Preprint di Matematica - n. 4 Maggio 2006
Dirichlet Problems for some
Hamilton-Jacobi Equations With
Inequality Constraints
Jean-Pierre Aubin
Alexandre M. Bayen
Patrick Saint-Pierre
SCUOLA NORMALE SUPERIORE
PISA
1
Dirichlet Problems for some Hamilton-Jacobi Equations With
Inequality Constraints
Jean-Pierre Aubin
1 2
, Alexandre M. Bayen
3
and Patrick Saint-Pierre
4
Abstract
We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for
a class of Hamilton-Jacobi equations. The hypograph of the “solution” is defined as the “capture basin”
under an auxiliary control system of a target associated with the initial and boundary conditions, viable
in an environment associated with the inequality constraint. From the tangential condition characterizing
capture basins, we prove that this solution is the unique “upper semicontinuous” solution to the Hamilton-
Jacobi-Bellman partial differential equation in the Barron/Jensen-Frankowska sense. We show how this
framework allows us to translate properties of capture basins into corresponding properties of the solutions
to this problem. For instance, this approach provides a representation formula of the solution which boils
down to the Lax-Hopf formula in the absence of constraints.
1
LASTRE (Laboratoire d’Applications des Syst`emes Tychastiques R´egul´es) 14, rue Domat, F-75005 Paris,
aubin.jp@gmail.com, http://lastre.asso.fr/aubin
2
Jean-Pierre Aubin thanks Giusepp e Da Prato for inviting him to the Scuola Normale di Pisa and acknowledges the
financial support provided through the European Community’s Human Potential Programme under contract HPRNCT-2002-
00281 (Evolution Equations for Deterministic and Stochastic Systems).
3
University of California at Berkeley, Department of Civil and Environmental Engineering. Davis Hall 711, UC Berkeley,
Berkeley, CA 94720-1710. bayen@ce.berkeley.edu
4
Universit´e Paris-Dauphine, D´epartement de Math´ematiques, CRVJC, Place du Mar´echal de Lattre de Tassig ny, 75775 Paris
Cedex 16, France
2
1 Introduction
1.1 Motivation
This article is motivated by macroscopic fluid models of highway traffic, following the pioneering work of
Lighthill, Whitham and Richards [61, 75]. In their original work, the authors modelled highway traffic flow
with a first order hyperbolic partial differential equation with concave flux function, called the Lighthill-
Whitham-Richards (partial differential) equation. This model is the seminal model for numerous highway
traffic flow studies available in the literature today [2, 42, 43, 60, 30, 82, 28]. It models the evolution of the
density of vehicles on a highway by a conservation law, in which the mathematical model of the flux function
inside the conservation law results from empirical measurements [57].
Solutions to such equations may have shocks (they are set-valued maps), which model abrupt changes in
vehicle density on the highway [2], and only model physical phenomena to a certain degree. Hence discontin-
uous selections of these solutions are looked after, for instance, the entropy solution [2] of Oleinik [70], which
is acknowledged to be the proper weak solution of this problem. There has been an extensive literature on
this problem, of which we single out the work of Bardos, Leroux and Nedelec [23, 80].
Very few results applicable to highway traffic are available for control of first order hyperbolic conservation
laws. Differential flatness [47] has been successfully applied to Burgers equation (and therefore to the
Lighthill-Whitham-Richards equation) in [72] order to avoid the formation of such shockwaves. This analysis
does not so far extend to the presence of shocks. Lyapunov based techniques have also bee n applied to the
Burgers equation [59]. Adjoint based methods have been successfully applied to networks of Lighthill-
Whitham-Richards equations in [54]; these results seem so far the most promising, but they do not have
guarantees to provide an optimal control policy. Questions of interest in controlling first-order partial
differential equations, and in particular, Lighthill-Whitham-Richards equations, are still open and difficult
to solve due to the presence of shocks occurring in the solutions of these partial differential equations.
