harraki2016 (5)

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]
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.
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 pis 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.
IMA Journal of Mathematical Control and Information Advance Access published October 18, 2016
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2I. EL HARRAKI AND A. BOUTOULOUT
around a constant reference trajectory in the space H3(0, 1)×H2(0, 1).InKhapalov (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)
y(0, t)=y(1, t)=0,
(1.1)
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 xand time t, and γ(t)as the gain of viscous (motion-
activated) 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 semi-
linear 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 H1
0(Ω) ×L2(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.
2. Approximate controllability of the linear wave equation
2.1. Problem statement
Let Ωbe an open bounded subset of Rnwith regular boundary ∂Ω. We consider the space H=L2(Ω)
with its inner product and the corresponding norm denoted by 
H. We define the unbounded operator
A=−Δwith the domain D(A)=H2(Ω) ∩H1
0(Ω). Let us also consider the space V=H1 0(Ω) with
the norm vV=A1
2vH,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 Uendowed 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:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ytt =Δy+v(x,t)y−u(t)¯
Bytin Q
y(x,0)=y0(x),yt(x,0)=y1(x)in Ω
y(ξ,t)=0onΣ,
(2.1)
where ¯ B=BB∗for B∈L(U,(D(A1/2))with adjoint operator B∗.
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CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS 3
Here v∈L∞(Q), and u(t)=α1for t∈(0, T1)
0 for t>T1,with α1a real positive number. In order to reach
a given target state (yd,0)with yd∈D(A), we will consider the following assumptions:
(H1)A−v1Iis a strictly positive operator where v1=−Δyd
yd
1G, with G={x∈Ω:yd(x)= 0}and
v1∈L∞(Ω).
(H2)For a fixed positive constant β, the function Hdefined by
H:λ∈C+={λ∈Csuch that Re(λ) > 0}−→λB∗(λ2I+A−v1I)−1B∈L(H)
is a bounded function on Cβ:={λ∈Csuch that Re(λ) =β}.
(H3)There exists a time T>0 and a constant C>0 such that
T
0
(B∗φ(t))t2
Udt ≥C(φ0,φ1)2
H×V∀(φ0,φ1)∈D(A)×V,
where φsatisfies
⎧
⎪
⎪
⎨
⎪
⎪
⎩
φtt =Δφ +v1(x)φ in Q
φ(x,0)=φ0(x),φt(x,0)=φ1(x)in Ω
φ(ξ,t)=0onΣ.
(2.2)
Remark 2.1 Note that the assumption (H2)is less restrictive than the boundedness of Bwhich 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×Hto system (2.1). Moreover,
if Z0∈D(˜
A1)then Z∈C(0, ∞;D(˜ A1)) ∩C1(0, ∞;V×H), where ˜ A1is the unbounded linear operator
given by
00
Δ−α1¯
B, (2.3)
with the domain
D(˜ A1)={(x,y)∈V×V,Δx−α1¯
By ∈H}. (2.4)
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4I. EL HARRAKI AND A. BOUTOULOUT
Proof. We write the system (2.1) in the form
Zt=˜
A1Z+˜ P1Z,Z0=y0
y1, (2.5)
with
Z=y
yt,Zt=y
ytt
and ˜ P1Z=0
vI.
Since ˜ A1is m-dissipative (see Ammari & Nicaise,2015), it generates a contraction semi-group on V×H
denoted by (S1(t)). Moreover, remarking that ˜
P1is a Lipschitz perturbation of the generator, we deduce
the existence of the solution (see Pazy,1983, page 183). 
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=˜
A2Z+Bu1,Z0=y0
y1, (2.6)
where
u1=B∗yt,˜
A2=00
Δ+v1I0and B=0
B.
It is clear that the operator ˜ A2generates a strongly continuous group of operator on V×H, denoted by
(S2(t))t∈Rand 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
Z(t)=S2(t)Z0+T1
0
S2(t−s)Bu1(s)ds.
Definition 2.1 We say that Bis 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∗S∗
2(t)Z02 Udt ≤CZ02 V×H.
Proposition 2.1 (see Tucsnak and Weiss,2009). If Bis 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 Bfor (S2(t))∗
t∈R
(see Ammari & Tucsnak,2001).
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CONTROLLABILITY OF THE WAVE EQUATION VIA MULTIPLICATIVE CONTROLS 5
Now, in order to prove the approximate controllability, we will use the following proposition:
Proposition 2.2 (see Attouch et al.,2006). Let v∈W1,p(Ω),1 ≤p<+∞. Then, for each i=1, ...,N
1v=0
∂v
∂xi
=0.
In other words, ∂v
∂xi
=0. a.e. in ˜ E={x∈Ω:v(x)=0},i=1, ...,N.
Theorem 2.1 Suppose that assumptions (H1)–(H3)hold, then for all initial state (y0,y1)∈V×Hand
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
y(T1)−ydV+yt(T1)H<ε. (2.7)
Proof. Let y=ψ+yd, from the system (2.1)wehave
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ψtt =Δψ +v(x,t)ψ −u(t)¯
Bψt+Δyd+v(x,t)ydin Q
ψ(x,0)=ψ0(x),ψt(x,0)=ψ1(x)in Ω
ψ(ξ,t)=0onΣ,
(2.8)
therefore, with the controls v(x,t)=v1(x)and u(t)=1wehave
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ψtt =Δψ +v1(x)ψ −¯
Bψt+Δyd+v1(x)ydin Q
ψ(x,0)=ψ0(x),ψt(x,0)=ψ1(x)in Ω
ψ(ξ,t)=0onΣ.
(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 ψsatisfies
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ψtt =(Δ +v1(x)I)ψ −¯
Bψtin Q
ψ(x,0)=ψ0(x),ψt(x,0)=ψ1(x)in Ω
ψ(ξ,t)=0onΣ.
(2.10)
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