Ann. I. H. Poincaré – AN 30 (2013) 879–915
www.elsevier.com/locate/anihpc
Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law
Vincent Perrollaz
1Université Pierre et Marie Curie – Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, F-75005, France Received 29 October 2011; received in revised form 21 October 2012; accepted 17 December 2012
Available online 11 January 2013
Abstract
In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space.
©2013 Elsevier Masson SAS. All rights reserved.
Keywords:Asymptotic stabilization; Conservation law; Entropy solution; Generalized characteristics; PDE control
1. Introduction
This paper is concerned with the asymptotic stabilization problem for a nonlinear scalar conservation law with a source term, on a bounded interval and in the framework of entropy solutions:
∂tu+∂xf (u)=g(t), u(0, x)=u0(x), u(t,0)=ul(t), u(t, L)=ur(t),
(t, x)∈(0, T )×(0,1). (1)
Hereuis the state andul,ur andgare the controls. For any regular strictly convex fluxf inC1(R;R)and any state
¯
u∈Rwe will provide explicit stationary feedback law forg,ur andulsuch that the stateu¯is asymptotically stable in theL1(0,1)norm and in theL∞(0,1)norm.
1.1. Generalities and previous results
Scalar conservation laws are used for instance to model traffic flow or gas networks, but their importance also consists in being a first step in the understanding of systems of conservation laws. Those systems of equations model a huge number of physical phenomena: gas dynamics, electromagnetism, magneto-hydrodynamics, shallow water theory, combustion theory, . . . , see[12, Chapter 2].
E-mail address:perrollaz@ann.jussieu.fr.
1 The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Project CISIFS, grant ANR-09-BLAN-0213-02.
0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2012.12.003
For equations such as (1), the Cauchy problem on the whole line is well posed in small time in the framework of classical solutions and with aC1initial value. However those solutions generally blow up in finite time: shock waves appear. Hence to get global in time results, a weaker notion of solution is called for. In[23]Oleinik proved that given a fluxf ∈C2such thatf>0 and anyu0∈L∞(R)there exists a unique weak solution to:
ut+ f (u)
x=0, x∈Randt >0, (2)
u(0, .)=u0, (3)
satisfying the additional condition:
u(t, x+a)−u(t, x)
a E
t forx∈R, t >0, anda >0. (4)
HereEdepends only on the quantities inf(f)and sup(f)taken on[−u0L∞,u0L∞]and not onu0. Later in[21], Kruzkov extended this global result to the multidimensional problem, with aC1 fluxf:R→Rn not necessarily convex:
ut+div
f (t, x, u)
=g(t, x, u), fort >0 andx∈Rn. (5)
This time the weak entropy solution is defined as satisfying the following integral inequality:
for all real numberskand all non-negative functionsφinCc1
R2
, (6)
R2
|u−k|φt+sgn(u−k)
f (u)−f (k)
∇φ+sgn(u−k)g(t, x, u)φ dt dx+
R
u0(x)φ(0, x) dx0. (7) The initial boundary value problem for Eq. (1) is also well posed as shown by Leroux in[19]for the one dimensional case with BV data, by Bardos, Leroux and Nédélec in[6] for the multidimensional case withC2 data and later by Otto in[24](see also[22]) forL∞data. However the meaning of the boundary condition is quite intricate and the Dirichlet condition may not be fulfilled pointwise a.e. in time. We will go into further details later.
Now for a general control system:
X˙ =F (X, U ), X(t0)=X0
(8) (Xbeing the state of the system belongs to the spaceX andUthe so called control belongs to the spaceU), we can consider two classical problems (among others) in control theory.
1. First the exact controllability problem which consists, given two statesX0andX1inX and a positive timeT, in finding a certain functiont∈ [0, T] →U (t)∈U such that the solution to (8) satisfiesX(T )=X1.
2. IfF (0,0)=0, the problem of asymptotic stabilization by a stationary feedback law asks to find a function of the stateX∈X→U (X)∈U, such that for any stateX0a maximal solutionX(t )of the closed loop system:
X(t )˙ =F
X(t ), U X(t )
,
X(t0)=X0, (9)
is global in time and satisfies additionally:
∀R >0, ∃r >0 such thatX0r ⇒ ∀t∈R, X(t )R, (10)
X(t )t→+∞−→ 0. (11)
The asymptotic stabilization property might seem weaker than exact controllability: for any initial stateX0, we can find T andU (t)such that the solution to (8) satisfiesX(T )=0 in this way we stabilize 0 in finite time. However this method suffers from a lack of robustness with respect to perturbation: with any error on the model, or on the initial state, the control may not act properly anymore. This motivates the problem of asymptotic stabilization by a stationary feedback law which is more robust. In fact in finite dimension, the asymptotic stabilization property automatically guarantees the existence of a Lyapunov function.