In order to alleviate the technical difficulties resulting from shocks present in solution of the Lighthill-
Whitham-Richards equation, an alternate formulation consists in considering the cumulated number of ve-
hicles, widely used in the transportation literature as well [67, 68, 69]. The cumulative number of vehicles
can be thought of as a primitive of the density over space. Formally, the evolution of the cumulated number
N(t, x) of vehicles is the solution of an Hamilton-Jacobi (partial differential) equation of the form
∂N(t, x)
∂t
+ ψ
∂N(t, x)
∂x
= ψ(v(t))
where the flux function ψ appearing in this Hamilton-Jacobi equation is in fact concave as shown by the
empirically measured flux function of the Lighthill-Whitham-Richards equation [61, 75, 23, 80]. The function
v(·) will be regarded as a control of the Hamilton-Jacobi equation in forthcoming studies. It could for example
model the inflow of vehicles at the entrance of a stretch of highway. It is a given datum in this paper.
The solution of this Hamilton-Jacobi equation has no shocks, but is not necessarily differe ntiable. It is only
upp e r semicontinuous. Actually, the non differentiability of the cumulated number of vehicles is closely
related to the presence of the sho cks of the solution to the Lighthill-Whitham-Richards equation (see for
instance [36, 37, 38]).
Since the Lighthill-Whitham-Richards equation and the Hamilton-Jacobi equation model the same physical
phenomenon and since both formulations are equivalently used in the highway transportation literature, we
single out in this paper the study of the evolution of the cumulated number of vehicles for benefiting of the
extensive knowledge of Hamilton-Jacobi equations for which control and viability techniques can be applied.
3
1.2 Contributions of the paper
We shall revisit this Hamilton-Jacobi equation by answering new questions:
• introducing a nontrivial right hand side,
• involve Dirichlet conditions,
• and above all, impose inequality constraints on the solution, for instance, upper bounds on the cumu-
lated number of vehicles, depending on time and space variables.
For this purpose, we suggest to use a novel point of view based on the concept of capture basin of a target
viable in an environment extensively studied in the framework of viability theory: Given a closed subset of a
finite dimensional vector space regarded as an environment, a closed subset of this environment considered
as a target and a control system, the viable capture basin is the subset of initial states of the environment
from which starts at least one evolution governed by the control system viable in the environment until the
finite time when it reaches the target (see Definition 9.3, p.25). It happens that the hypograph of the solution
to the Hamilton-Jacobi equation satisfying initial and Dirichlet conditions as well as inequality constraints
is the capture basin of an auxiliary target (involving initial and boundary conditions) viable in an auxiliary
environment (involving inequality constraints) under an auxiliary control system (involving the flux function
of the Hamilton-Jacobi equation).
Hence, anticipating on this property, we define the viability hyposolution of the Dirichlet problem for such
an Hamilton-Jacobi equation with constraints from this property of being a viable capture basin (see Def-
inition 3.1, p.6). Then we proceed by translating properties of viable c apture basins (see [7] for instance)
in the language of partial differential equations for this particular case. We shall prove that the viability
hyposolution
1. is the unique generalized solution in the B arron/Jensen-Frankowska sense
5
(a weaker concept of
viscosity solution introduced by Crandall, Evans and Lions in [41, 40] for continuous solutions adapted
to the case when solution is only semicontinuous): Theorem 8.1, p.22,
2. is equivalently the unique upper semicontinuous solution in the contingent Frankowska sense: Theo-
rem 7.1, p.19,
3. satis fies the sup-linearity prop erty and depends “hypo-continuously” of the initial and Dirichlet con-
ditions,
4. is represented by the Lax-Hopf formula (see Theorem 4.1, p.8) in the absence of inequality constraints, a
more involved represe ntation formula (s ee Theorem 4.5, p.11) in the presence of inequality constraints,
upp e r estimates (maximum principe, see Proposition 4.3, p.10) and lower estimates (s ee Proposition 4.4,
p.10).
