 We say that B is an admissible control operator for (S2 (t))∗t∈R , if for each T1 > 0 there
exists a constant C > 0 such that for all Z0 ∈ D(A) × V
T1
0
B ∗ S2∗ (t)Z0 2U dt ≤ CZ0 2V ×H .
Proposition 2.1 (see Tucsnak and Weiss, 2009). If B is an admissible control operator for (S2 (t))∗t∈R
then,
Z ∈ C([0, T1 ], V × H), ∀T1 > 0.
Remark 2.3 Note that the assumption (H2 ) implies the admissibility of the operator B for (S2 (t))∗t∈R
(see Ammari & Tucsnak, 2001).
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Since Ã1 is m-dissipative (see Ammari & Nicaise, 2015), it generates a contraction semi-group on V × H
denoted by (S1 (t)). Moreover, remarking that P̃1 is a Lipschitz perturbation of the generator, we deduce
the existence of the solution (see Pazy, 1983, page 183).
5
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
Now, in order to prove the approximate controllability, we will use the following proposition:
Proposition 2.2 (see Attouch et al., 2006). Let v ∈ W 1,p (Ω), 1 ≤ p < +∞. Then, for each i = 1, . . . , N
1v=0
In other words,
∂v
= 0.
∂xi
∂v
= 0. a.e. in Ẽ = {x ∈ Ω : v(x) = 0}, i = 1, . . . , N.
∂xi
y(T1 ) − yd V + yt (T1 ) H < ε.
(2.7)
Proof. Let y = ψ + yd , from the system (2.1) we have
⎧
ψ = Δψ + v(x, t)ψ − u(t)B̄ψt + Δyd + v(x, t)yd
⎪
⎪
⎨ tt
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x)
⎪
⎪
⎩
ψ(ξ , t) = 0
in
Q
in
Ω
(2.8)
on Σ,
therefore, with the controls v(x, t) = v1 (x) and u(t) = 1 we have
⎧
ψ = Δψ + v1 (x)ψ − B̄ψt + Δyd + v1 (x)yd
⎪
⎪
⎨ tt
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x)
⎪
⎪
⎩
ψ(ξ , t) = 0
in
Q
in
Ω
on
Σ.
(2.9)
It follows from (H1 ) that for all x ∈ G
Δyd + v1 (x)yd = 0.
Moreover, using the Proposition 2.2 we have ∇yd = 0 a.e. in Ω \ G, then taking the second order
derivative, we obtain
Δyd + v1 (x)yd = 0 a.e. x ∈ Ω,
hence, we deduce that ψ satisfies
⎧
ψ = (Δ + v1 (x)I)ψ − B̄ψt
in Q
⎪
⎪
⎨ tt
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x) in Ω
⎪
⎪
⎩
ψ(ξ , t) = 0
on Σ.
(2.10)
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Theorem 2.1 Suppose that assumptions (H1 )–(H3 ) hold, then for all initial state (y0 , y1 ) ∈ V × H and
for every ε > 0, there exists a time T1 > 0 and two multiplicative controls v(x, t) and u(t) such that the
solution of (2.1) satisfies
6
I. EL HARRAKI AND A. BOUTOULOUT
Under the assumptions (H1 )–(H3 ), Theorem 2.4 (see Ammari & Tucsnak, 2001) ensures the existence
of some positive constant C such that
ψ(t) 2V + ψt (t) 2H ≤
thus, the estimate (2.7) holds for T1 > max(
C
(ψ0 , ψ1 )2D(Ã ) ,
1
1+t
4C(ψ0 , ψ1 )2D(Ã ) − ε 2
1
, 0) which gives the approximate
ε2
controllability.
•
The above results can be extended to other state spaces with more regularity.
•
The strict positivity of A − v1 I holds for all v1 < −ρ1 where ρ1 is the first non-positive eigenvalue
of Δ.
2.3. Example
We study the approximate null controllability of the equation modelling the vibrations of a string.
Suppose that the string is of length π, and we denote by δb the Dirac mass concentrated at the point b.
