Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws

1
Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation
laws
Sébastien Blandin, Xavier Litrico, Maria Laura Delle Monache, Benedetto Piccoli and Alexandre Bayen
Abstract— We consider the problem of Lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law
with strictly convex flux in one dimension of space, around a uniform
equilibrium. We show that for a specific class of boundary conditions,
the solution to the initial-boundary value problem for an initial
condition with bounded variations can be approximated arbitrarily
closely in the L1 norm by a piecewise smooth solution with finitely
many discontinuities. The constructive method we present designs
explicit boundary conditions in this class, which guarantee Lyapunov
stability of the weak entropy solution to the initial-boundary value
problem. The stabilization result accounts for the proper treatment
of boundary conditions in the weak sense. We also design a greedy
controller obtained by maximizing the instantaneous decrease rate of
the Lyapunov function, and illustrate the limitations of such simple
controllers. Finally, we design improved boundary feedback controller
which guarantees Lyapunov asymptotic stability while accounting for
proper treatment of weak boundary conditions. Controllers performance is illustrated on numerical benchmarks using the Godunov
scheme.
Index Terms— Lyapunov stabilization, control, conservation laws
MSC: 93D05, 35L65
I. I NTRODUCTION
A. Motivation
The conservation principle is one of the most fundamental modeling principles for physical systems. Statements of conservation
of mass, momentum, energy are at the center of modern classical
physics. For distributed dynamical systems, this principle can be
written in conservation law form with the use of partial differential
equations (PDE). The problem of well-posedness of the partial
differential equation is concerned with the existence, uniqueness,
and continuous dependence of the solution to the problem data [17].
First existence results for scalar conservation laws in one dimension of space date back to [29]. For hyperbolic systems of
conservation laws in one dimension of space, existence results were
provided in [18] with the introduction of the random choice method.
Existence and uniqueness in the scalar case for several dimensions
of spaces were proven in [24], and to this date constitute the only
general results known on well-posedness for several dimensions
of space. Existence and uniqueness for n × n hyperbolic systems
of conservation laws in one-dimension of space was shown only
Sébastien Blandin is a Research Scientist, IBM Research, Singapore
(email: [email protected]).
Xavier
Litrico
is
the
Director
of
LyRE,
R&D
center
of
Lyonnaise
des
Eaux,
Bordeaux,
France
([email protected]).
Maria Laura Delle Monache is Postdoctoral Researcher at the
Department of Mathematical Sciences, Rutgers University, Camden
([email protected]).
Benedetto Piccoli is the Joseph and Loretta Lopez Chair Professor of
Mathematics, Department of Mathematical Sciences, and Director, Center
for Computational and Integrative Biology, Rutgers University, Camden
([email protected]).
Alexandre Bayen is the Chancellor’s Professor, Department of Electrical Engineering and Computer Sciences, Department of Civil and
Environmental Engineering, University of California, Berkeley (e-mail:
[email protected]).
recently, see [11], [12] for systems with genuinely nonlinear
or linearly degenerate characteristic families, and [5], [8] for the
general case of systems of strictly hyperbolic conservation laws.
The global well-posedness of solutions to hyperbolic systems of
conservation laws in several dimensions of space is still largely
open.
The keystone of well-posedness results, and a standard argument
for constructive existence proofs, is the consideration of a functional
space in which small variations in the problem data, i.e. initial
condition for the Cauchy problem, initial and boundary conditions
for the initial-boundary value problem (IBVP), create only small
variations in the tentative solution.
The control problem is posed from a different perspective, and
in different terms. In the case of boundary control [23], the control
problem consists of an objective trajectory for the system in a
given functional space, and the knowledge of an initial condition.
The problem of control [13] or stabilization [22], [7], [31], [14]
consists in the existence and design of a controller, i.e. boundary
conditions in the case of boundary control or stabilization, for which
the solution to the partial differential equation stays in a domain
prescribed by the objective trajectory.
In this article we propose to show Lyapunov stability of the
solution to the initial-boundary value problem associated with a
scalar conservation law under suitably designed boundary conditions. We consider a scalar conservation law with smooth strictly
concave or convex flux in one dimension of space. This partial
differential equation is called the Burgers equation [20] in the case
of a quadratic convex flux, and in the case of a concave flux,
corresponds to the Lighthill-Whitham-Richards (LWR) [28], [32]
PDE in its various forms, used in particular for macroscopic traffic
flow modeling. The problem is defined in the following section.
B. Problem statement
Consider the scalar conservation law in one dimension of space
∂t u + ∂x f (u) = 0
(1)
.
on the domain Ω = { (t, x)| t ≥ 0 and a ≤ x ≤ b}. The flux
function f (·) is assumed to be smooth (infinitely differentiable)
and strictly convex or strictly concave1 . The initial-boundary value
problem (IBVP) for (1) in Ω with initial condition u0 : (a, b) 7→ R,
and boundary conditions ua , ub : R+ 7→ R, reads
∂t u + ∂x f (u) = 0
(2)
u(0, x) = u0 (x)
(3)
u(t, a) = ua (t), u(t, b) = ub (t).
(4)
The Lyapunov boundary stabilization problem can now be formulated.
Definition 1: Given a stationary solution u∗ to the PDE (1),
the Lyapunov (resp. asymptotic) boundary stabilization problem
1 this is equivalent to the condition of genuine nonlinearity of the
