Some results on the boundary control of systems of conservation laws
Olivier Glass
Ceremade, Université Paris-Dauphine
ANalysis and COntrol on NETworks: trends and perspectives, Padova, March 9–11, 2016
Introduction
I We discuss control problems of one-dimensional hyperbolic systems of conservation laws:
ut+ f (u)x = 0, f : Ω ⊂ Rn→ Rn, (SCL) satisfying the (strict) hyperbolicity condition that at each point
df has n distinct real eigenvalues λ1< · · · < λn.
I Typically: compressible fluid flows, fluid through a canal, traffic flow, etc.
Characteristic fields
I Corresponding to thecharacteristic speeds λ1< · · · < λn, the Jacobian A(u) := df (u) has n right eigenvectors ri(u).
I We denote(`i)i =1,...,n the left eigenvectorsof df (u) satisfying
`i· rj = δij.
I The characteristic families will be supposed to be genuinely non-linear (GNL) orlinearly degenerate (LD), that is:
∇λi· ri 6= 0 for all u in Ω or ∇λi· ri ≡ 0.
Controllability problem
I Domain: [0, T ] × [0, L].
I Stateof the system u(t, ·) ∈ BV (0, L)
I Control: the “boundary data” onone or bothsides. When one controls on one side only, the boundary condition on the other side is fixed.
I Exact controllability: given u0and u1, can we find aboundary control for the system driving u0 to u1?
I Controllability to constant states: given u0and given u1 a constant state, can we find a boundary control for the system driving u0to u1?
Reformulation of the controllability problem
One can reformulate the controllability problem as follows.
I Exact controllability: given u0and u1, can we find asolutionof the system driving u0 to u1?
I Controllability to constant states: given u0and given u1 a constant state, can we find asolutionof the system driving u0to u1?
L 0
t T
Stabilization problem
I For this problem, we will suppose moreover that thecharacteristic speedsarestricly separated from 0:
λ1< · · · < λm< 0 < λm+1< · · · < λn.
I We will be interested in boundary conditions put in the following form:
u+(t, 0) u−(t, L)
= Gu+(t, L) u−(t, 0)
with
u+:= (um+1, . . . , un) and u−:= (u1, . . . , um).
I We consider anequilibrium pointu of the system. To simplify, we fix u = 0.
I The question is todesign G so that u becomes anasymptotically stable point for the resultingclosed-loop system.
Stabilization problem
I We recall that a point u is calledstable when for any neighborhood V of u, there exists a neighborhood U of u such that any trajectory of the system starting from u stays in V for all t ≥ 0.
I It is called asymptotically stablewhen moreover any trajectory starting from U satisfies u(t, ·) → u as t → +∞.
I It is called exponentially stablewhen any trajectory starting from some neighborhood U of u satisfies
ku(t, ·)k ≤ C exp(−γt)ku(0, ·)k for all t ≥ 0, for some fixed γ > 0 and C > 0.
Two types of solutions
I Hyperbolic systems of conservation laws developsingularities in finite time.
I So one can either work with regular solutions (C1)with small C1-norm (for small time), or withweak entropy solutions.
I This is the framework in which we work here: entropy solutions “à la Glimm”, that is, ofbounded variationin x , with small total variation.
Entropy conditions
Definition
Anentropy/entropy flux couplefor a hyperbolic system of conservation laws (SCL) is defined as a couple of regular functions (η, q) : Ω → R satisfying:
∀u ∈ Ω, Dη(u) · Df (u) = Dq(u).
Definition
A function u ∈ L∞(0, T ; BV (0, L)) ∩ Lip(0, T ; L1(0, L)) is called an entropy solutionof (SCL) when, for any entropy/entropy flux couple (η, q), with ηconvex, one has in the sense of measures
η(u)t+ q(u)x ≤ 0, that is, for all ϕ ∈ D((0, T ) × (0, L)) with ϕ ≥ 0,
Z
(0,T )×(0,L)
η(u(t, x ))ϕt(t, x ) + q(u(t, x ))ϕx(t, x ) dx dt ≥ 0.
Previous results in the classical case
I Seminal work by Cirina (1969)
I Theorem (Li-Rao, 2002): Consider
∂tu + A(u)ux = F (u),
such that A(u) has n distinct real eigenvalues such that
λ1(u) < · · · < λk(u)≤ −c < 0 < c ≤λk+1(u) < · · · < λn(u), and
F (0) = 0.
Then for all φ, ψ ∈ C1([0, 1]) such that kφkC1+ kψkC1< ε, there exists a solution u ∈ C1([0, T ] × [0, 1]) such that
u|t=0= φ, and u|t=T = ψ.