Precisely, we consider the following initial and boundary value problems:
⎧
ytt (x, t) = yxx (x, t) + v(x, t)y(x, t) + u(t)yt (b, t)δb
⎪
⎪
⎨
y(x, 0) = y0 (x), yt (x, 0) = y1 (x)
⎪
⎪
⎩
y(0, t) = y(π, t) = 0.
x ∈ (0, π ), t > 0
x ∈ (0, π )
(2.11)
Our previous consideration suggests that the natural well-posedness space for (2.11) is X = V ×H, where
H = L 2 (0, π ) and V = H01 (0, π ). We, now, show that problem (2.11) fits into the abstract framework.
We define
B: C
−→
V
k
−→
kδb ,
V
−→
C
y
−→
y(b).
(2.12)
hence by direct calculation, we see that
B∗ :
(2.13)
π
We consider the operator A : V −→ V defined by Au, v =
0
ux vx dx ∀(u, v) ∈ V 2 , therefore
the operator Au = −uxx with the domain D(A) = H 2 (0, π ) ∩ H01 (0, π ) is unbounded, self-adjoint
and positive from L 2 (0, π ) into itself. With these definitions, the problem (2.11) enters in the abstract
framework from above section. In order to use the Theorem 2.1, we recall the following proposition (see
Ammari & Nicaise, 2015), which ensures that the hypothesis (H2 ) holds true for our application:
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Remark 2.4
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
7
Proposition 2.3 (Ammari & Nicaise, 2015). Consider the system (2.11). Let γ > 0, k ∈ (0, π ) a fixed
positive constant and Cγ = {λ ∈ C such that Re(λ) = γ }. Then for all λ ∈ C+
H(λ) = λB∗ (λ2 I + A)−1 B
is a linear mapping from C into itself, and we have H(λ)l = h(λ)l, ∀l ∈ C where
h(λ) =
sinh(λk) sinh(λ(k − π ))
.
sinh λπ
Denote by S the set of all numbers ρ ∈ (0, 1)\Q such that if [a0 , a1 , . . . , an , . . .] is the expansion of
ρ as a continued fraction, then an is bounded. We recall also (see Ammari & Tucsnak, 2001) that for
T > 0, there exists a constant C > 0 such that the solution φ of
⎧
φtt = Δφ
in Q
⎪
⎪
⎨
φ(x, 0) = φ0 (x), φt (x, 0) = φ1 (x) in Ω
⎪
⎪
⎩
φ(ξ , t) = 0
on Σ,
(2.14)
satisfies
0
T
(φt (k, t))2 dt ≥ C (φ0 , φ1 ) 2H×V ∀(φ0 , φ1 ) ∈ V × H,
b
∈ S.
π
We conclude that for all initial states (y0 , y1 ) ∈ V × H and for every ε > 0, there exists a time T1 > 0
such that the bilinear controls v(x, t) = 0 and u(t) = 1 give the approximate null controllability.
if
3. Controllability of the semi-linear wave equation
3.1. Approximate controllability
In this section, we study the semi-linear version of (2.1) with bounded damping. Let us consider the
following system
⎧
y = Δy + v(x, t)y − u(t)B̄yt − F(y) in Q
⎪
⎪
⎨ tt
y(x, 0) = y0 (x), yt (x, 0) = y1 (x)
in Ω
⎪
⎪
⎩
y(ξ , t) = 0
on Σ,
(3.1)
where B̄ = BB∗ with B ∈ L(U, H). Our result concerns the approximate controllability of the equilibrium
states (yd , 0) with yd ∈ D(A). We will construct a bilinear control serving first to treat a more easier
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Moreover, H(λ) is bounded on Cγ .
8
I. EL HARRAKI AND A. BOUTOULOUT
non-linear function in the system generated by ψ = y − yd , and second, to ensure the exponential decay
of the system energy. We need to introduce the following assumptions:
(P1 ) F : H −→ H is C 1 globally Lipschitz, monotone and such that F(0) = 0.