characteristic field.
2
consists in the existence of boundary conditions ua , ub depending
on initial condition u0 with bounded variations such that the
following is true. The IBVP (2)-(3)-(4) is well-posed and its solution
is (resp. asymptotically) stable in the sense of Lyapunov at u∗ .
Note that Lyapunov (resp. asymptotic) stabilization consists in the
existence of a positive definite function decreasing (resp. vanishing)
in time along trajectories of the system.
Stationary solutions to the PDE include constant solutions and
solutions with a single stationary jump discontinuity, called shock.
In this article we address the case of constant solutions.
The well-posedness of the IBVP (2)-(3)-(4) is critical to the
definition of the problem, since the design of arbitrary boundary
conditions can make the problem ill-posed (see [36] for an illustration on the LWR equation in the case of traffic). This would
lead to a discrepancy between the control implemented and its
realized value in the system, and a divergence between the desired
trajectory of the system and its real trajectory. In the case of traffic,
it corresponds for instance to installing a green traffic light at the
location of a traffic jam with the intended goal that cars in the jam
adopt the corresponding free-flow speed. For well-posedness of the
IBVP (2)-(3)-(4), the PDE (1) and the boundary conditions (4) must
be understood in the weak sense. The weak formulation is presented
in Section II-A.
The remainder of the article is organized as follows. Section II
defines the notations used later in the article and states useful
lemmas. Section III contains the proof of well-posedness of the
initial-boundary value problem with piecewise-smooth increasing
datum with negative gradient concentrated at a finite number of
locations. Preparatory derivations involving the Lyapunov function
candidate can be found in Section IV. In Section V, we show
stability of the Lyapunov function for weak entropy solutions to
the scalar conservation law remaining in the special class with a
finite number of shocks. In Section VI we design a controller that
maximizes the instantaneous decrease of the Lyapunov function
identified previously, but highlight several configurations in which
asymptotic stability is not achieved by this controller. In Section VII
we design a new controller which guarantees asymptotic stability,
and guarantees the existence of solution to the IBVP in the
special class of solutions with a finite number of shocks. Moreover
we propose a nonlocal control to improve the convergence time.
Numerical examples are proposed in Section VIII, and promising
research avenues related to this work in Section IX.
II. P RELIMINARIES
It is well-known that jump discontinuities can arise in finite time
in solutions to conservation laws [17]. Thus classical solutions do
not exist in general, and it is necessary to consider a more general
formulation of the conservation law.
1) Weak entropy solution to the Cauchy problem: The weak
formulation of the conservation law is obtained by considering
derivatives in the sense of distribution.
Definition 2: A function u : [0, T ] × R 7→ R is a weak solution
to the Cauchy problem (2)-(3) if for any continuously differentiable
function φ with compact support contained in (−∞, T ) × R,
Z TZ ∞
Z ∞
(u φt + f (u) φx ) dx dt +
u0 (x) φ(0, x)dx = 0,
−∞
−∞
and t 7→ u(t, ·) is continuous from [0, T ] into L1loc .
σ ∆u = ∆f (u),
(5)
(6)
where ∆u = ur − ul is the jump in u, with ur , respectively ul , the
value of the right, respectively left, limit of u at the jump location.
To isolate a unique weak solution to a Cauchy problem associated
with the conservation law, an additional admissibility condition
(see Section 4.5 of [15]) is required. Different conditions have
been proposed in the literature. In the scalar case, one of the first
admissibility conditions, due to Oleinik [29], states that for a shock
joining a left state ul and a right state ur , the following inequality
must be satisfied for all u between ul and ur :
f (u) − f (ul )
f (u) − f (ur )
≥σ≥
,
u − ul
u − ur
(7)
where σ is the Rankine-Hugoniot speed (6). Kruzkhov [24] showed
that it was sufficient to satisfy the entropy inequality condition for
a specific family of entropy-entropy flux pairs in the scalar case,
yielding the Kruzkhov entropy condition. The Lax admissibility
condition [25], which exhibits a convenient geometric interpretation, states that for a shock joining a left state ul and a right state
ur , the following inequality must be satisfied:
λ(ul ) ≥ σ ≥ λ(ur )
(8)
where λ(u) is the characteristic speed of u (i.e. f 0 (u)), and σ
is the Rankine-Hugoniot speed (6). For the case of systems one
requires that condition (8) holds for a genuinely nonlinear or
linearly degenerate i-th characteristic family with λ replaced by
the i-th eigenvalue λi of the Jacobian matrix DF (u). In the scalar
case, for a convex flux, these formulations have been proven to be
equivalent (see Section 2.1 of [26]). The Lax admissibility condition
allows the selection of a particular weak solution.
Definition 3: A function u : [0, T ]×R 7→ R is the weak entropy
solution to the Cauchy problem (2)-(3) if it is a weak solution
(definition 2), that satisfies the Lax admissibility condition (8).
The definition of weak conditions to the IBVP requires a corresponding statement of weak boundary conditions, presented in the
following section.
2) Weak boundary conditions: The first statement of weak
boundary conditions was introduced in [6] in the scalar case
in multiple dimensions of space, with C 2 flux and C 2 initial
and boundary datum, using a vanishing viscosity method. In one
dimension, this formulation reads:
max sgn (u(t, a) − ua (t)) (f (u(t, a)) − f (k)) = 0
(9)
min sgn (u(t, b) − ub (t)) (f (u(t, b)) − f (k)) = 0
(10)
k∈[α,β]
A. Weak solutions to the initial-boundary value problem
0
Given that u is smooth around a jump discontinuity, integrating
the weak formulation (5) yields the Rankine-Hugoniot relation [17]
defining the speed σ of propagation of jump discontinuities