In the framework of entropy solutions, only a few results exist for the exact controllability problem see[2,3,1, 7,14,15,17]. In all cases the control act only at the boundary points. Furthermore many of those results show that boundary controls are not sufficient to reach many states. However with an additional controlg(t)as in (1) and with f (z)=z22 (Burgers equation), Chapouly showed in[8]that in the framework of classical solutions, any regular state is reachable from any regular initial data and in any time (note that in this context, the controls also had to prevent the blow up of the solution, which will not be a concern for entropy solutions). It was shown in[25]that the same kind of improvement also occur in the framework of entropy solutions.
The aim of this paper is to investigate the problem of asymptotic stabilization with this additional control on the right-hand side. To the author’s knowledge it is the first result about asymptotic stabilization through a closed loop feedback law in the framework of entropy solutions. However, it should be noted that in the framework of classical solutions the problem has been studied extensively, see for example:[18],[4]or[5]among many others.
As for the physical significance ofg, it can be seen as a pressure field in the case where the Burgers equation is con- sidered as a one dimensional isentropic Euler compressible equation. Eq. (1) is also a toy model for the compressible Euler–Poisson system:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
∂tρ+∂xm=0, ρ(0, .)=ρ0, ρ(t,0)=ρl(t ), ρ(t, L)=ρr(t),
∂tm+∂x m2
ρ +p(ρ)
=ρ∂xV−m, m(0, .)=m0, m(t,0)=ml(t ), m(t, L)=mr(t),
−∂xx2 V =ρ, V (t,0)=Vl(t ), V (t, L)=Vr(t),
(12)
where the controls areρl,ρr,ml,mr,VlandVr. Indeed once we takeg(t)=Vr(t)−LVl(t)we get∂xV =g(t)+Aρ(t) withAthe following linear integral operator:
Aθ (x)= x 0
θ (z) dz− 1 L
L 0
y 0
θ (z) dz dy.
So we have to deal with a hyperbolic system controlled by the boundary data and an additional source term depending only on the time variable.
1.2. Results
In this article, functions in BV(R)will be considered continuous from the left in order to prevent ambiguity on the representative of theL1equivalence class.
We will also suppose that the fluxf is aC1strictly convex function.
We will make use of the following notation:
∀α, β∈R, I (α, β)=
min(α, β),max(α, β)
. (13)
We are interested in the following equation:
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
∂tu+∂x
f (u)
=g(t) on(0,+∞)×(0,1),
u(0, .)=u0 on(0,1),
sgn u
t,1−
−ur(t) f
u t,1−
−f (k)
0 ∀k∈I
ur(t), u t,1−
, dta.e., sgn
u t,0+
−ul(t) f
u t,0+
−f (k)
0 ∀k∈I
ul(t), u t,0+
, dta.e.
(14)
We recall that following[19]and[6]a functionuinL∞((0,+∞);BV(0,1))is an entropy solution of (14) when it satisfies the following inequality for everykinRand every non-negative functionφinCc1(R2):
+∞
0
1 0
|u−k|φt+sgn(u−k)
f (u)−f (k)
φx+sgn(u−k)g(t)φ dx dt+ 1 0
u0(x)−kφ(0, x) dx
+
+∞
0
sgn
ur(t)−k
f (k)−f u
t,1−
φ(t,1)−sgn
ul(t)−k
f (k)−f u
t,0+
φ(t,0) dt
0. (15)
Remark 1.
• Let us recall that with a convex flux, an entropy solution satisfy (see[11]or[12, Chapter 11]):
∀(t, x)∈(0,+∞)×(0,1), u t, x−
u
t, x+
. (16)
• It should be noted that ifuinL∞((0,+∞);BV(0,1))∩Lip([0,+∞);L1(0,1))the exists a unique representative usuch that:
u∈Lip
[0,+∞);L1(0,1)
and ∀t0, u(t, .)∈BV(0,1).
Thus the traces of uat x=0 andx=1 are taken as the limit of this representative for every time t and the boundary conditions in (14) hold almost everywhere and not necessarily everywhere. This will make the analysis more delicate as will be seen in Section3.
Here the functionsg,ul andur will not depend on the time but on the stateu(t, .). Their value will be prescribed by a closed loop feedback law.
Consideru¯∈R. It is clear that if we defineuby:
∀(t, x)∈R+×(0,1), u(t, x)= ¯u,
thenuis an entropy solution of (14) for initial and boundary data equal tou. In the following we will provide two¯ feedback laws and two corresponding results (in the respective casesf(u)¯ =0 andf(u) >¯ 0), such that the previous stationary solution is asymptotically stable. Note that the casef(u) <¯ 0 can be directly deduced from the positive case using the transformation:
X=1−x, F (z)=f (−z),
U (t, X)= −u(t,1−x).
Iff(u) >¯ 0, we will use the following stationary feedback laws:
∀W∈L1(0,1), G1(W )=f(u)¯
2 W− ¯uL1(0,1), (17)
∀W∈L1(0,1), ul(W )=ur(W )= ¯u. (18) In the system (14) we will replace:g(t)byG1(u(t, .)),ul(t)byul(u(t, .))and finallyur(t)byur(u(t, .))to obtain a closed loop system.