(P2 ) Assume that there exists a time T > 0 and a constant C > 0 such that
0
T
B∗ Φ(t) 2U dt ≥ C (Φ0 , Φ1 ) 2H×V ∀(Φ0 , Φ1 ) ∈ H × V ,
⎧
Φtt = ΔΦ + q(t, x)Φ
in Q
⎪
⎪
⎨
Φ(x, 0) = Φ0 (x), Φt (x, 0) = Φ1 (x) in Ω
⎪
⎪
⎩
Φ(ξ , t) = 0
on Σ,
(3.2)
with q ∈ L ∞ (Q).
(P3 ) A − αI is a strictly positive operator where α = v1 + v2 , v1 =
v1 ∈ L ∞ (Ω) and v2 ∈ L ∞ (Ω).
−Δyd
F(yd )
1G , v 2 =
1G ,
yd
yd
Theorem 3.1 Assume that assumptions (P1 )–(P3 ) hold, then for all initial states (y0 , y1 ) ∈ V × H and
for every ε > 0, there exists a time T1 > 0 and two multiplicative controls v(x, t) and u(t) such that the
solution of (3.1) satisfies
y(T1 ) − yd V + yt (T1 ) H < ε.
(3.3)
Proof. Let y = ψ + yd . Then from the system (3.1) we have
⎧
ψ = Δψ + v(x, t)ψ − u(t)B̄ψt + Δyd + v(x, t)yd − F(ψ + yd ) in Q
⎪
⎪
⎨ tt
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x)
in Ω
⎪
⎪
⎩
ψ(ξ , t) = 0
on Σ.
(3.4)
Taking the controls v(x, t) = α(x) and u(t) = 1, the system (3.4) becomes
⎧
⎪
⎪ψtt = Δψ + α(x)ψ − B̄ψt − F(ψ + yd ) + v2 (x)yd + Δyd + v1 (x)yd
⎨
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x)
⎪
⎪
⎩
ψ(ξ , t) = 0
For all x ∈ G we have
Δyd + v1 (x)yd = 0.
in
Q
in
Ω
on Σ.
(3.5)
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where Φ ∈ C(0, T , H) ∩ C 1 (0, T , V ) is the mild solution of
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
9
Moreover, from Proposition 2.2 and taking the second order derivative, we get
Δyd + v1 (x)yd = 0 a.e. in Ω.
Then (3.5) yields
in
Q
in
Ω
(3.6)
on Σ.
Let f (ψ) = F(yd + ψ) − v2 (x)yd , then f (0) = 0, furthermore the monotony of F ensures that
f (w), w ≥ 0 for all w ∈ H. The idea is to use a stabilization argument (see Haraux, 1989; Tebou, 2009)
to achieve the controllability result. Since A − αI is strictly positive we denote by δ the best positive
constant such that
v21 = (A − αI)1/2 v2H ≥ δ 2 v2H ,
(3.7)
and the second step consists of using the observability of (3.2) to prove the existence of some real positive
numbers M and σ such that
ψ(t)21 + ψt (t)2H ≤ M exp(−σ t)(ψ0 21 + ψ1 2H ).
Thus, let us consider φ the solution of
⎧
φtt = Δφ + α(x)φ − f (φ)
in Q
⎪
⎪
⎨
φ(x, 0) = φ0 (x), φt (x, 0) = φ1 (x) in Ω
⎪
⎪
⎩
φ(ξ , t) = 0
on Σ,
(3.8)
with φ(0) = ψ0 and φt (0) = ψ1 . Set
E1 (t) =
1
(φ(t)21 + φt (t)2H ) + S(φ(t)),
2
(3.9)
where
u
S(u) =
f (s)ds.
0
Let ψ1 = ψ − φ, then from (3.6) and (3.8) we have
⎧
ψ = Δψ1 + α(x)ψ1 − B̄ψt − f (ψ) + f (φ) in Q
⎪
⎪
⎨ 1tt
ψ1 (x, 0) = 0, ψ1t (x, 0) = 0
in Ω
⎪
⎪
⎩
ψ1 (ξ , t) = 0
on Σ.