I Manyworks since in this context ofsemi-global solutions, in particular in the case ofnetworks! See e.g. the book of Li Ta-Tsien on the subject.
Some results in the case of entropy solutions
I Several works on the convex scalar case:
I Ancona and Marson (1998),
I Horsin (1998),
I Perrollaz (2011),
I Adimurthi-Gowda-Goshal (2013),
I Andreianov-Donadello-Marson (2015),
I Adimurthi-Goshal-Marcati (2016),
I Several works on the system case:
I Bressan-Coclite (asymptotic result and a counterexample, 2002),
I Ancona-Coclite (Temple systems, 2002),
I Ancona-Marson (one-side open loop stabilization, 2007),
I G. (2007, 2014),
I Andreianov-Donadello-Ghoshal-Razafison (2015, triangular system),
I T. Li-L. Yu (2015, LD systems),
I Coron-Ervedoza-G.-Goshal-Perrollaz (2015).
Some controllability results
Bressan and Coclite’s counterexample
I Bressan and Coclite (2002): for a class of systems containing DiPerna’s system:
( ∂tρ + ∂x(ρu) = 0,
∂tu + ∂x
u2
2 +γ−1K2 ργ−1
= 0,
there are initial conditions ϕ ∈ BV ([0, 1]) of arbitrary small total variation such that any entropy solution u remaining of small total variation satisfies: for any t, u(t, ·)is not constant. 6= C1case!
I But these systems are very particular: the interaction of two shocks in one family generates a shock in the opposite family.
I What happens when the interaction of two shocks in one family generatesa rarefactionin the opposite family?
Isentropic Euler equation
Theorem (G., 2007)
Consider u0= (ρ0, v0) and u1= (ρ1, v1) two states in R+∗× R. There exist ε > 0 and T = T (u0, u1) > 0, such that for any u0∈ BV ([0, 1]) satisfying:
ku0− u0k ≤ ε and TV (u0) ≤ ε,
there is an entropy solution u = (ρ, v ) of the isentropic Euler equation:
∂tρ + ∂x(ρv ) = 0,
∂t(ρv ) + ∂x(ρv2+ κργ) = 0, in [0, T ] × [0, L] such that
u|t=0= u0, and u|t=T = u1.
Remark. Saint-Venant’s equations (a.k.a. shallow water) correspond to γ = 2.
The p-system
Theorem (G., 2007)
Consider u0= (τ0, v0) and u1= (τ1, v1) two states in R+∗× R. There exist ε > 0, and T = T (u0, u1) > 0, such that for any u0∈ BV ([0, 1]) satisfying:
ku0− u0k ≤ ε1 and TV (u0) ≤ ε, there is an entropy solution u = (τ, v ) of the p-system
∂tτ − ∂xv = 0,
∂tv + ∂x(κτ−γ) = 0, in [0, T ] × [0, L] such that
u|t=0= u0, and u|t=T = u1.
I In both cases, there are more general sufficient conditions of a BV final state to be reachable.
Full Euler equation
Theorem (G., 2014)
Let u0:= (ρ0, v0, P0) ∈ R3 with ρ0, P0> 0. Let η > 0. There exists ε > 0 such that for any u0= (ρ0, v0, P0) ∈ BV (0, L; R3) such that
ku0− u0kL∞(0,L)+ TV (u0) ≤ ε,
for any u1= (ρ1, v1, P1) with ρ1, P1> 0, there exist T > 0 and a weak entropy solution u = (ρ, v , P) of the full Euler system
∂tρ + ∂x(ρv ) = 0,
∂t(ρv ) + ∂x(ρv2+ P) = 0,
∂t
γ − 1
2 ρv2+ P
+ ∂x
γ − 1
2 ρv3+ γPv
= 0.
such that
u|t=0= u0 and u|t=T = u1, and
TV (u(t, ·)) ≤ η, ∀t ∈ (0, T ).
Full Euler equation (Lagrangian case)
I As in the isentropic case, the full Euler equation can be expressed in Lagrangian coordinates:
∂tτ − ∂yv = 0,
∂tv + ∂yP = 0,
∂t(e +v22) + ∂y(Pv ) = 0.
I In that case a specific convex entropy couple is of particular importance: (−S , 0) where S is thephysical entropy:
S := log
P
(γ − 1)ργ
= log Pτγ γ − 1
.
I Hence an entropy solution u of the above system satisfies
∂tS (u) ≥ 0.