5
H´el`ene Frankowska proved that the epigraph of the value function of an optimal control problem—assumed to be only lower
semicontinuous—is semipermeable (i.e., invariant and backward viable) under a (natural) auxiliary system. Furthermore, when
it is continuous, she proved th at its epigraph is viable and its hypograph invariant [50, 51, 53]. By duality, she proved that
the latter property is equivalent to the fact that the value function is a viscosity solution of the associated Hamilton–Jacobi
equation in the sense of Crandall and Lions. See also [25, 21, 8] for more details. Such concepts have been extended to solutions
of systems of first-order partial differential equations without boundary conditions by H´el`ene Frankowska and one of the authors
(see [13, 14, 15, 16, 17, 18] a nd chapter 8 of [5]). See also [10, 11].
4
1.3 Outline of the paper
In order to make the paper more readable by postponing the technical difficulties, we chose to begin by
stating the problem and the main assumptions which will not be repeated all along the paper. We next define
the viability hyposolution to the non homogeneous Dirichlet/initial value problem for our class of Hamilton-
Jacobi equations under inequality constraint as the capture basin of a target summarizing the Dirichlet/initial
data viable in a target associated with inequality constraints. Then, we translate the properties of capture
basins to the viability hyposolution, starting with a general representation formula providing Lax-Hopf
formulas in the absence of inequality constraints. We next check that the viability hyposolution satisfies the
Dirichlet and initial conditions as well as the inequality constraints. The last three sections are devoted to the
proof that the viability hyposolution is a solution to the Hamilton-Jacobi partial differential equation in two
equivalent dual generalized sense by translating both the Viability Theorem and the Invariance Theorem
characterizing the capture basin in terms of either tangential conditions or normal conditions, as it was
done in a long series of papers by H´el`ene Frankowska. Using tangential conditions, we express the viability
hyposolution as a solution to the Hamilton-Jacobi partial differential equation couched in terms of contingent
hypoderivatives, whereas using normal conditions, we characterize it in terms of superdifferentials, as it was
done independently by Barron/Jensen and Frankowska, in the spirit of nonsmooth analysis and viscosity
solutions. The presence of inequality constraints complicates the technical formulation of the concept of
solution at points where the solution touches the constraint, above all in the sup erdifferentials formulation,
justifying the reason why we conclude this paper by this dual characterization. An appendix gathers some
definitions, notations and basic prerequisites of viability theory and c onvex analysis for the convenience of
readers who are not familiar with these topics.
2 Statement of the Problem
We set X := R
n
. Let us consider
1. A concave function ψ : X 7→ R satisfying growth conditions
∀ v ∈ X, β − σ
A
(v) ≤ ψ(v) ≤ δ − σ
A
(v)
for some compact convex subset A ⊂ X, where σ
A
(v) := sup
u∈A
hu, vi is the support function of A
and where β ≤ δ,
2. A bounded continuous function v : R
+
7→ Dom(ψ),
3. An upper semicontinuous initial datum N
0
: X 7→ R
+
. We set N
0
(0, x) := N
0
(x) and N
0
(t, x) := −∞
if t > 0.
4. A closed subset K ⊂ X with nonempty interior Int(K) =: Ω and boundary ∂K =: Γ,
5. An upper sem icontinuous boundary datum γ : R
+
× X 7→ R, satisfying
6
∀ x ∈ ∂K, N
0
(x) = γ(0, x) and ∀ t ≥ 0, ∀ x ∈ Int(K), γ(t, x) = −∞
6. A Lipschitz function b : R
+
× X 7→ R ∪ {−∞} setting the upper constraint.
6
This is not m and ator y. We can take any function such that Dom(γ) ⊂ K is strictly contained in K, an instance which may be
useful for defining “guards” in impulse or hybrid systems, for instance. Boundary conditions are obtained when Dom(γ) = ∂K.