We will need to distinguish between two possible behaviors off as follows:
Definition 1.
• We say thatf if of type I if there existsu∗such that:
f u∗
=0. (19)
The Burgers equation has a flux of type I.
• We say thatf is of type II otherwise. In this case we have either
∀z∈R, f(z) >0, (20)
or
∀z∈R, f(z) <0. (21) The fluxf (z)=ezis of type II.
Remark 2.If the fluxf is of type I, we can deduce since it also strictly convex:
z→+∞lim f (z)= lim
z→−∞f (z)= +∞. (22)
Iff(u)¯ =0, guarantees the existence ofuˆ= ¯usuch thatf (u)¯ =f (u). We can then reformulate the boundary condi-ˆ tion of (14) as follows (we describe the casef(u) >¯ 0):
u t,1−
∈
u∗; +∞
dta.e., (23)
u t,0+
∈(−∞,uˆ] ∪ { ¯u} dta.e. (24)
We can now announce the following result.
Theorem 1.For anyu0inBV(0,1), the closed loop system(14)whereul,ur andgare given by the feedback laws (18)and(17)has a unique entropy solution u. It is global in time, belongs to the spaceL∞((0,+∞);BV(0,1))∩ Lip([0,+∞);L1(0,1))and continuously depends on the initial data. Furthermore if the fluxf is of type I we have:
• There exist two positive constantsC1andC2depending only onu¯such thatusatisfies:
∀t0, u(t, .)− ¯u
L1(0,1)C1e−
f(u)¯
2 tu0− ¯uL1(0,1), (25)
∀t0, u(t, .)− ¯u
L∞(0,1)C2e−f
(¯u)
2 tu0− ¯uL∞(0,1). (26)
• There exists a certain timeT depending only onu¯such thatuisC1on(T ,+∞)× [0,1].
On the other hand if the fluxf is of type II we have the following properties.
• The solutionusatisfies the following stabilization estimate:
∀t0, u(t, .)− ¯u
L∞(0,1)e
f(u)¯
f(u¯−u0− ¯uL∞(0,1))e−f
(u)¯
2 tu0− ¯uL∞(0,1). (27)
• There exists a certain timeTdepending onu¯andu0− ¯uL∞(0,1)such thatuisC1on(T,+∞)× [0,1].
Remark 3.
• In Section4, we will provide explicit formulas forC1,C2,C3,T andT.
• It is interesting to see that a feedback using the L1 norm actually provides a control in the L∞norm. On the other hand a feedback relying on theL∞norm might be problematic due to the impossibility of taking the limit in.L∞(0,1)with only a pointwise convergence and also due to the lack of time regularity ofu(t, .)L∞(0,1)for an entropy solution of the open loop system.
Let us now suppose thatf(u)¯ =0, we introduce the following auxiliary function A:
A(z)=
f (u¯+z)−f (u)¯
2 if 0z1,
f(u¯+1)
2 (z−1)+f (u¯+1)2−f (u)¯ ifz1. (28)
We will use once again:
∀W∈L1(0,1), ul(W )=ur(W )= ¯u,
for the boundary terms but the stationary feedback law for the source term will now be:
∀W∈L1(0,1), G2(W )=A
W− ¯uL1(0,1)
, (29) and as before we will replaceg(t)byG2(u(t, .))in (14). This allows us to prove the following.
Theorem 2.The closed loop system(14)whereul,ur andgare provided by the feedback laws(18)and(29)has the following properties.
• For anyu0inBV(0,1)there exists a unique entropy solutionu. It is global in time, belongs to the space L∞
(0,+∞);BV(0,1)
∩Lip
[0,+∞);L1(0,1) , and depends continuously on the initial data.
• The solution satisfies:
u(t, .)− ¯u
L∞(0,1)t→+∞−→ 0. (30)
• If additionally α=inf
z∈Rf(z) >0,
then we have a globally Lipschitz functionRsuch that:
R(0)=f(1+ ¯u) 2α
2e
e−1+A−1
e(f(1+ ¯u))2 4α(e−1)
, (31)
∀t0, u(t, .)− ¯u
L∞(0,1)R
u0− ¯uL∞(0,1)
. (32)
Remark 4.The last property is weaker than stability, thus we do not have asymptotic stability ofu. However taking¯ cpositive and adjusting A as follows:
A(z)=
f (u¯+z)−f (u)¯
2 if 0zc,
f(u¯+c)
2 (z−c)+f (u¯+c)2−f (u)¯ ifzc, (33)
we can see that f(u¯2+c)tends to 0 withcand thereforeR(0)can be as small as we want.
