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DISSIPATIVE BOUNDARY CONDITIONS FOR
ONE-DIMENSIONAL NONLINEAR HYPERBOLIC
SYSTEMS
Jean-Michel Coron, Georges Bastin, Brigitte d’Andréa-Novel
To cite this version:
Jean-Michel Coron, Georges Bastin, Brigitte d’Andréa-Novel. DISSIPATIVE BOUNDARY CON-
DITIONS FOR ONE-DIMENSIONAL NONLINEAR HYPERBOLIC SYSTEMS. SIAM Journal on
Control and Optimization, Society for Industrial and Applied Mathematics, 2008, 47 (3), pp.1460-
1498. �hal-00923596�
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM J. CONTROL OPTIM.
c
2008 Society for Industrial and Applied Mathematics
Vol. 47, No. 3, pp. 1460–1498
DISSIPATIVE BOUNDARY CONDITIONS FOR ONE-DIMENSIONAL
NONLINEAR HYPERBOLIC SYSTEMS
∗
JEAN-MICHEL CORON
†
, GEORGES BASTIN
‡
, AND BRIGITTE D’ANDR
´
EA-NOVEL
§
Abstract. We give a new sufficient condition on the boundary conditions for the exponential
stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on
the construction of an explicit strict Lyapunov function. We compare our sufficient condition with
other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.
Key words. nonlinear hyperbolic systems, boundary conditions, stability, Lyapunov function
AMS subject classifications. 35F30, 35F25, 93D20, 93D30
DOI. 10.1137/070706847
1. Introduction. We are concerned with the following one-dimensional n × n
nonlinear hyp erbolic system:
(1.1) u
t
+ F (u)u
x
=0,x∈ [0, 1],t∈ [0, +∞),
where u :[0, ∞) × [0, 1] → R
n
and F : R
n
→M
n,n
(R), M
n,n
(R) denoting, as
usual, the set of n × n real matrices. We consider the case where, p ossibl y after an
appropriate state tran sformation , F (0) is a diagonal matrix with distinct and nonzero
eigenvalues:
F (0) := diag (Λ
1
, Λ
2
,...,Λ
n
),(1.2)
Λ
i
> 0 ∀i ∈{1,...,m},
Λ
i
< 0 ∀i ∈{m +1,...,n},
Λ
i
=Λ
j
∀(i, j) ∈{1,...,n}
2
such that i = j.(1.3)
In (1) an d in what follows, diag (Λ
1
, Λ
2
,...,Λ
n
) denotes the diagonal matrix whose
ith element on the diagonal is Λ
i
.
Our concern is to analyze the asymptotic behavior of the classical solutions of the
system under the following boundary condition:
(1.4)
u
+
(t, 0)
u
−
(t, 1)
= G
u
+
(t, 1)
u
−
(t, 0)
,t∈ [0, +∞),
where the map G : R
n
→ R
n
vanishes at 0, while u
+
∈ R
m
, u
−
∈ R
n−m
are defined
by requiring that u := (u
tr
+
,u
tr
−
)
tr
. The problem is to find the map G such that
∗
Received by the editors October 30, 2007; accepted for publication (in revised form) January 16,
2008; published electronically May 7, 2008.
http://www.siam.org/journals/sicon/47-3/70684.html
†
Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie and Institut Universitaire de
France, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France (coron@ann.jussieu.fr). This author’s
research was partially supported by the “Agence Nationale de la Recherche” (ANR), Project C-QUID,
number BLAN-3-139579.
‡
Center for Systems Engineering and Applied Mechanics (CESAME), Universit´e Catholique de
Louvain, 4 Avenue G. Lemaˆıtre, 1348 Louvain-la-Neuve, Belgium (Georges.Bastin@uclouvain.be).
§
Centre de Robotique,
´
Ecole des Mines de Paris, 60 boulevard Saint Michel, 75272 Paris Cedex
06, France (brigitte.dandrea-novel@ensmp.fr).
1460
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
DISSIPATIVE BOUNDARY CONDITIONS 1461
the boundary condition (1.4) is dissipative, i.e., implies that the equilibrium solution
u ≡ 0 of system (1.1) with the boundary condition (1.4) is exponentially stable.
This problem has been considered in the literature for more th an 20 years. To
our knowledge, the first results were published by Slemrod in [21] and Greenberg and
Li in [9] for the special case of 2 ×2 (i.e., u ∈ R
2
) systems. A generalization to n ×n
systems was given by the Li school. Let us mention in particular [17] by Qin, [25] by
Zhao, and [14, Theorem 1.3, page 173] by Li. All these results rely on a systematic
use of direct estimates of the solutions and their derivatives along the characteristic
curves. They give rise to sufficient dissipative boundary conditi ons which are kinds of
“small gain conditions.” Th e weakest sufficient condition [14, Theorem 1.3, page 173]
is formulated as follows: ρ(|G
′
(0)|) < 1, where ρ(A) denotes the spectral radius of
A ∈M
n,n
(R) an d |A| denotes the matrix whose elements are the absolute values of
the elements of A ∈M
n,n
(R).
In th is paper we follow a different approach, which is based on a Lyapunov sta-
bility analysis. The special case of 2 × 2 systems and F (u) diagonal has recently
been treated in our p reviou s paper [6]. In the present paper, by using a more general
strict Lyapunov function (see section 4), we get a new weaker dissipative boundary
condition, stated as follows:
Inf {ΔG
′
(0)Δ
−1
;Δ∈D
n,+
} < 1,
where denotes the usual 2-norm of matrices in M
n,n
(R) and D
n,+
denotes the
set of diagonal matrices whose elements on the diagonal are strictly positive.
Moreover, our proof is rather elementary, and the existence of a strict Lyapunov
function may be useful for studying robustness issues.
Our paper is organized as follows. In section 2, after some mathematical pre-
liminaries, a precise technical definition of our new dissipative boundary condition is
followed by the statement of our exponential stability theorem. Section 3 is then de-
voted to a discussion of the optimality properties of our dissipative b ound ary condition
and t o a comparison of this condition with other stability criteria from the literature,
namely the criterion [14, Theorem 1.3, p, 173] mentioned above and a stability cri-
terion for linear hyperbolic systems due to Silkowski. The proof of our exponential
stability theorem, including the Lyapunov stability analysis, is thoroughly given in
section 4. The paper ends with two appendices, where some technical properties of
our dissipative boundary conditi on are given.
2. A sufficient condition for exponential stability. For
x := (x
1
,...,x
n
)
tr
∈ C
n
,
|x| denotes the usual Hermitian norm of x:
|x| :=
n
i=1
|x
i
|
2
.
For n ∈ N \{0} and m ∈ N \{0}, we denote by M
n,m
(R) the set of n × m real
matrices. We define, for K ∈M
n,m
(R),
K := max{|Kx|; x ∈ R
n
, |x| =1} = max{|Kx|; x ∈ C
n
, |x| =1},
and, if n = m,
ρ
1
(K):=Inf{ΔKΔ
−1
;Δ∈D
n,+
},(2.1)
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
1462 J.-M. CORON, G. BASTIN, AND B. D’ANDR
´
EA-NOVEL
where D
n,+
denotes the set of n × n real diagonal matrices with strictly positive
diagonal elements.
For ε, let B
ε
be the open ball of R
n
of radius ε. We assume that, for some ε
0
> 0,
F : B
ε
0
→M
n,n
(R) is of class C
2
and that there exists m ∈{0,...,n} and n real
numbers Λ
1
,...,Λ
n
such that
Λ
i
> 0 ∀i ∈{1,...,m} and Λ
i
< 0 ∀i ∈{m +1,...,n},(2.2)
F (0) = diag (Λ
1
,...,Λ
n
),(2.3)
Λ
i
=Λ
j
∀(i, j) ∈{1,...,n}
2
such that i = j.(2.4)
For u ∈ R
n
, u
+
∈ R
m
and u
−
∈ R
n−m
are defined by requiring
u =
u
+
u
−
.
As mentioned in the introduction, we are mainly interested in analyzing the
asymptotic convergence of the classical solutions of the following Cauchy problem:
u
t
+ F (u)u
x
=0,x∈ [0, 1],t∈ [0, +∞),(2.5)
u
+
(t, 0)
u
−
(t, 1)
= G
u
+
(t, 1)
u
−
(t, 0)
,t∈ [0, +∞),(2.6)
u(0,x)=u
0
(x),x∈ [0, 1].(2.7)
Concerning G, we assume that G : B
ε
0
→ R
n
is of class C
2
and satisfies G(0)=0. We
define F
+
(u) ∈M
m,n
(R), F
−
(u) ∈M
(n−m),n
(R), G
+
(u) ∈ R
m
, and G
−
(u) ∈ R
n−m
by requiring
F (u)=
F
+
(u)
F
−
(u)
,G(u)=
G
+
(u)
G
−
(u)
.
Regarding the existence of the solutions to the Cauchy problem (2.5)–(2.7), we
have the following proposition.
Proposition 2.1. There exists δ
0
> 0 such that, for every u
0
∈ H
2
((0, 1), R
n
)
satisfying
|u
0
|
H
2
((0,1),R
n
)
δ
0
and the compatibility conditions
u
0
+
(0)
u
0
−
(1)
= G
u
0
+
(1)
u
0
−
(0)
,(2.8)
(2.9) F
+
(u
0
(0))u
0
x
(0) =
G
′
+u
+
u
0
+
(1)
u
0
−
(0)
F
+
(u
0
(1))u
0
x
(1)
+
G
′
+u
−
u
0
+
(1)
u
0
−
(0)
F
−
(u
0
(0))u
0
x
(0),
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
DISSIPATIVE BOUNDARY CONDITIONS 1463
(2.10) F
−
(u
0
(1))u
0
x
(1) =
G
′
−u
+
u
0
+
(1)
u
0
−
(0)
F
+
(u
0
(1))u
0
x
(1)
+
G
′
−u
−
u
0
+
(1)
u
0
−
(0)
F
−
(u
0
(0))u
0
x
(0),
the Cauchy problem (2.5)–(2.7) has a unique maximal classical solution
u ∈ C
0
([0,T),H
2
((0, 1), R
n
))
with T ∈ [0, +∞]. Moreover, if
|u(t, ·)|
H
2
((0,1),R
n
)
δ
0
∀t ∈ [0,T),
then T =+∞.
For a proof of this proposition, see, for instance, [12] by Kato, [13, pp. 2–3] by
Lax, [16, pp. 35–43] by Majda, or [20, pp. 106–114] by Serre. Actually [12, 13, 16, 20]
deal with R instead of [0, 1], but th e proofs given there can be adapted to treat this
new case. See also [15, pp. 96–107] by Li and Yu for the well-posedness of the Cauchy
problem (2.5)–(2.7) in the framework of functions u of class C
1
. Let us briefly explain
how to adapt these proofs in order to get, for example, the existence of a solution
u ∈ C
0
([0,T],H
2
((0, 1), R
n
)) to the Cauchy problem (2.5)–(2.7) if m = n (just to
simplify the notation), for T ∈ (0, +∞) given, and for every u
0
∈ H
2
((0, 1), R
n
)
satisfying the compatibility conditions (2.8)–(2.9) (when m = n, condition (2.10)
disappears) and such that |u
0
|
H
2
((0,1),R
n
)
is small enough (the smallness depending
on T in general). We first deal with the case where
T ∈ (0, min{Λ
−1
1
,...,Λ
−1
n
}).
The basic ingredient is the following fixed point method, which is related to the one
given in [15, page 97] (see also the pioneering works [12] and [13, pp. 2–3], where the
authors deal with R instead of [0, 1]). For R>0 and for u
0
∈ H
2
((0, 1), R
n
) satisfying
the compatibility conditions (2.8)–(2.9), let C
R
(u
0
) be the set of
u ∈ L
∞
((0,T),H
2
((0, 1), R
n
)) ∩ W
1,∞
((0,T),H
1
((0, 1), R
n
))
∩ W
2,∞
((0,T),L
2
((0, 1), R
n
))
such that
|u|
L
∞
((0,T ),H
2
((0,1),R
n
))
R,
|u|
W
1,∞
((0,T ),H
1
((0,1),R
n
))
R,
|u|
W
2,∞
((0,T ),L
2
((0,1),R
n
))
R,
u(·, 1) ∈ H
2
((0,T), R
n
) and |u(·, 1)|
H
2
((0,T ),R
n
)
R
2
,
u(0, ·)=u
0
,
u
t
(0, ·)=−F (u
0
)u
0
x
.