(3.10)
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⎧
ψ = Δψ + α(x)ψ − B̄ψt − F(ψ + yd ) + v2 (x)yd
⎪
⎪
⎨ tt
ψ(x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x)
⎪
⎪
⎩
ψ(ξ , t) = 0
10
I. EL HARRAKI AND A. BOUTOULOUT
Now, we set
E2 (t) =
1
(ψ1 (t)21 + ψ1t (t)2H ).
2
(3.11)
Multiplying (3.10) by ψ1t and integrating over Ω we obtain
dE2
=
dt
−(f (ψ) − f (φ))ψ1t dx −
Ω
B̄ψt ψ1t dx.
(3.12)
According to (P1 ), and the Cauchy–Schwartz inequality, one derives for 0 ≤ t ≤ T
t
E2 (t) ≤
t
M1 ψ1 (s)H ψ1t (s)H ds +
B̄ψ1 (s)H ψ1t (s)H ds,
0
(3.13)
0
then
M1
E2 (t) ≤
δ
0
t
1
ψ1 (s)1 ψ1t (s)H ds +
2
t
(B̄ψ1 (s)2H + ψ1t (s)2H )ds .
(3.14)
0
From the boundedness of the operator B, we get
E2 (t) ≤
M1
δ
t
E2 (s)ds +
0
1
2
T
K 2 B∗ ψ1 (s)2U ds +
0
t
E2 (s)ds.
(3.15)
0
Applying Gronwall lemma we get
E2 (t) ≤ K exp(δ1 )
2
T
B∗ ψ1 (s)2U ds,
(3.16)
0
TM1 + T δ
with δ1 =
.
δ
Now we set
E3 (t) =
1
(ψ(t)21 + ψt (t)2H ) + S(ψ(t)),
2
and we observe that
E1 (0) = E3 (0).
(3.17)
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Ω
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
11
By noticing that φt = Φ solves a hyperbolic problem with a bounded potential that depends on both
space and time variables, then using the observability estimate provided by (P2 ) we have
T
E3 (0) ≤ C
B∗ φt (s)2U ds.
(3.18)
0
Then for C > 2C, we have
T
B
ψ1t (s)2U ds
T
+C
0
B∗ ψt (s)2U ds.
(3.19)
0
By considering (3.16) and the boundedness of operator B, we get the existence of a positive constant K
such that
T
E3 (0) ≤ K
B∗ ψt (s)2U ds.
(3.20)
0
We, now, multiply (3.6) by ψt and integrate over Ω we have
E3 (T ) − E3 (0) = −
T
B∗ ψt (s)2U ds.
0
It follows that E3 is a non-increasing function of the time variable and we deduce that
E3 (T ) ≤ K E3 (0) − K E3 (T ).
Moreover, in view to the semi-group property we deduce that
ψ(t)21 + ψt (t)2H ≤ M exp(−σ t)(ψ1 2H + ψ0 21 ),
with
M=
K +1
K +1
1
.
and σ = log
K
T
K
Finally, by the equivalence of norms .V and .1 (see Tucsnak and Weiss, 2009), we get the approximate
controllability for any time T1 satisfying (P2 ) and the following estimate:
M exp(−σ T1 )(ψ1 2H + ψ0 2V ) <
which gives the approximate controllability.
ε2
,
4
(3.21)
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E3 (0) ≤ C
∗
12
I. EL HARRAKI AND A. BOUTOULOUT
3.2. Extension results
Now we will see that the approximate controllability result proved in the last subsection can be extended
to other state spaces and generalizes the same result obtained in Ouzahra (2014) for the linear wave
equation. First, we define the spaces D(Ak ) = {y ∈ D(Ak−1 ) such that Ay ∈ D(Ak−1 )}, k ≥ 1 and
D(A0 ) = L 2 (Ω). In the sequel, we fix k and we consider the unbounded operator Ak = A in the space
Hk = D(Ak−1 ) with D(Ak ) = D(Ak ) as domain. It is easily seen that Hk is a Hilbert space for the scalar
product
k
Aj u, Aj vH ,
j=1
and that Ak is self-adjoint, positive and coercive (see Brezis, 1983). Let HK denote the corresponding
norm to the inner product .Hk . We also consider the space Vk = D(A1/2
k ) with the norm vVk =
1
Ak2 vHk , v ∈ Vk . Let yd ∈ D(Ak ) and consider the following assumptions:
(Pk1 ) F : Hk −→ Hk is C 1 globally Lipschitz and monotone such that F(0) = 0.