k∈[γ,δ]
for almost all t > 0, and where α = min(u(t, a), ua (t)),
β = max(u(t, a), ua (t)), γ = min(u(t, b), ub (t)), δ =
max(u(t, b), ub (t)), and sgn denotes the sign function. For the case
of systems of conservation laws, the interested reader is referred to
[9],[34]. In the scalar case, at a left boundary a, the corresponding
statement of weak boundary conditions derived from the structure
of the solution to the Riemann problem is the following.
Definition 4: A function u : Ω 7→ R satisfies the boundary
condition ua at a if for almost every time t (in the sense of Lebesgue
measure), the solution to the Riemann problem centered at a with
initial data
(
ua (t)
if x < a
(11)
u(t, a)
if x > a
3
either does not contain any wave (when left and right initial states
are the same), or contains waves with non positive speeds (a wave
with zero speed is allowed). Notice that for weak solutions we
consider the condition will hold for all times except a finite number.
B. Well-posedness of the initial-boundary value problem
In [6], a solution satisfying (5) in the scalar case is constructed
using a vanishing viscosity method for the weak boundary conditions statement (9)-(10) and is shown to be the admissible solution
according to Kruzkhov entropy condition [24].
More recently, an existence result for n × n systems using
wavefront tracking was proposed in [1]. The standard Riemann
semigroup (SRS) method, introduced in [10] for the Cauchy problem associated with a Temple system [37] of conservation laws,
was extended to the IBVP in [2], [3], with the boundary conditions
statement from definition 4. In [3], it is shown for a n × n system
that if the SRS exist, its trajectories coincide with wavefront tracking solutions. Uniqueness and continuous dependence is obtained
for the case of non-characteristic conditions, and uniqueness for
the characteristic case. The SRS is constructed for 2 × 2 system
in [2], and for the case of n × n system in [16]. The stability
of the IBVP with two boundaries was established via vanishing
viscosity for 2 × 2 systems in [35]. We state in the scalar case for
a static boundary the main result of [2] for characteristic boundary
conditions, obtained for 2 × 2 systems with continuous boundary
(see theorem C of [2]). A general result for n × n systems was
established in [4].
Theorem 1: [2] Let f be a smooth map such that the equation (1)
is strictly hyperbolic with characteristic field linearly degenerate or
genuinely nonlinear (i.e. f is linear, convex or concave). For every
δ > 0 there exists L > 0 and a continuous semigroup S defined
for data in L1 ∩ BV with total variation bounded by δ, such that
•
•
•
The map t 7→ u(t, ·) yields a weak solution to the IBVP (2)(3)-(4).
For piecewise constant initial and boundary data, the trajectories of the semigroup coincide with the solution to the IBVP
obtained by piecing together the standard solutions to the
Riemann problems at the points of discontinuity of the initial
condition and at the boundary.
For initial data u00 , u000 , boundary data u0a , u00a , u0b , u00b in L1 ∩
BV with total variation bounded by δ, let u0 , u00 denote the
corresponding trajectories of the semigroup S, and t0 , t00 > 0,
then
ku0 (t0 , ·)−u00 (t00 , ·)k1 ≤ L(|t0 −t00 |+ku00 −u000 k1 +ku0a −u00a k1
+ ku0b − u00b k1 ).
We also refer the interested reader to the work of Otto [30] for
the case where the entropy solution does not have traces at the
boundary, see also [33]. In the following section we use this result
in the case of a left and a right boundary to show that we can restrict
our Lyapunov analysis to the case of piecewise smooth data.
III. A PPROXIMATION OF SOLUTION TO INITIAL - BOUNDARY
VALUE PROBLEM BY PIECEWISE SMOOTH SOLUTION
In this section, we present results on the approximability of the
solution to an IBVP with initial condition in BV by the solution
to an IBVP with piecewise smooth solution at all times. We show
that the solution to the IBVP with BV data can be approximated
arbitrarily closely in the L1 norm by the solution to an IBVP with
piecewise smooth data. We define the required regularity class used
throughout the article.
Definition 5: We note PWS+ the class of piecewise smooth
functions f : R 7→ R such that
0
• f is positive (where defined).
0
• f has only downward jumps (i.e. f seen as measure has only
negative Dirac masses).
We now state the approximability result.
Theorem 2: Consider T > 0, a < b, and let u0 : (a, b) 7→ R,
ua , ub : (0, T ) 7→ R be functions with bounded total variation. For
every > 0, there exists u0 : (a, b) 7→ R in PWS+ and piecewise
constant boundary data ua , ub : (0, T ) 7→ R such that
kua (t) − ua (t)k1 ≤ ,
kub (t) − ub (t)k1 ≤ and the solution u to the IBVP for equation (1) and data (u0 , ua , ub )
and the solution u to the IBVP for equation (1) and data
(u0 , ua , ub ) satisfy:
∀ 0 ≤ t ≤ T, ku(t, ·) − u (t, ·)k1 ≤ .
Proof: In the compact domain [a, b], we can approximate
the initial condition condition u0 arbitrarily closely in the L1
sense by a piecewise constant function u0 with a finite number
of discontinuities and lower total variation. Also we can replace
the upward jumps of u0 with smooth increasing functions without
increasing the total variation. We can also approximate boundary
data by piecewise constant functions of lower variation.
Since u0 has only positive derivative, no new shock can form
in the solution inside the domain. Moreover, a finite number of
shocks will be introduced by the boundary conditions. Therefore
the solution u will remain in the class PWS+ .
Using the continuous dependence result of Theorem 1, the
resulting trajectories u, u of the semigroup can be made arbitrarily
close in the L1 norm by controlling the distance between the initial
conditions.
Under suitable boundary conditions, the solution to the IBVP with
piecewise smooth data is piecewise smooth:
Theorem 3: Let T, δ > 0, a < b, and let u0 : (a, b) 7→ R be
in PWS+ and ua , ub : (0, T ) 7→ R be piecewise constant. Let u
denote the solution to the IBVP (5)-(3)-(11). At all times 0 ≤ t ≤
T , u(t, ·) is piecewise smooth.