Full Euler equation (Lagrangian case)
Theorem (G., 2014)
Suppose that γ ∈ 1,53. Let η > 0. Let u0:= (τ0, v0, P0) ∈ R3 with τ0, P0> 0 and let u1= (τ1, v1, P1) with τ1, P1> 0, such that
S (u1) > S (u0).
There exist ε > 0 and T > 0 such that for any u0= (τ0, v0, P0) in BV (0, L; R3) such that
ku0− u0kL∞(0,L)+ TV (u0) ≤ ε,
there is u = (τ, v , P) a weak entropy solution of the full Euler equation in Lagrangian coordinates such that
u|t=0= u0 and u|t=T = u1, and
TV (u(t, ·)) ≤ η, ∀t ∈ (0, T ).
A remark
I These systems can be recast as
∂tρ + ∂x(ρv ) = 0,
∂tv + v ∂xv + (∂xP)/ρ = 0,
∂tS + v ∂xS = 0,
and
∂tτ − ∂yv = 0,
∂tv + ∂yP = 0,
∂tS = 0, in the context of regular solutions.
I Hence such a result for in the Lagrangian casecannot be truefor regular solutions.
I It is the only example (that I know of) where the situation is better from the point of view of controllability in the context of entropy solutions than in the context of regular solutions.
A general idea of the proof
I Solutions are constructeddirectlyusing a wave-front tracking approach (cf. Dafermos, DiPerna, Bressan, . . . ):
I one constructs a sequence of approximations of a solutions,
I these approximations are piecewise constant functions on R+× [0, L]
where the discontinuities are straight lines separating states
connected by shocks or rarefactions (or sometimes “artificial fronts”),
>J
* S
>
J (
R
* C
0 L
t
A general idea of the proof
I One has tosend fronts from the boundaryin an appropriate manner.
I The main purpose of fronts sent from the boundary is to avoid the proliferation of shocksinside the domain (at the core of Bressan and Coclite’s counterexample).
I For that one relies on cancellations effects: in the systems above, the interaction of two shocks in a family generates ararefaction in the opposite family.
I One can use these artificially created rarefactions to annihilate shockspresent in the domain.
Open problems
I These results concern controllabilityfrom both sides. Managing boundariesis a clear next step (ongoing work with F. Ancona and T. K. Nguyen)
I Equivalent on networksis very important from both points of view of theory and applications. What can be adapted from the
“classical” theory?
A work in that direction in the scalar case has been done by Ke Wang.
I Would there be a“general” conditionon f indicating when constants can be reached and when not?
A feedback stabilization result
(with J.-M. Coron, S. Ervedoza, S. Goshal and
V. Perrollaz )
A simple framework
I Here we consider2 × 2 systems of conservation laws:
ut+ f (u)x = 0 in [0, +∞) × [0, L],
with characteristic speeds λ1< λ2and satisfying the conditions:
I each characteristic field is GNL,
I velocities arepositive: 0 < λ1< λ2.
I Theboundary conditionsare as follows:
u(t, 0) = Ku(t, L), where K is a 2 × 2 (real) matrix.
I The goal is to find conditions on K suchensuring the (exponential) stability of the system.
Stabilization problem
I One would like to extend to the realm of BV entropy solutions stabilization results that were obtained in the context of classical solutions, such as
I Slemrod, Greenberg-Li, . . .
I Bastin-Coron, Bastin-Coron-d’Andrea-Novel, Bastin-Coron-d’Andrea-Novel-de Halleux-Prieur, Bastin-Coron-Krstic-Vazquez, . . .
I Leugering-Schmidt, Dick-Gugat-Leugering, Gugat-Herty,. . .
I Ta-Tsien Li, Tie Hu Qin, . . .
I Many others!
I In particular, studies in an interval are preliminary to studying problems innetworks.
I For instance in Bastin-Coron-d’Andrea-Novel-de Halleux-Prieur, the stabilization of the 2 × 2 Saint-Venant system on a network of 2 intervals is transformed in a stabilization problem for a 4 × 4 system on a single interval.
Main result
Theorem (Coron-Ervedoza-G.-Goshal-Perrollaz)
Suppose the above assumptions satisfied. If the matrix K satisfies
α∈(0,+∞)inf max|`1(0) · Kr1(0)| + α|`2(0) · Kr1(0)|,
α−1|`1(0) · Kr2(0)| + |`2(0) · Kr2(0)| < 1, then there exist positive constants C , ν, ε0> 0, such that for every u0∈ BV (0, L) satisfying
|u0|BV ≤ ε0,
there exists an entropy solution u of the system in L∞(0, ∞; BV (0, L)) satisfying u(0, ·) = u0(·) and the boundary conditions for almost all times, such that
|u(t)|BV ≤ C exp(−νt)|u0|BV, t ≥ 0.