(Pk2 ) Assume that there are a time T and a constant C > 0 such that
T
B∗ Φ(t) 2U dt ≥ C (Φ0 , Φ1 ) 2
Hk ×Vk
0
, ∀(Φ0 , Φ1 ) ∈ Hk × Vk ,
where Φ is the mild solution of
⎧
Φtt = ΔΦ + q(t, x)Φ
in Q
⎪
⎪
⎨
Φ(x, 0) = Φ0 (x), Φt (x, 0) = Φ1 (x) in Ω
⎪
⎪
⎩
Φ(ξ , t) = 0
on Σ,
(3.22)
with q ∈ L ∞ (Q).
(Pk3 ) A − αI is a strictly positive operator where α = v1 + v2 , v1 =
v1 ∈ W k−1,∞ (Ω) and v2 ∈ W k−1,∞ (Ω).
−Δyd
F(yd )
1G , v2 =
1G ,
yd
yd
Following the same steps as in the previous subsection we get the next theorem.
Theorem 3.2 Assume that assumptions (Pk1 )–(Pk3 ) hold, then for every ε > 0 and for all initial states
(y0 , y1 ) ∈ Vk × Hk , there exists a time T1 > 0 and two multiplicative controls v(x, t) and u(t) such that
the solution of (3.1) satisfies
y(T1 ) − yd Vk + yt (T1 ) Hk < ε.
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u, v =
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
13
3.3. Exact controllability
In this part we consider the following problem of exact controllability for the one-dimensional wave
equation (Ω = (0, L)):
⎧
y = yxx + v(x, t)y − u(t)B̄yt − F(y) in Ω × (0, T2 )
⎪
⎪
⎨ tt
(3.23)
y(x, 0) = y0 (x), yt (x, 0) = y1 (x)
in Ω
⎪
⎪
⎩
y(0, t) = y(1, t) = 0.
(P4 )
Suppose that |yd | ≥ β > 0, a.e. in ω.
Theorem 3.3 Let assumptions (P1 )–(P4 ) hold, then for all initial states (y0 , y1 ) ∈ V × H, there exists a
time T2 and two multiplicative controls v(x, t) and u(t) such that the solution of (3.23) satisfies
y(T2 ) = yd , yt (T2 ) = 0.
Proof. Step 1: Approximate controllability.
Taking the controls v(x, t) = α(x) and u(t) = 1, the system (3.23) becomes
⎧
y = Δy + α(x)y − B̄yt − F(y) in Q
⎪
⎪
⎨ tt
y(x, 0) = y0 (x), yt (x, 0) = y1 (x) in Ω
⎪
⎪
⎩
y(ξ , t) = 0
on Σ.
(3.24)
(3.25)
Then using Theorem 3.1, we deduce that for every ε, there exists a time T1 such that
(y(T1 ) − yd , yt (T1 )) V×H < ε.
(3.26)
Step 2: Null controllability of a close candidate. Let ψ = y − yd , for t ∈ (T1 , T2 ) we replace the
control v(x, t) with λ(x, t) + α(x) and we take u(t) = 0, then the system (3.23) becomes
⎧
ψtt = Δψ + λ(x, t)(ψ + yd ) + α(x)ψ − f (ψ) in Q1
⎪
⎪
⎨
ψ(x, T1 ) = y(T1 ), ψt (x, T1 ) = yt (T1 )
in Ω
⎪
⎪
⎩
ψ(ξ , t) = 0
on Σ1 ,
(3.27)
where Q1 = Ω × (T1 , T2 ) and Σ1 = ∂Ω × (T1 , T2 ). Now, in order to prove that (3.23) is exactly
controllable, it suffices to prove that (3.27) is exactly null controllable. Our strategy consists of studying
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The idea consists first, to exploit our result of subsection (3.1) in order to make the state closer to the
desired one at a time T1 . Then we will use the results of the exact controllability with an additive control
of linear system. Indeed we will construct a time T2 and controls that guarantee the exact reachability of
the target state for an auxiliary linearized system. Finally, we will use the Schauder fixed point theorem
to achieve the proof. To achieve the exact controllability, we take B = χω with ω is a subset of the hole
domain Ω, and we need this additional assumption.