In the next section, we present the Lyapunov stability analysis for
functions in PWS+ .
IV. LYAPUNOV ANALYSIS
In this section, we propose a Lyapunov function and compute
its derivative. In the following we call ũ = u − u∗ where u∗
is a constant, hence stationary, solution around which we want to
stabilize the system, and u is the solution to the IBVP associated
with the scalar conservation law (1). Following the results from
section III, we assume that u is in PWS+ .
A. Lyapunov function candidate
We consider the classical Lyapunov function candidate [21], [23]:
Z
Z
1 b 2
1 b
V (t) =
ũ (x, t) dx =
(u(x, t) − u∗ )2 dx,
(12)
2 a
2 a
where u is a weak solution to the scalar conservation law. From
definition 3, we have t 7→ u(t, ·) continuous from [0, T ] to L1 ,
and the function V (·) is well-defined and continuous. We index the
jump discontinuities of u(t, ·) in increasing order of their location
at time t by i = 0, . . . , N (t), including for notational purposes the
boundaries a, b, with x0 (t) = a and xN (t) = b. The Lyapunov
function candidate can be rewritten as:
N (t)−1 Z x
i+1 (t)
1 X
V (t) =
ũ2 (t, x) dx.
(13)
2 i=0 xi (t)
4
From Theorem 3, we know that for all integer i ∈ [0, N (t)),
the function u(t, ·) is smooth in the domain (xi (t), xi+1 (t)), thus
∂t u(t, ·) exists and is continuous for t such that xi (t) < xi+1 (t).
Since discontinuity trajectories are differentiable with speed given
by the Rankine-Hugoniot relation (6), it follows that at any time t
such that N (t) is constant in a neighborhood of t and the boundary
trace is continuous, the function V (·) is differentiable.
B. Differentiation of the Lyapunov function candidate
In this section, we compute the derivative of the Lyapunov
function candidate (12), at any time t such that N (t) is constant in a
neighborhood and the boundary trace is continuous. Differentiating
expression (13) yields:
In equation (15) we gather the first four terms that depend on the
boundary trace of the solution, and the last two terms that depend
on the shock dynamics inside the domain. In the following section,
we analyze the stability properties of the internal terms.
C. Internal stability
The last two terms of equation (15) correspond to jump discontinuity in the solution and are neither observable nor controllable
from the boundaries. We now show that these terms have a
stabilizing effect on the Lyapunov function candidate (12).
Proposition 1: Given a constant solution u∗ to the scalar conservation law (1), we have the following inequality
N (t)−1 Z x
"
#
N (t)−1
i+1 (t)
X
ũ(t, xi −) + ũ(t, xi +)
dV
1 X
∗
∗
∗
∆i ũf (ũ + u ) − F (ũ + u ) −
∆i f (ũ + u ) ≤ 0
(t) =
∂t ũ2 dx
2
i=1
dt
2 i=0 xi (t)
(16)
N (t)−1 1 X
dxi
dxi+1
i.e. the jump discontinuity dynamics of the solution u to the
+
(t) − ũ2 (t, xi (t)+)
(t) .
ũ2 (t, xi+1 (t)−)
2 i=0
dt
dt
IBVP, contributes to the decrease of the Lyapunov function candi(14) date (12).
Proof: In order to show that the term (16) is negative, we
As detailed at the end of section IV-A, the term under the sum
show that each term in the sum is negative. If we note ul , ur the
is smooth, and we can write ∂t ũ2 = 2 ũ ∂t ũ. Since u satisfies the
value of u on the left and on the right of the jump discontinuity,
conservation law (1), we have ∂t ũ = −∂x f (ũ+u∗ ). The derivative
respectively, and ũl , ũr the corresponding reduced variables, we
of the Lyapunov function can be written as:
want to show that
N (t)−1 Z x
X
i+1 (t)
dV
(t) = −
ũ ∂x f (ũ + u∗ ) dx
dt
x
(t)
i
i=0
1
+
2
(ũr f (ũr + u∗ ) − F (ũr + u∗ ))
N (t)−1 dxi+1
dxi
ũ2 (t, xi+1 (t)−)
(t) − ũ2 (t, xi (t)+)
(t) .
dt
dt
X
i=0
Equivalently, in the original state variable u = ũ + u∗ , we have:
By integrating by parts the sum terms, we obtain:
N (t)−1
X
dV
x
(t)
(t) = −
[ũ f (ũ + u∗ )]xii+1
(t)
dt
i=0
N (t)−1 Z x
X
i+1 (t)
+
f (ũ + u∗ ) ∂x ũ dx
i=0
xi (t)
N (t)−1 +
1 X
2 i=0
ũ2 (t, xi+1 (t)−)
dxi+1
dxi
(t) − ũ2 (t, xi (t)+)
(t) .
dt
dt
and if we note F (·) a primitive function of the flux function f (·)
we have:
dV
(t) = ũ(t, a) f (ũ(t, a) + u∗ ) − ũ(t, b) f (ũ(t, b) + u∗ )
dt
− F (ũ(t, a) + u∗ ) + F (ũ(t, b) + u∗ )
N (t)−1 X
1 dxi
+
∆i (ũf (ũ + u∗ ) − F (ũ + u∗ )) −
(t) ∆i ũ2 ,
2 dt
i=1
where ∆i is defined around the discontinuity xi (t) as in equation (6). Using the Rankine-Hugoniot relation, defined in equation (6), to write the speed of the jump discontinuity dxi (t)/dt
as a function of the left and right jump values we obtain:
dV
(t) = ũ(t, a) f (ũ(t, a) + u∗ ) − ũ(t, b) f (ũ(t, b) + u∗ )
dt
− F (ũ(t, a) + u∗ ) + F (ũ(t, b) + u∗ )
N (t)−1
+
X
∆i (ũf (ũ + u∗ ) − F (ũ + u∗ ))
i=1
N (t)−1
X ũ(t, xi −) + ũ(t, xi +)
−
∆i f (ũ + u∗ ).
2
i=1
− (ũl f (ũl + u∗ ) − F (ũl + u∗ ))
ũl + ũr
−
(f (ũr + u∗ ) − f (ũl + u∗ )) ≤ 0.
2
(15)
[((ur − u∗ ) f (ur ) − F (ur )) − ((ul − u∗ ) f (ul ) − F (ul ))]
ul + ur − 2 u∗
−
(f (ur ) − f (ul )) ≤ 0.
2
This can be rewritten as:
1
(17)
F (ul ) − F (ur ) + (ur − ul ) (f (ur ) + f (ul )) ≤ 0,
2
which can be obtained from the Oleinik condition(7), here by
integration the left inequality of (7) between ul and ur . Thus
any solution satisfying the Oleinik entropy condition benefits from
stability of the jump discontinuity dynamics.
V. W ELL - POSED BOUNDARY STABILITY
In this section, we motivate and define the control space and
propose a stabilizing controller.
A. Control space
Definition 6: Let us denote smin , smax the minimal and maximal speed of the waves composing the solution to the Riemann
problem at the boundary. The control space at the left boundary is
the set of pairs (ul , ur ) such that either no wave is generated by
the Riemann problem (ul = ur ), or smin ≥ 0 and smax > 0. The
control space at the right boundary is the set of pairs (ul , ur ) such
that either no wave is generated by the Riemann problem (ul = ur ),
or smin < 0 and smax ≤ 0.
Proposition 2: Let m denote the minimizer of the strictly
convex flux function f . The control spaces Ca , Cb at the left and
right boundaries, respectively, are characterized as the set of pairs
(ul , ur ) such that one of the following properties is satisfied
5
summarized in the following formula:
.
Ca = (ul , ur )


