HAL Id: hal-01139980
https://hal.archives-ouvertes.fr/hal-01139980
Submitted on 7 Apr 2015
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Internal controllability of rst order quasilinear
hyperbolic systems with a reduced number of controls
Fatiha Alabau-Boussouira, Jean-Michel Coron, Guillaume Olive
To cite this version:
Fatiha Alabau-Boussouira, Jean-Michel Coron, Guillaume Olive. Internal controllability of rst order
quasilinear hyperbolic systems with a reduced number of controls. SIAM Journal on Control and Op-
timization, Society for Industrial and Applied Mathematics, 2017, 55 (1), pp.300-323. �hal-01139980�
INTERNAL C ONTROLLABILITY OF FIRST ORDER QUASILINEAR
HYPERBOLIC SYSTEMS WITH A REDUCED NUMBER OF
CONTROLS
FATIHA ALABAU-BOUSSOUIRA, JEAN-MICHEL CORON AND GUILLAUME OLIVE
∗
Abstract. In this paper we investigate the exact controllability of n × n first order quasilinear
hyperbolic systems by m < n internal controls that are localized in space in some part of the
domain. We distinguish two situations. The first one is when the equations of the system have
the same speed. In this case, we can use the method of characteristics and obtain a simple and
complete characterization for linear systems. Thanks to a linear test this also provides some sufficient
conditions for the local exact controllability around the trajectories of semilinear systems. However,
when the speed of the equations are not anymore the same, we see that we encounter the problem of
loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve
this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system,
we first use the notion of algebraic solvability due M. Gromov. However, in contrast with this prior
article w here a standard fixed point argument could be used to treat the nonlinearities, we use here
a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss
of derivatives.
Key words. Quasilinear hyperbol ic systems, exact internal controllability, controllability of
systems, algebraic solvability.
AMS subject classifications. 35L50, 93B05, 93C10.
1. Introduction. In this paper we investigate the exact controllability of n ×
n first order quasilinear hyperbolic systems by m < n internal controls that are
localized in space in some part of the domain. While the controllability of quasilinear
hyperbolic systems by boundary controls has been intensively studied, [Cir69, LR02,
LR03, Wan06, Zha09, LRW10], to our knowledge there are no equivalent results for
the internal controllability. On the other hand, the controllability of systems of PDEs
with a reduced number of controls has been a challenging problem for the last decades,
see for ins tance [AB03, ABL12, AB13, DLRL14, AB14] for linear hyperbolic systems
and [Zha09] for quasilinear hyperbolic systems, [BGBPG02], [GBPG05], [Gue07] and
the survey [AKBGBdT11] for linea r pa rabolic systems, [CGR10, CL14] for nonlinea r
parabolic systems, [CG09b] for Stokes e quations, [FCGIP06], [CG0 9a] and [CL14] for
Navier-Stokes equations. Let us also point out that, in many of these articles, the
general stra tegy is to start with a controllability result in the case where there are as
many controls as the number of equations a nd then to try to remove some of these
controls by a suitable procedure. We also follow this general strategy here.
In [LR03], the authors introduced a constructive metho d to control quasilinear
systems of n equations by n boundary controls. This proficient method is based on
existence and uniqueness res ults of semi-global solutions [LJ01] (i.e. with large time
and small data) that they apply to several mixed initial-boundary value problems,
using also the equivalent roles of the time and the space . As we shall see below, using
a method of extension of the domain (as it is often used in the parabolic framework),
we can recover this result for the internal controllability, that is we can prove the
controllability of n × n quasilinear systems by n internal controls. The situation is
more complicated when we have less controls than equations. Indeed, even though
some results are known for the boundary controllability of n × n quasilinea r systems
∗
Jean-Michel Coron and Guillaume Olive were suppor ted by the ERC advanced grant 266907
(CPDENL) of the 7th Research Framework Programme (FP7)
1
2 F. Alabau-Boussouira, J.-M. Coron and G. Olive
with m < n controls [Zha 09], the extension method is not anymore applicable in this
context. Thus, we need to develop direct metho ds to solve the problem of internal
controllability.