The feedback laws (17) and (29) act in two steps. In the first step the controlguniformly increases the stateu(t, .) and therefore the characteristic speedf(u(t, .))(in the case wheref(u)¯ 0) to eventually reach a point where the speed is everywhere positive on(0,1)(the same speed profile as the target stateu). It should be noted that we may¯ potentially increaseu(t, .)− ¯uduring this part. Once such a speed profile is reached the feedback loop increases the speedf(u)more than the stateuand we have stabilization towardu. This is the same strategy as the return method¯ of J.-M. Coron[9,10].
The paper will be organized as follows. In Section2, we will prove using a Banach fixed point theorem that the closed loop systems of both Theorem1and Theorem2has a unique maximal entropy solution which furthermore is global in time, one might consider looking at the articles[26]and[27]where related questions are considered (note that a Lax–Friedrichs scheme with a discreteu(t, .)− ¯uL1(0,1)would have also provided existence). In Section3, we will adapt the result of[11]and describe the influence of the boundary conditions on the generalized characteristics touching the boundary points. In Section4we prove Theorem1. Finally in Section5we will prove Theorem2.
2. Cauchy problem for the closed loop system
In this section, we will prove the following result which will imply the first part of Theorem1and Theorem2about existence uniqueness and continuous dependence on the initial data for the closed loop systems.
Proposition 2.1.Foru¯inR,u0inBV(0,1)andgaC1function onRwhich is globally Lipschitz(with constantLG) and satisfiesg(0)=0, there exists a unique entropy solution of:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
∂tu+∂x f (u)
=gu(t, .)− ¯u
L1(0,1)
on(0,+∞)×(0,1),
u(0, .)=u0 on(0,1),
sgn u
t,1−
− ¯u f
u t,1−
−f (k)
0 ∀k∈I
¯ u, u
t,1−
, dta.e., sgn
u t,0+
− ¯u f
u t,0+
−f (k)
0 ∀k∈I
¯ u, u
t,0+
, dta.e.
(34)
Furthermore two solutionsuandvof(34)for two initial datau0andv0satisfy:
∀t0, u(t, .)−v(t, .)
L1(0,1)u0−v0L1(0,1)eLGt.
It is crucial that the boundary data is independent ofu, so that we can use a fixed point theorem on the source term of the equation. This proposition implies the first parts of Theorem1and Theorem2because both take the form of (34).
Thanks to[19,20,6], we know that for anyu0in BV(0,1)and any functionh inC0(R+)there exists a unique entropy solutionvinL∞loc((0,+∞);BV(0,1))∩Liploc([0,+∞);L1(0,1))to
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
∂tv+∂x f (v)
=h(t), on(0,+∞)×(0,1),
v(0, .)=u0, on(0,1),
sgn v
t,1−
− ¯u f
v t,1−
−f (k)
0, ∀k∈I
¯ u, v
t,1−
, dta.e., sgn
v t,0+
− ¯u f
v t,0+
−f (k)
0, ∀k∈I
¯ u, v
t,0+ , dta.e.
(35)
We now have the following key estimate, which is a classical result of Kruzkov[21]when there is no boundary.
Lemma 1.Ifvandv˜are entropy solutions of (35)with respective source termshandh˜and respective initial datau0 andu˜0then we have:
∀T 0, v(T , .)− ˜v(T , .)
L1(0,1)u0− ˜u0L1(0,1)+ T 0
h(s)− ˜h(s)ds. (36)
Proof. Following the method of Kruzkov [21], we takeψ a non-negative function inCc1(R4). Since for any(˜t ,x)˜ inR2,ψ (.,t , .,˜ x)˜ is inCc1(R2)we get withk= ˜v(t ,˜ x)˜ in (15):
+∞
0
1 0
v(t, x)− ˜v(t ,˜ x)˜ ψt(t,t , x,˜ x)˜ +sgn
v(t, x)− ˜v(˜t ,x)˜ f
v(t, x)
−f
˜ v(t ,˜ x)˜
ψx(t,t , x,˜ x)˜
+sgn
v(t, x)− ˜v(t ,˜ x)˜
h(t)ψ (t,t , x,˜ x) dx dt˜ + 1 0
u0(x)− ˜v(˜t ,x)˜ ψ (t,t , x,˜ x) dx˜ +
+∞
0
sgn
¯
u− ˜v(t ,˜ x)˜ f
v(˜˜ t ,x)˜
−f v
t,1−
ψ (t,t ,˜ 1,x)˜
−sgn
¯
u− ˜v(˜t ,x)˜ f
˜ v(t ,˜ x)˜
−f v
t,0+
ψ (t,t ,˜ 0,x) dt˜
0.