ul = ur
ul ≥ m and

u ≥ m and
l
s.t.


ul = ur
ul ≤ m and

u ≥ m and
l
.
Cb = (ul , ur ) s.t.
ur ≥ m
ur ≤ m
and
ur ≤ m
ur ≤ m and
f (ul ) > f (ur ),
(18)
f (ul ) < f (ur ).
(19)
Using the characterization of the control space introduced in this
section, we show in the following section that the system is
stabilizable.
B. Lyapunov stabilization
In this section, we prove that there exist boundary conditions
in the control space (18)-(19) such that the candidate Lyapunov
function (12) is strictly decreasing.
Lemma 1: Let g : u 7→ (u − u∗ ) f (u) − F (u), with f a smooth
strictly convex function, and F a primitive of f . Let m denote
the minimum of f . The function g is smooth on the real line, and
satisfies the following properties:
g is strictly increasing in (−∞, min(m, u∗ )), strictly decreasing in (min(m, u∗ ), max(m, u∗ )), and strictly increasing in
(max(m, u∗ ), +∞).
• For u > v such that f (u) = f (v), we have g(u) > g(v).
Proof: The fact that g is smooth results from the smoothness
of f . The first property is obtained by computing the derivative
g 0 (u) = (u − u∗ ) f 0 (u) of g, and noting that f is strictly convex
with minimum at m. To prove the second property, let us consider
u > v such that f (u) = f (v). The difference g(v) − g(u) reads
g(v) − g(u) = F (u) − F (v) + (v − u) f (u) that is strictly negative
by strict convexity of f .
The function g is represented for the case of the Burgers flux
function in figure 1 with the arbitrary choice of g(m) = 0.
g(u)
g(u)
•
0
−1
u*
0
u
1
0
−1
0
u
u*
1
Fig. 1. Representation of the variations of g: for a Burgers flux function
f in the case u∗ < 0 (left) and in the case u∗ > 0 (right). The points
u = m (m = 0 in this case) and u = u∗ (u∗ = ±0.5 in this case) are
local extrema of g.
Theorem 4: Let V (·) denote the candidate Lyapunov function (12) for the PDE (1). There exist boundary conditions
ua (·), ub (·) in the control spaces (18) (19), respectively, such that
the following holds. If the corresponding solution to the IBVP is in
the class PWS+ then the function V (·) is strictly decreasing, thus
the solution is stable in the sense of Lyapunov.
Proof: The set of boundary values (ua (t), ub (t)) which
guarantee stabilization can be computed using Lemma 1 and are