Rewriting the condition
Denoting for p ∈ [1, ∞)
k(x1, . . . , xn)kp:=
n
X
i =1
|xi|p
!1/p
, k(x1, . . . , xn)k∞:= max
i =1...n|xi| kMkp := max
kxkp=1kMxkp for M ∈ Rn×n, one defines
ρp(K ) := inf{k∆K ∆−1kp, ∆ diagonal with positive entries}.
It is easy to check that
α∈(0,+∞)inf max|`1(0) · Kr1(0)| + α|`2(0) · Kr1(0)|,
α−1|`1(0) · Kr2(0)| + |`2(0) · Kr2(0)| = ρ1(K ), so that the condition can be written as ρ1(K ) < 1.
Analogous conditions
I For the same question for classical solutions in Cm-norm(m ≥ 1), a sufficient condition is:
ρ∞(K ) < 1.
Cf. T. H. Qin, Y. C. Zhao, T. Li and Bastin-Coron.
I In the case ofSobolev spaces Wm,p([0, L])with m ≥ 2 and p ∈ [1, +∞], a sufficient condition is:
ρp(K ) < 1.
Cf. Coron-d’Andréa-Novel-Bastin for p = 2, Coron-Nguyen for general p.
Remarks
I One can actually show that
ρ1(K ) = ρ∞(K ).
I The known results on the existence of a standard Riemann semigroup for initial-boundary problem do not seem to cover our situation exactly and uniqueness of solutions in the spirit of Bressan-LeFloch or Bressan-Goatin seems open.
Cf. Amadori, Amadori-Colombo, Colombo-Guerra, Donadello-Marson, . . .
A general idea
I The result relies on a Lyapunov function.
I This Lyapunov function is mainly inspired by two sources:
I Lyapunov functions constructed in the classical case, cf.
Coron-Bastin-d’Andrea-Novel, Coron-Bastin, . . .
I Glimm’s functional used to construct entropy solutions in BV
I One constructs solutions using the wave-front trackingapproach (here, DiPerna’s approach since we consider 2 × 2 systems)
1. Lyapunov functions in the classical case
An example from Coron-Bastin-d’Andrea-Novel: for the system
∂ta + c(a, b)∂xa = 0,
∂tb − d (a, b)∂xb = 0,
with c, d > 0, one finds a Lyapunov functional of the form
L = Z L
0
a2(x )e−µxdx + Z L
0
b2(x )e+µxdx
+ C1 Z L
0
(∂xa)2(x )e−µxdx + Z L
0
(∂xb)2(x )e+µxdx
!
+ C2
Z L
0
(∂xx2a)2(x )e−µxdx + Z L
0
(∂xxb)2(x )e+µxdx
! .
This is connected to J.-M. Coron’s Lyapunov function forstabilization of the incompressible Euler equationin 2-D simply connected domains (see also G. in the multi-connected case).
2. Glimm’s functional
I One works with front-tracking approximations
I Glimm’s functional is obtained as follows: consider
V (τ ) = X
α wave at time t
|σα| ; Q(τ ) = X
α,β approaching waves
|σα|.|σβ|,
σαbeing the strengthof the wave.
I Analyzing wave interactions one shows that for some C > 0, if TV (u0) is small enough, thenV (t) + CQ(t)is non-increasing and deduces bounds in L∞t BVx.
Our Lyapunov functional
Our Lyapunov functional is as follows:
J := V + CQ where
V (U) =
n
X
i =0
(|σi ,1| + |σi ,2|) e−γxi,
Q(U) = X
(xi,σi)
|σi|e−γxi X
(xj,σj) approaching (xi,σi)
|σj|e−γxj
! ,
for suitable constants, where
I σi ,k is the strength of the k-wave at xi (σi when there is no ambiguity, i.e. for i ≥ 1),
I x1, . . . , xn are the discontinuities in (0, L),
I u(t, 0+) = Ψ2(σ0,2, Ψ1(σ0,1, Ku(t, L−))).
Our Lyapunov functional
Analyzing in particularinteractions of fronts with the boundary, one shows that for suitable constants and provided that
TV (u0) is small enough, one has for proper ν > 0:
J(t) ≤ J(0) exp(−νt).
This allows to construct approximations and the solutions globally in time and to get the result.
Open problems
I Considering a less particular case: speeds with different signs, n × n systems, nonlinear boundary conditions, non GNL characteristic fields, etc.
I Equivalent on networks! We believe this result to be sufficiently similar to the classical case to yield close results on networks in the entropy case.
I What about source terms?