We start the study with linear systems of equations with the same velocity. In this
case, we ca n apply the method of characteristics and obtain a complete and simple
characterization of the exact controllability. We show that the linear system can be
viewed as a par ametrized family of ODEs that are controlled independently. The
difficulty is actually to prove that this is enough to build a smooth (C
1
) control for
the linear hyperbolic system. Moreover, since we look for controls of the hyperbolic
system that are localized in some part of the doma in, a nonstandard conditio n on the
supports of the ODEs also appears and needs to be handled. Another key point of
the proof is the e xplicit formula of the HUM control for ODEs.
Using then a standard fix ed point ar gument we can obtain sufficient conditions for
the local exact controllability around the trajectories of semilinear systems. However,
when the e quations do not have the same speed anymore and the nonlinea rity is
stronger, that is when we consider quasilinear s ystems, the standard linear test fails
because of a loss of derivatives. To solve this problem, we need to use a fixed point of
Nash-Moser type. We propose to use the fixed point theorem of M. Gromov [Gro86,
Section 2.3.2, Main Theorem], which is bas ed on the notion of a lgebraic solvability
for partial differential operators. The metho d consists in, first controlling the n × n
system by n controls, and then to eliminate a certain number o f controls through the
algebraic solvability. The use of the Gromov a lgebraic solvability in the framework
of the control theory was introduced in [Cor07, Pages 13-15] in the framework of
the control of linear or dinary differential equations (however it does not lead to new
results in this case) and in [C L 14] for a for a Navier-Stokes control system. In this last
case, the parabolicity allows to have smooth controls, as shown in [CL14], and thus
to avoid the problem of loss of derivatives. The difference between the pre sent work
and [CL14], where the algebraic solvability was the difficult task (the fixed point was
standard), is that, following the algebraic solvability s tep, we show how to apply the
fixed point theorem of Gromov to obtain the controllability of the quasilinear system.
Last, but not least, this method is probably not optimal with the regularity obtained,
which leaves some challenging problems.
2. Systems of equations with the same velocity.
2.1. Linear syste ms. Let us consider the following linear hyperbolic system
with periodic boundary conditions:
y
t
+ y
x
+ A(t, x)y = B(t, x)Θ, (t, x) ∈ [0, T ] × [0, L],
y(t, L) = y(t, 0), t ∈ [0, T ],
y(0, x) = y
0
(x), x ∈ [0, L].
(2.1)
In (2.1), T > 0 is the control time, L > 0 is the leng th of the doma in. A and B
are time and space dependent matrices of size n × n and n × m, respectively, where
n ∈ N
∗
denotes the number of equations of the system and m ∈ N
∗
the number of
controls (with possibly m < n). y
0
is the initial data and y(t, ·) : [0, L] −→ R
n
is the
state at time t ∈ [0, T ]. Finally, Θ(t, ·) : [0, L] −→ R
m
is the distributed control at
time t ∈ [0, T ], that we look subject to the constraint
supp Θ ⊂ [0, T ] × [a, b], (2.2)
Controllability of first order hyperbol ic systems 3
where here, and in what follows, the interval [a, b], with 0 ≤ a < b ≤ L, is fixed.
Throughout this article, for k ∈ N and p ∈ N
∗
, we denote by C
k
L
([0, T ] × [0, L])
p
(resp. C
k
L
([0, L])
p
) the Banach space of functions y ∈ C
k
L
([0, T ] × [0, L])
p
(resp. y ∈
C
k
L
([0, L])
p
) that are x L-periodic, that is
∂
i
x
y(t, 0) = ∂
i
x
y(t, L), ∀t ∈ [0, T ], ∀i ∈ J0, kK. (2.3a)
resp. y
(i)
(0) = y
(i)
(L), ∀i ∈ J0, kK.
(2.3b)
All along Section 2.1 we assume that A ∈ C
1
L
([0, T ] × [0, L])
n×n
, B ∈ C
1
L
([0, T ] ×
[0, L])
n×m
. These assumptions are made for regularity purposes, see below.
We recall that, for every T > 0, there exists C > 0 such that, for every Θ ∈
C
1
L
([0, T ] × [0, L])
m
and every y
0
∈ C
1
L
([0, L])
n
, there ex ists a unique classica l glo bal
solution y ∈ C
1
L
([0, T ] × [0, L])
n
to (2.1), and this solution satisfies the estimate
kyk
C
1
≤ C
ky
0
k
C
1
+ kΘk
C
1
.