[m, u(t, b)) × {u(t, b)}









(m, +∞) × (−∞, m)





s.t. g(ua (t)) < g(ub (t))






{u(t, a)} × {u∗ }




{u(t, a)} × {u(t, b)}














{u(t, a)} × {u∗ }



if
(u(t, a), u(t, b)) ∈ [m, +∞) × (m, +∞)
if
(u(t, a), u(t, b)) ∈ [m, +∞) × (−∞, m]
if
(u(t, a), u(t, b)) ∈ (−∞, m) × (−∞, m]
if (u(t, a), u(t, b)) ∈ (−∞, m) × (m, +∞)
and g(u(t, a)) < g(u(t, b))
if (u(t, a), u(t, b)) ∈ (−∞, m) × (m, +∞)
and g(u(t, a)) ≥ g(u(t, b))
(20)
We highlight that the boundary values (20) only provide stability,
and not asymptotic stability in general. In the following section we
instantiate a greedy controller which maximizes the instantaneous
decrease of the Lyapunov function, and also illustrate that the
greedy controller may fail to provide asymptotic stability.
In Section VII we then design an improved controller which
guarantees asymptotic stability and we show that the solution
resulting from these boundary controls and an initial condition in
the class PWS+ remains in PWS+ .
VI. M AXIMIZING INSTANTANEOUS LYAPUNOV FUNCTION
DECREASE RATE
In this section, we characterize the values of the control, in the
control space, that minimize the Lyapunov function derivative. A
greedy boundary control (ua (t), ub (t)) maximizing the instantaneous decrease of the Lyapunov function, read as follows (for the
case u∗ < m):

{m} × {u(t, b)}









{m} × {u∗ }








∗

{u(t, a)} × {u }
if (u(t, a), u(t, b)) ∈ [m, +∞) × (m, +∞)
if (u(t, a), u(t, b)) ∈ [m, +∞) × (−∞, m]
if
(u(t, a), u(t, b)) ∈ (−∞, m) × (−∞, m]



{u(t, a)} × {u(t, b)} if (u(t, a), u(t, b)) ∈ (−∞, m) × (m, +∞)




and g(u(t, b)) > g(u∗ )









{u(t, a)} × {u∗ }
if (u(t, a), u(t, b)) ∈ (−∞, m) × (m, +∞)



and g(u(t, b)) ≤ g(u∗ )
(21)
While the greedy controller (21) maximizes the instantaneous
decrease of the Lyapunov function, we illustrate in the following
example that asymptotic stability may not be obtained. We also
illustrate the naive control (ua (t) = u∗ , ub (t) = u∗ ) may
not provide asymptotic stability, and several key factors of the
mechanisms involved for both controllers.
Example 1: We first consider an example where the greedy
controller fails to provide asymptotic stability, and where the brute
force controller creates oscillations. Without loss of generality, we
choose u∗ < m. Given 0 < ∆ < (a+b)/2 and k > 0, we consider
the following initial datum on (a, b):
u0 (x) =