Integrating the above inequality int ,˜ x˜(which is possible since we have a compact support) and we get:
+∞
0
+∞
0
1 0
1 0
v(t, x)− ˜v(t ,˜ x)ψ˜ t(t,t , x,˜ x)˜ +sgn
v(t, x)− ˜v(t ,˜ x)˜ f
v(t, x)
−f
˜ v(t ,˜ x)˜
ψx(t,t , x,˜ x)˜
+sgn
v(t, x)− ˜v(t ,˜ x)˜
h(t)ψ (t,t , x,˜ x) dx d˜ x dt d˜ t˜+
+∞
0
1 0
1 0
u0(x)− ˜v(t ,˜ x)ψ (0,˜ t, x,˜ x) dx d˜ x d˜ t˜ +
+∞
0
1 0
+∞
0
sgn
¯
u− ˜v(˜t ,x)˜ f
˜ v(t ,˜ x)˜
−f v
t,1−
ψ (t,t ,˜ 1,x)˜
−sgn
¯
u− ˜v(˜t ,x)˜ f
˜ v(t ,˜ x)˜
−f v
t,0+
ψ (t,t ,˜ 0,x) dt d˜ x d˜ t˜
0. (37)
Reversing the role ofvandv˜we also have:
+∞
0 +∞
0
1 0
1 0
v(t, x)− ˜v(˜t ,x)˜ ψt˜(t,t , x,˜ x)˜ +sgn
v(t, x)− ˜v(t ,˜ x)˜ f
v(t, x)
−f
v(˜˜ t ,x)˜
ψx˜(t,t , x,˜ x)˜
+sgn
˜
v(t ,˜ x)˜ −v(t, x)h(˜˜ t )ψ (t,t , x,˜ x) dx d˜ x dt d˜ t˜+ +∞
0
1 0
1 0
u˜0(x)˜ −v(t, x)ψ (t,0, x,x) d˜ x dx dt˜ +
+∞
0
1 0
+∞
0
sgn
¯
u−v(t, x) f
v(t, x)
−f
˜ v
˜ t ,1−
ψ (t,t , x,1)˜
−sgn
¯
u− ˜v(t, x) f
v(t, x)
−f
˜ v
˜ t,0+
ψ (t,t , x,˜ 0) dt dx dt˜
0. (38)
And finally adding (37) and (38) we get:
+∞
0 +∞
0
1 0
1 0
v(t, x)− ˜v(˜t ,x)˜ ψt(t,t , x,˜ x)˜ +ψt˜(t,t , x,˜ x)˜ +sgn
v(t, x)− ˜v(t ,˜ x)˜
h(t)− ˜h(˜t )
ψ (t,t, x,˜ x)˜ +sgn
v(t, x)− ˜v(˜t ,x)˜ f
v(t, x)
−f
˜
v(t ,˜ x)˜
ψx(t,t , x,˜ x)˜ +ψx˜(t,t , x,˜ x)˜
dx dx dt d˜ t˜ +
+∞
0
1 0
1 0
u˜0(x)˜ −v(t, x)ψ (t,0, x,x) d˜ x dx dt˜ + +∞
0
1 0
1 0
u0(x)− ˜v(˜t ,x)ψ (0,˜ t , x,˜ x) dx d˜ x d˜ t˜ +
+∞
0
1 0
+∞
0
sgn
¯
u−v(t, x) f
v(t, x)
−f
˜ v
˜ t ,1−
ψ (t,t, x,˜ 1)
−sgn
¯
u− ˜v(t, x) f
v(t, x)
−f
˜ v
˜ t,0+
ψ (t,t , x,˜ 0) dt dx dt˜ +
+∞
0
1 0
+∞
0
sgn
¯
u− ˜v(t ,˜ x)˜ f
v(˜˜ t ,x)˜
−f v
t,1−
ψ (t,t ,˜ 1,x)˜
−sgn
¯
u− ˜v(t ,˜ x)˜ f
v(˜˜ t ,x)˜
−f v
t,0+
ψ (t,t,˜ 0,x) dt d˜ x d˜ t˜
0. (39)
Now considerφa non-negative function inCc1(R2)andρa non-negative, even,C∞function with support in[−1;1]
and satisfying1
−1ρ(x) dx=1. We define the family(ψn)of non-negative functions inCc1(R4)by:
ψn(t,t , x,˜ x)˜ =n2φ t+ ˜t
2 ,x+ ˜x 2
ρ
n(t− ˜t ) ρ
n(x− ˜x)
. (40)
It is clear that for allninNand all(t,t , x,˜ x)˜ inR4:
∂tψn(t,t , x,˜ x)˜ +∂t˜ψn(t,t , x,˜ x)˜ =n2∂1φ t+ ˜t
2 ,x+ ˜x 2
ρ
n(t− ˜t ) ρ
n(x− ˜x) ,
∂xψn(t,t , x,˜ x)˜ +∂x˜ψn(t,t , x,˜ x)˜ =n2∂2φ t+ ˜t
2 ,x+ ˜x 2
ρ
n(t− ˜t ) ρ
n(x− ˜x) . We substituteψnin (39) and letntend to infinity. We will do so term by term.