This well-posedness result follows from the classical theory of linear hyperbolic systems
using the method of characteristics [LY85]. Note that the kind of boundary conditions
we co nsider is nonlocal but, as already noticed in [CBdN08] (see also [LRW1 0]), it can
always be reduced to more standard (i.e. lo cal) boundary conditions by introducing
the enlarged system satisfied by (y, ˜y) where ˜y(t, x) = y(t, L − x).
Definition 2.1. We say that System (2.1) is exactly controllable at time T > 0
if, for every y
0
∈ C
1
L
([0, L])
n
and for every y
1
∈ C
1
L
([0, L])
n
, there exists a control Θ ∈
C
1
L
([0, T ] ×[0, L])
m
that satisfies the constraint (2.2) and is such that the corresponding
solution y ∈ C
1
L
([0, T ] × [0, L])
n
to (2.1) satisfies
y(T, x) = y
1
(x), ∀x ∈ [0, L].
2.1.1. The extended characteristics. Let us now introduce an important tool
when dealing with hyperbolic systems, namely the characteristics of the system. In
our case (speed 1 on each equatio n), the characteristic X of System (2.1) passing
through the p oint (t
0
, x
0
) ∈ [0, T ] × [0, L] is the straight line
X(t, t
0
, x
0
)
def
= t − t
0
+ x
0
, t ∈ [0, T ].
However, in this pa per, the crucial tools we need are the extended characteristics
X : [0, T ] × [0, L) −→ [0, L], defined by (see Fig. 2.1 below):
X(t, x)
def
=
X (t, 0, x) if t ∈ [0, τ
0
(x, L)] ,
X (t, τ
k−1
(x, L), 0) if t ∈ (τ
k
(x, 0), τ
k
(x, L)] , k ∈ J1, k
max
(x, 0)K,
where, for e very k ∈ N and c ∈ [0, L], we intro duce the functions
τ
k
(x, c)
def
=
0 if c − x + kL ∈ (−∞, 0),
c − x + kL if c − x + kL ∈ [0, T ],
T if c − x + kL ∈ (T, +∞),
4 F. Alabau-Boussouira, J.-M. Coron and G. Olive
and k
min
(x, c) (resp. k
max
(x, c)) denotes the smallest (resp. greatest) integer k ∈ N
such that c − x + kL > 0 (resp. c − x + kL < T ). More precisely, denoting by ⌊·⌋ the
floor function and ⌈·⌉ the ceiling function,
k
min
(x, c)
def
= ⌊
−c + x
L
⌋ + 1 =
0 if x ∈ [0, c),
1 if x ∈ [c, L),
k
max
(x, c)
def
= ⌈
T − c + x
L
⌉ − 1 =
⌈
T −c
L
⌉ − 1 if x ∈ [0, p(c)] ,
⌈
T −c
L
⌉ if x ∈ (p(c), L) ,
where
p(c)
def
=
⌈
T − c
L
⌉ −
T − c
L
L.
Note that τ
k
(x, 0) = τ
k−1
(x, L) for every k ≥ 1 and τ
k
max
(x,0)
(x, L) = T , so that
X(t, x) is indeed defined for every t ∈ [0, T ].
x
t
0
L
T
a
b
x
X(·, x)
[ ]
( ]
( ]
τ
0
(x, L) τ
1
(x, L)
τ
1
(x, a) τ
1
(x, b)
Fig. 2.1: The extended characteristics
X(·, x).
For 0 ≤ a < b ≤ L, we lis t below some properties of these functions, under
the essential assumption that every extended characteristic
X crosses the domain
[0, T ] × [a, b] at some time, that is
T > L − (b − a).
1. k
min
(x, b) ≤ k
max
(x, a).
2. τ
k
(x, a) < τ
k
(x, b) for every k ∈ Jk
min
(x, b), k
max
(x, a)K if x 6= b and x 6= p(a).
3. τ
k
(x, b) ≤ τ
k+1
(x, a) for every k ∈ Jk
min
(x, b), k
max
(x, a) − 1K.
4.
X(t, x) ∈ (a, b) for every t ∈ (τ
k
(x, a), τ
k
(x, b)) for every k satis fying
k ∈ Jk
min
(x, b), k
max
(x, a)K.