m





ū
 −k




ū + k

)
if x ∈ (a, a+b
2
if x ∈ ( a+b
+
(2 p) ∆,
2
p ∈ {0, · · · , b−a
− 1}
4∆
if x ∈ ( a+b
+
(2
p+
2
p ∈ {0, · · · , b−a
−
1}
4∆
a+b
2
1) ∆,
+ (2 p + 1) ∆)
a+b
2
+ (2 p + 2) ∆)
where u∗ < m < ū and f (ū) = f (u∗ ). This case corresponds to
the first row of equation (21), hence the applied boundary controls
6
is (ua (t) = m, ub (t) = u(t, b)). Since the characteristic speed
of m is zero, the right boundary value converges towards m but
never reaches it, hence the system remains in the configuration
characterized by the first row of equation (21), and converges to
the steady state m over the interval (a, b), not reaching the target
u∗ < m. Hence we have stability but not asymptotic stability.
For the same example, one may note that the brute force control
(ua (t) = u∗ , ub (t) = u∗ ) has no action on the system when
u(t, b) = ū + k since the control values are outside of the
control space. When u(t, b) = ū − k, the brute force control
induces slow backward moving shock waves (ū − k, u∗ ) from
the right boundary which interact with fast forward moving shock
waves (ū + k, ū − k) coming from the initial datum, and create
slow forward moving shock waves (ū + k, u∗ ), hence we observe
large oscillations at the right boundary (irrespective of the size
of the oscillations in the initial datum). Eventually all the waves
generating by oscillating initial datum exit the domain and the naive
control produces backward moving shock waves (m, u∗ ) which
yield convergence.
VII. LYAPUNOV ASYMPTOTIC STABILITY
In this section we design an improved controller which guarantees asymptotic stability and show that the associated solution to
the corresponding IBVP remains in the class PWS+ for initial data
u0 in the same class.
A. Controller design
The asymptotic convergence issue highlighted in the Example 1
stems from the fact that if the system reaches a configuration
corresponding to the first row of equation (21), given the prescribed
control values, it may remain in that configuration, which grants
stability but prevents asymptotic stability for u∗ < m. Without
loss of generality, we focus on the case u∗ < m. We define û > m
by f (û) = f (u∗ ), ǔ > m by g(ǔ) = g(u∗) and ū = û+ǔ
. We
2
propose the following values for the controller (ua (t), ub (t)):
ua (t) = min{u(t, a), m},
ub (t) = ψ(u(t, b)),
(22)
where ψ(α) = χ]−∞,ū] (α) u∗ + (1 − χ]−∞,ū] (α))α and χI is the
indicator function of the interval I. It is convenient to describe the
control according to different cases as for the greedy control, thus
we get:

{m} × {u(t, b)}







∗


{m} × {u }
if (u(t, a), u(t, b)) ∈ [m, +∞) × (ū, +∞)