+∞
0
1 0
1 0
u˜0(x)˜ −v(t, x)ψn(t,0, x,x) d˜ x dx dt˜
=
+∞
0
1 0
1 0
u˜0(x)˜ −v(t, x)n2φ t
2,x+ ˜x 2
ρ(nt )ρ
n(x− ˜x)
dx dx dt˜
=
+∞
0
n
−n
1 0
u˜0
X− δx 2n
−v δt
n, X− δx 2n
φ δt
2n, X
ρ(δt)ρ(δx) dX dδxdδt,
after the change of variable(t, x,x)˜ →(δt=nt, δx=n(x− ˜x), X=x+ ˜2x). And since 1
0
u˜0
X−δx 2n
−v δt
n, X−δx 2n
φ δt
2n, X
dXn−→→+∞
1 0
u˜0(X)−u0(X)φ(0, X) dX dδtdδxa.e., we obtain:
+∞
0
1 0
1 0
u˜0(x)˜ −v(t, x)ψn(t,0, x,x) d˜ x dx dt˜ n−→→+∞1 2
1 0
u˜0(X)−u0(X)φ(0, X) dX. (41) Note that the 12factor comes from integratingδt from 0 to+∞and becauseρis even.
The same type of reasoning implies:
+∞
0
1 0
1 0
u0(x)− ˜v(t ,˜ x)ψ˜ n(0,t , x,˜ x) dx d˜ x d˜ t˜n−→→+∞1 2
1 0
u˜0(X)−u0(X)φ(0, X) dX, (42)
+∞
0
+∞
0
1 0
1 0
v(t, x)− ˜v(t ,˜ x)∂˜ tψn(t,t , x,˜ x)˜ +∂˜tψn(t,t , x,˜ x)˜
dx dx dt d˜ t˜
n→+∞−→
+∞
0
1 0
v(T , X)− ˜v(T , X)∂Tφ(T , X) dX dT , (43)
+∞
0
+∞
0
1 0
1 0
sgn
v(t, x)− ˜v(t ,˜ x)˜ f
v(t, x)
−f
v(˜˜ t ,x)˜
∂tψn(t,t , x,˜ x)˜ +∂x˜ψn(t,t , x,˜ x)˜
dx dx dt d˜ t˜
n→+∞−→
+∞
0
1 0
sgn
v(T , X)− ˜v(T , X) f
v(T , X)
−f
˜
v(T , X)
∂Xφ(T , X) dX dT . (44)
In the derivation of (44) the fact that(w, z)→sgn(z−w)(f (z)−f (w))is Lipschitz nearz=wis crucial so the lack of regularity of(w, z)→sgn(z−w)prevents the same argument to work for the remaining terms. However it is clear that
sgn
v(t, x)− ˜v(˜t ,x)˜
h(t)− ˜h(˜t )
ψ (t,t, x,˜ x)˜ h(t )− ˜h(˜t )ψ (t,t, x,˜ x).˜ So we get:
lim sup
n→+∞
+∞
0 +∞
0
1 0
1 0
sgn
v(t, x)− ˜v(˜t ,x)˜
h(t)− ˜h(˜t )
ψn(t,t , x,˜ x) dx d˜ x dt d˜ t˜
+∞
0
1 0
h(T )− ˜h(T )φ(T , X) dX dT . (45)
It only remains to control the boundary terms:
+∞
0
1 0
+∞
0
sgn
¯
u−v(t, x) f
v(t, x)
−f
˜ v
˜ t ,1−
ψn(t,t , x,˜ 1)
−sgn
¯
u− ˜v(t, x) f
v(t, x)
−f
˜
vt ,˜ 0+
ψn(t,t , x,˜ 0) dt dx dt,˜ and
+∞
0
1 0
+∞
0
sgn
¯
u− ˜v(˜t ,x)˜ f
˜ v(t ,˜ x)˜
−f v
t,1−
ψn(t,t ,˜ 1,x)˜
−sgn
¯
u− ˜v(t ,˜ x)˜ f
v(˜˜ t ,x)˜
−f v
t,0+
ψn(t,t ,˜ 0,x) dt d˜ x d˜ t.˜ But ifφhas a support inR×(0,1)then fornlarge enough we have:
ψn(., ., .,0)=ψn(., ., .,1)=ψn(., .,1, .)=ψ (., .,0, .)=0,
so both boundary terms tend to 0. Combining this with (41), (42), (43), (44) and (45), we see that for every non- negative functionφinCc1(R×(0,1)):
+∞
0
1 0
v(t, x)− ˜v(t, x)φt(t, x)+sgn
v(t, x)− ˜v(t, x) f
v(t, x)
−f
˜ v(t, x)
φx(t, x)
+h(t)− ˜h(t)φ(t, x) dx dt+ 1 0
u0(x)− ˜u0(x)φ(0, x) dx
0. (46)
A density argument shows that the previous estimate holds for any Lipschitz function φ with compact support in R×(0,1). Now forT >0 andn∈N∗, we defineαnandβnas follows:
αn(t)=
⎧⎪
⎨
⎪⎩
1 fortT ,
0 fortT +1n,
1−n(t−T ) forT tT+1n,
βn(x)=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1 forx∈ [1n,1−1n], 2nx−1 forx∈ [2n1,n1], 2n(1−x)−1 forx∈ [1−1n,1−2n1]
0 otherwise.