14
I. EL HARRAKI AND A. BOUTOULOUT
the linearized version of (3.27), and by a fixed point method we will regain the original problem. So,
consider the system
⎧
ψtt = Δψ + λ(x, t)(ψ + yd ) + α(x)ψ − N(z)ψ
⎪
⎪
⎨
ψ(x, T1 ) = y(T1 ), ψt (x, T1 ) = yt (T1 )
⎪
⎪
⎩
ψ(ξ , t) = 0
in
Q1
in
Ω
(3.28)
on Σ1
with
1
N(z) =
f (rz)dr and z ∈ C((T1 , T2 ); H).
0
To find the bilinear control, we will use some controllability results with additive control (see Zuazua,
1993; Zhang, 2000). Let η be the solution of the following system:
⎧
ηtt = Δη + α(x)η + Bv(x, t) − N(z)η
⎪
⎪
⎨
η(x, 0) = η0 (x), ηt (x, 0) = η1 (x)
⎪
⎪
⎩
η(ξ , t) = 0
in
Q1
in
Ω
on
Σ1 .
(3.29)
Using (P1 )–(P2 ), for a given initial state (η0 , η1 ) ∈ X = V × H and terminal state (z0 , z1 ) ∈ X,
for any given z ∈ C([0, T ]; H), there exists a control v ∈ L 2 (0, T ; U), such that we have the following
assertions:
•
The mild solution η ∈ C([0, T ]; V) ∩ C 1 ([0, T ]; H) of (3.29) satisfies
η(T ) = z0 , η(T ) = z1 ,
with v(t) = B∗ Φ and Φ(t) satisfying
⎧
Φtt = ΔΦ + α(x)Φ − N(z)Φ
⎪
⎪
⎨
Φ(x, 0) = Φ0 , Φt (x, 0) = Φ1
⎪
⎪
⎩
Φ(ξ , t) = 0
•
in
Q1
in
Ω
on
Σ1 .
(3.30)
For any t ∈ (0, T1 ) and any z ∈ C((0, T1 ); H) it holds that
v(t) U ≤ C( (η0 , η1 ) X + (z0 , z1 ) X ).
Taking T1 as the initial instant and (ψ(T1 ), ψt (T1 )) as initial state, the system (3.29) becomes
⎧
ηtt = Δη + α(x)η − N(z)η + Bv(x, t) in Q1
⎪
⎪
⎨
η(x, T1 ) = ψ(T1 ), ηt (x, T1 ) = ψt (T1 ) in Ω
⎪
⎪
⎩
η(ξ , t) = 0
on Σ.
(3.31)
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CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
15
We deduce that there exists a time T2 ≥ T1 and a control v such that the system (3.31) is null
controllable. Furthermore, the fact that the linear operator B is bounded gives the following estimate:
Bv(t) H ≤ C (ψ(T1 ), ψ(T1 )) X ,
(3.32)
with v = B∗ Φ ∈ L 2 (T1 , T2 ; U). The idea is to exploit the null controllability of the linear system (3.31)
by searching the control λ on (T1 , T2 ) such that
λ(x, t)(ψ + yd ) = BB∗ Φ(x, t),
η(t)
ηt (t)
= S(t −T1 )
ψ(T1 )
ψt (T1 )
t
+
S(t −s)
T1
0
Bv(s)
t
ds +
S(t −s)
T1
0
α(x)η(s) − N(z)η(s)
ds,
(3.33)
with S(t) on X is a semi-group of contractions. Hence we have for some positive constant L
t
(η, ηt )(t) X ≤ (ψ(T1 ), ψt (T1 ) X +
t
Bv(s) H ds + L
η(s) H ds.