{u(t, a)} × {u∗ }







{u(t, a)} × {u(t, b)}
if (u(t, a), u(t, b)) ∈ (−∞, m) × (−∞, ū]
if (u(t, a), u(t, b)) ∈ [m, +∞) × (−∞, ū]
if (u(t, a), u(t, b)) ∈ (−∞, m) × (ū, +∞)
(23)
B. Existence of solution for asymptotically stabilizing controller
To apply Theorem 4, we need to define boundary controls ua ,
ub , which will guarantee the solution to the corresponding IBVP
to remain in the class PWS+ for initial data u0 in the same class.
We prove such result for the boundary control providing asymptotic
stability of the Lyapunov function (12). For simplicity we assume
u∗ < m being the other case similar.
Theorem 5: Consider an initial datum u0 in PWS+ and the
boundary controls given by formula (21). Then the corresponding
IBVP admits a unique solution which is in the class PWS+ for all
times.
Proof: We will show that for u0 in PWS+ , the boundary
controls are well defined and piecewise smooth in time. Moreover
the solution remains in the class PWS+ .
First we enumerate the following cases:
• Case 1. u(t, a) ≥ m, u(t, b) > ū.
• Case 2. u(t, a) ≥ m, u(t, b) ≤ ū.
• Case 3. u(t, a) < m, u(t, b) ≤ ū.
• Case 4a. u(t, a) < m, u(t, b) > ū.
In Case 1. the boundary controls generate a rarefaction wave from
the left boundary, no wave on the right boundary and the traces
verify u(t+, a) = m and u(t+, b) = u(t, b) > m (where u(t+, ·)
indicates the limit at time t from the above).
In Case 2. the boundary controls generate a rarefaction wave from
the left boundary, a rarefaction wave or a shock on the right
boundary and the traces verify u(t+, a) = m and u(t+, b) = u∗ .
In Case 3. the boundary controls generate no wave from the left
boundary, a rarefaction wave or a shock on the right boundary and
the traces verify u(t+, a) = u(t, a) < m and u(t+, b) = u∗ .
In Case 4. the boundary controls generate no wave from the left
and right boundary and the traces verify u(t+, a) = u(t, a) < m
and u(t+, b) = u(t, b) > ū > m.
From this analysis we verify that the control is well defined for
constant traces and the corresponding solution remains in the class
PWS+ . It remains to verify that the solution is well defined and
remains in the same class for waves interacting with the boundary
and we proceed again by cases.
In case 1: if wave interact with the left boundary, then no wave is
produced, while for the right boundary a shock may arise and we
transition to case 2. In case 2: if wave interact with the left or right
boundary, then no wave is produced and we transition to case 1 or
3. In case 3: if wave interact with the left or right boundary, then
no wave is produced and we remain in case 3 or transition to 4.
In case 4: if wave interact with the left boundary, then no wave is
produced, while for the right boundary a shock may arise and we
transition to case 3.
C. Non-local controls
As we noticed the greedy control may not stabilize the system
to u∗ , while the brute force control ua ≡ ub ≡ u∗ may overshoot
(and not be in the control space) and produce oscillations. Finally,
control (23) stabilizes the system, but the stabilization time can
be far from optimal (and the same is for the brute force ones).
Therefore, in this Section, we show a nonlocal control unl
a,b which
fast stabilizes the system to u∗ . We use the term nonlocal to indicate
that this control will depend not only on the values of the traces
u(t, a) and u(t, b) but on the value of the whole solution u.
We focus again, for simplicity, on the case u∗ < m. Let A =
supx∈[a,b] u0 (x) and  < m be such that f (Â) = f (A). For every
U < Â we define:
T1 (U ) =
(b − a) (A − U )
,
f (U ) − f (A)
and set unl
a as in (23), while:
U
unl
b (t) =
u∗
T0 (U ) = T1 −
(b − a)
|f 0 (U )|
0 ≤ t ≤ T0
.
T0 < t < +∞
(24)
(25)
The meaning of this control is as follows. First we send a large
shock (u0 (a), U ) with negative speed to move the system in the
zone u < m and then apply the stabilizing control. Notice that
T1 is computed as the maximal time taken by the big shock
to cross the interval [a, b], while T0 is the time at which the
characteristic corresponding to u∗ should start from b to reach a
7
at time T1 . These choices will guarantee the desired effect. Notice
also that T0 is a safe choice, but smaller values may give a better
performance. In the following section we present numerical results
of the implementation of the boundary control proposed.
VIII. N UMERICAL EXAMPLES
Time = 0.5
2
u0 (x) = 1 + 0.5 sin(10 ∗ x).
1.5
1
0.5
Density
1
0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
Space
Time = 0.5
2
1
0.5
Density
Density
1
0.5
0
0.9
1
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0
-1
-1
-1.5
-1.5
-2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
Space
Time = 0.5
2
0.5
Space
Time = 2
2
1.5
1.5
1
1
0.5
0.5
Density
Density
0.8
-0.5
0
0
-0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
Space
Time = 0.5
2
0.5
Space
Time = 2
2
1.5
1.5
1
1
0.5
0.5
Density
Density
0.7
Time = 2
2
-2
0
-0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
Space
Time = 0.5
2
0.5
Space
Time = 2
2
1.5
1.5
1
1
0.5
0.5
Density
Density
0.6
1.5
-0.5
IX. C ONCLUSION
0.5
Space
1.5
In this article, we introduced a new technique for Lyapunov
boundary stabilization results of weak entropy solutions to scalar
conservation laws.
We proved that under suitable regularity of the initial and
boundary data, the solution to the initial-boundary value problem
could be considered to be piecewise regular with a finite number
of discontinuities. This allows the use of functional analysis tools
available for smooth functions.
We computed the derivative of the Lyapunov function candidate,
and showed that the solution to the scalar conservation law with
strictly convex flux is asymptotically stabilizable in the sense of
Lyapunov, using the definition of the Oleinik entropy condition.
We then showed how a greedy control, maximizing instantaneous
decrease of the Lyapunov function, may not asymptotically stabilize
the system. Therefore we designed a stabilizing control and a
nonlocal ones which improve the stabilization time.
0
-0.5
(26)
In figure 2 we present the evolution of the system under the greedy
boundary control, the brute force control, the stabilizing control
and the nonlocal ones with U = −2 and two different choices of
the switching time T0 , see (25). We notice that the greedy control,
as shown in Example 1, allows all oscillations to exit from right
boundary but the solution does not converge to the steady state u ≡
u∗ = −1 (see Figure 2, first row). On the other side the brute force
control does guarantee convergence to the steady state, but allows
strong oscillations on the right boundary (see Figure 2, second row,
first figure) and at time t = 2 the solution presents a large shock
connecting a positive state to the steady state u∗ but still at space
position x = 0.8 (see Figure 2, second row, second figure). The
stabilizing control also stabilizes and, moreover, avoids oscillations
(see Figure 2, third row, first figure). However, at time t = 2 the
solution presents a large shock connecting a positive state to the
steady state u∗ but still at space position x = 0.9 (see Figure 2, third
row, second figure). The nonlocal control guarantees convergence
and avoids oscillations. For the choice of switching time T0 = 1.5
(following (24)), at time t = 2 the solution presents a rarefaction
wave connecting u = −2 to u∗ and traveling backward (see Figure
2, fourth row, second figure). For the choice of switching time
T0 = 1, at time t = 2 the solution is already almost at the steady
state, with a little rarefaction still exiting the left boundary (see
Figure 2, fifth row, second figure). The downside of the nonlocal
control is the fact that a large shock is introduced in the solution,
which may be unfeasible for some specific application.
Time = 2
2
1.5
Density
In this section, we present numerical results obtained for
a benchmark scenario. The numerical scheme used is the standard Godunov scheme [19] with 100 cells in space and a time
discretization satisfying the tight Courant-Friedrich-Levy (CFL)
condition [27]. We consider the flux function u 7→ u2 /2, the
equilibrium state u∗ = −1, and the space domain [0, 1] with the
oscillating initial condition:
0
-0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
0.1
0.2
0.3
0.4
0.5
Space
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
Space
Fig. 2. Performance of the various controllers for an oscillating initial data.
Plots report solutions at time t=0.5 and at time t=2. First row: greedy control.
Second row: brute force control. Third row: stabilizing control. Fourth row:
nonlocal control with T0 = 1.5. Fifth row: nonlocal control with T0 = 1.
8
ACKNOWLEDGMENTS
The authors would like to thank Professor Miroslav Krstic for suggesting
to work on this problem during his visit at UC Berkeley and for fruitful
conversation leading to the genesis of this work. At this time the second
author was a visiting scholar at UC Berkeley with financial support from
CEMAGREF and from the France-Berkeley fund project. The authors would
also like to thank their colleague Professor Daniel Work from UIUC for
numerous fruitful discussions on a related topic.
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