Takingφ=αn(t)βn(x)in (46) and lettingntend to infinity we end up with:
u0− ˜u0L1(0,1)−v(T , .)− ˜v(T , .)
L1(0,1)+ T 0
h(t )− ˜h(t)dt
+ T 0
sgn v
t,0+
− ˜v t,0+
f v
t,0+
−f
˜ v
t,0+
−sgn v
t,1−
− ˜v t,1−
f v
t,1−
−f
˜ v
t,1− dt
0. (47)
Now for three numbersa, b, cwe have:
∀k∈I (a, b)∩I (a, c)∩I (b, c), sgn(c−b)
f (c)−f (b)
=sgn(c−a)
f (c)−f (k)
+sgn(b−a)
f (b)−f (k) .
Applying this identity with(a= ¯u, b=v(t,1−), c= ˜v(t,1−))or(a= ¯u, b=v(t,0+), c= ˜v(t,0+))and using the boundary conditions of (35) we obtain:
sgn v
t,0+
− ˜v t,0+
f v
t,0+
−f
˜ v
t,0+
0 dta.e., sgn
v t,1−
− ˜v t,1−
f v
t,1−
−f
˜ v
t,1−
0 dta.e.
Finally substituting those inequalities in (47) provides (36). 2 We defineXas the following function space:
X= α∈C0
R+ αX :=sup
t0
α(t )e−2LGt
<+∞
,
whereLGis the Lipschitz constant ofg(see2.1). Consider the operatorF which toα∈X associates the function v(t, .)− ¯uL1(0,1), withvthe entropy solution of:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
∂tv+∂x f (v)
=g α(t )
on(0,+∞)×(0,1),
v(0, .)=u0 on(0,1),
sgn v
t,1−
− ¯u f
v t,1−
−f (k)
0 ∀k∈I
¯ u, v
t,1−
, dta.e., sgn
v t,0+
− ¯u f
v t,0+
−f (k)
0 ∀k∈I
¯ u, v
t,0+ , dta.e.
(48)
Then we have the following result.
Lemma 2.The operatorFhas the following properties.
• For any functionαinX, the functionF(α)belongs toX.
• The operatorFis12-Lipschitz onX.
Proof. We takeαinX andvthe entropy solution of (48). The constant functionu¯is a solution of (35) with source term equal to 0 and initial data equal tou. So using (36) we have:¯
F(α)(T )=v(T , .)− ¯u
L1(0,1)u0− ¯uL1(0,1)+ T 0
gα(t )dt.
Therefore for anyT 0, we have:
e−2LGTF(α)(T )u0− ¯uL1(0,1)+ T 0
e−2LGTg α(t )
dt
u0− ¯uL1(0,1)+ T 0
LGe−2LG(T−t )α(t )e−2LGtdt
u0− ¯uL1(0,1)+ αX T 0
LGe−2LG(T−t )dtu0− ¯uL1(0,1)+αX 2 . It follows thatF(α)is inX.
In order to prove the second assertion let us considerα,β inX andvα,vβ the corresponding entropy solutions of (48). Using (36) we see that for any non-negativeT:
F(α)(T )−F(β)(T )=vα(T , .)− ¯u
L1(0,1)−vβ(T , .)− ¯u
L1(0,1) vα(T , .)−vβ(T , .)
L1(0,1)
T 0
g α(t )
−gβ(t)dt.
But for anyT 0:
e−2LGTF(α)−F(β)(T ) T 0
LGe−2LG(T−t )α(t )−β(t)e−2LGtdt α−βX
T 0
LGe−2LG(T−t )dt α−βX
2 . 2
Let us now go back to the proof of Proposition2.1. Applying the Banach fixed point theorem toF, we see that (34) has a unique entropy solutionusuch thatu(T , .)− ¯uL1(0,1)is inX. But ifvis an entropy solution of (34) and if we use (36) we have:
v(T , .)− ¯u
L1(0,1)u0− ¯uL1(0,1)+ T 0
LGv(t, .)− ¯u
L1(0,1)dt.
Using Gronwall’s lemma we obtain:
v(T , .)− ¯u
L1(0,1)u0− ¯uL1(0,1)eLGT. Thus the application:
T →v(T , .)− ¯u
L1(0,1),
is inXand thereforev=u. Using Lemma1and Gronwall’s lemma we have that foruandvthe entropy solutions to (34) for initial datau0andv0:
∀t0, u(t, .)−v(t, .)
L1(0,1)u0−v0L1(0,1)eLGt. This concludes the proof of Proposition2.1.