T1
(3.34)
T1
Using (3.32) we deduce that there exists a positive constant K such that
t
(η(t), ηt (t)) X ≤ (ψ(T1 ), ψt (T1 ) X +K (ψ(T1 ), ψt (T1 ) X +L
(η(s), ηt (s)) X ds, (3.35)
T1
then from (3.26) we get
t
(η, ηt )(t) X ≤ (1 + K)ε + L
(η(s), ηt (s) X ds.
(3.36)
T1
Moreover the Gronwall lemma gives
(η, ηt )(t) X ≤ K1 ε,
(3.37)
where K1 is a real positive. Since the embedding H 1 (Ω) → L ∞ (Ω) is continuous [Ω = (0, L)] (see
Adams, 1975), and since the injection V → H is continuous we deduce that there exists K > 0
such that
η(t)L∞ (Ω) ≤ K ε.
Using (P4 ) and taking ε ≤
β
in (3.26) we deduce that
2K
|η(t) + zd | ≥ |zd | − |η(t)| > 0 a.e. in ω for all t ∈ (T1 , T2 ).
Downloaded from http://imamci.oxfordjournals.org/ at Ryerson University on October 19, 2016
for this aim we prove that ψ + yd = 0 a.e. x ∈ ω. So let us consider the solution of (3.31), which is
given for t ∈ (T1 , T2 ) by the variation of constants formula
16
I. EL HARRAKI AND A. BOUTOULOUT
It follows that the solution of (3.31) with the control v on (T1 , T2 ) and the one of (3.28) with the control
λ are identical. Hence, the system (3.28) is exactly null controllable. Then, one can deduce the control
for the linearized problem as follows
λ(x, t) =
BB∗ Φ(x, t)
ψ(x, t) + yd (x)
in
ω × (T2 , T1 ).
(3.38)
N : L ∞ ((T1 , T2 ) × (0, L)) −→
z
L ∞ ((T1 , T2 ) × (0, L))
−→
ψ,
(3.39)
where ψ ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H) is the mild solution of the system (3.28) with the control λ.
Hence it suffices to prove that N has a fixed point. The map N is continuous from L ∞ ((T1 , T2 ) × (0, L))
into L ∞ ((T1 , T2 ) × (0, L)). Furthermore, taking onto account Theorems 2.53–2.55 (see Coron, 2007,
pages 68–72), we get the existence of M > 0 such that
N : L ∞ ((T1 , T2 ) × (0, L)) ⊂ E,
where
E = ψ ∈ C([T1 , T2 ]; V) ∩ C 1 ([T1 , T2 ]; H) such that ψ C([T1 ,T2 ];V) < M, ψt C 1 ([T1 ,T2 ];H) < M .
Note that E is a convex subset of L ∞ ((T1 , T2 )×(0, L)). Moreover, by a theorem due to Jacques Simon
(see Simon, 1987) E is a compact subset of L ∞ ((T1 , T2 ) × (0, L)). Hence, by the Schauder fixed-point
theorem, N has a fixed point, which gives the null controllability of (3.27). This concludes the proof. 4. Conclusion
In this work, we have investigated the problem of controllability of multidimensional bilinear wave
equation with damping. First, the approximate controllability result has been obtained for the equilibrium
states, and the time of approximate controllability has been explicitly determined. The semi-linear wave
equation is also studied. This result extends the one of Khapalov (2010) in the one-dimensional case
and those of Ouzahra (2014) for the homogeneous case. The problem of the semi-linear wave equation
with unbounded damping is of great importance. However, our arguments are based on stabilization
results, and to the best of the author’s knowledge, the stabilization of the semi-linear in the presence of
an unbounded damping is an open problem, thus our method does not work in this case. This leaves the
issue of the controllability of the semi-linear wave with unbounded damping an open problem.
Acknowledgement
This work has been carried out with a grant from Hassan II Academy of Sciences and Technology.
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In the last step, we use a fixed point method (see Zuazua, 1993; Coron, 2007). For the initial state
(ψ(T1 ), ψt (T2 )) ∈ X and for any given z ∈ C((T1 , T2 ); H) there exists a control λ ∈ L 2 (0, T ; V ) such
that the solution of the linearized system (3.28) is null controllable. Thus, we can define the map
CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS
17
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