3. Generalized characteristics and boundary conditions
We begin by recalling a few definitions and results from[11]. We will refer in this section to the system:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
∂tu+∂x f (u)
=h(t) on(0,+∞)×(0,1),
u(0, .)=u0 on(0,1),
sgn u
t,1−
− ¯u f
u t,1−
−f (k)
0 ∀k∈I
¯ u, u
t,1−
, dta.e., sgn
u t,0+
− ¯u f
u t,0+
−f (k)
0 ∀k∈I
¯ u, u
t,0+
, dta.e.,
(49)
whereu0∈BV(0,1),h∈C0(R+),u¯∈Randuis the unique entropy solution. Following[11]we introduce the notion of generalized characteristic.
Definition 2.
• Ifγ is an absolutely continuous function defined on an interval(a, b)⊂R+and with values in(0,1), we say that γ is a generalized characteristic of (49) if:
˙ γ (t )∈I
f u
t, γ (t )− , f
u
t, γ (t )+
dta.e.
This is the classical characteristic ODE taken in the weak sense of Filippov[13].
• A generalized characteristicγ is said to be genuine on(a, b)if:
u
t, γ (t )+
=u
t, γ (t )−
dta.e.
We recall the following results from[11].
Theorem 3.
• For any(t, x)in(0,+∞)×(0,1)there exists at least one generalized characteristicγdefined on(a, b)such that a < t < bandγ (t)=x.
• Ifγ is a generalized characteristics defined on(a, b)then for almost alltin(a, b):
˙ γ (t )=
f(u(t, γ (t)) ifu(t, γ (t )+)=u(t, γ (t )−),
f (u(t,γ (t)+))−f (u(t,γ (t)−))
u(t,γ (t )+)−u(t,γ (t )−) ifu(t, γ (t )+)=u(t, γ (t )−).
• Ifγ is a genuine generalized characteristics on(a, b), then there exists aC1functionv defined on(a, b)such that:
u
b, γ (b)+
v(b)u
b, γ (b)− , u
t, γ (t )+
=v(t )=u
t, γ (t )−
∀t∈(a, b), u
a, γ (a)−
v(a)u
a, γ (a)+
. (50)
Furthermore(γ , v)satisfy the classical ODE equation:
γ (t )˙ =f v(t )
,
˙
v(t )=h(t), ∀t∈(a, b). (51)
• Two genuine characteristics may intersect only at their endpoints.
• Ifγ1andγ2are two generalized characteristics defined on(a, b), then we have:
∀t∈(a, b)
γ1(t)=γ2(t) ⇒ ∀st, γ1(s)=γ2(s) .
• For any(t, x)inR+×(0,1)there exist two generalized characteristicsχ+andχ−called maximal and minimal and associated tov+andv−by(51), such that ifγ is a generalized characteristic going through(t, x)then
∀st, χ−(s)γ (s)χ+(s), χ+andχ−are genuine on{s < t}, v+(t)=u
t, x+
and v−(t)=u t, x−
.
Note that in the previous theorem, every property dealt only with the interior of R+× [0,1]. We will now be interested in the influence of the boundary conditions on the generalized characteristics. Following the method of[11], we begin with a few technical identities.
Lemma 3.
• Ifχis a Lipschitz function defined on[a, b]and satisfying:
∀t∈(a, b), 0χ (t ) <1, (52)
we have:
χ (b)
0
u(b, x) dx−
χ (a)
0
u(a, x) dx
= b a
χ (t )h(t ) dt+ b a
f u
t,0+
−f u
t, χ (t )+
+ ˙χ (t )u
t, χ (t )+
dt. (53)
• Ifχis a Lipschitz function defined on[a, b]and such that:
∀t∈(a, b), 1χ (t ) >0, we have:
1 χ (b)
u(b, x) dx− 1 χ (a)
u(a, x) dx
= b a
1−χ (t )
h(t) dt+ b a
f u
t, χ (t )−
−f u
t,1−
− ˙χ (t )u
t, χ (t )−
dt. (54)
• Finally ifχ1andχ2are two Lipschitz functions defined on[a, b]and satisfying:
∀t∈(a, b), 0< χ1(t) < χ2(t) <1, the following holds:
χ2(b) χ1(b)
u(b, x) dx−
χ2(a) χ1(a)
u(a, x) dx= b a
h(t )
χ2(t)−χ1(t) dt
+ b a
f u
t, χ1(t)−
−f u
t, χ2(t)+
+ ˙χ2(t)u
t, χ2(t)+
− ˙χ1(t)u
t, χ1(t)−
dt. (55)
Proof. We begin with the proof of (53). We may prove the equality when (52) holds on[a, b]and then extend it since both sides of (53) are continuous inaandb. For >0 we define the following two functions:
ψ(t, x)=
⎧⎨
⎩
1 whent∈ [a, b]and 0xχ (t ), 1−x−χ (t) whent∈ [a, b]andχ (t )xχ (t )+,
0 otherwise,
∀t∈R+, ρ(t)=
⎧⎪
⎪⎨
⎪⎪
⎩
1 whena+tb,
b−t
whenb−tb,
t−a
whenata+,
0 otherwise.
The productρψ is Lipschitz, non-negative and has compact support inR+× [0,1]. Sinceuis a weak solution of (49) we have:
