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AIMS’ Journals
Volume X, Number 0X, XX 200X pp. X–XX
RECONSTRUCTION OF INDEPENDENT SUB-DOMAINS FOR A
CLASS OF HAMILTON JACOBI EQUATIONS AND ITS
APPLICATION TO PARALLEL COMPUTING
Adriano Festa
ENSTA ParisTech
828, Boulevard des Mar´echaux,
91120 Palaiseau, FRANCE
Abstract. A previous knowledge of the domains of dependence of an Hamil-
ton Jacobi equation can be useful in its study and approximation. Information
of this nature are, in general, difficult to obtain directly from the data of
the problem. In this paper we introduce formally the concept of independent
sub-domains discussing their main properties and we provide a constructive
implicit representation formula. Using such results we propose an algorithm
for the approximation of these sets that will be shown to be relevant in parallel
computing of the solution.
1. Introduction. A classic, powerful approach to optimal control problems con-
sists in solving a verification partial differential equation in the Hamilton Jacobi
form obtained using the Bellman’s Dynamic Programming principle. One remark-
able advantage of this approach, compared to the optimality conditions study, is
the ability to provide global minima and closed-loop optimal controls; on the other
hand, the study and the approximation of the value function of the problem which
is an unavoidable technical step, is often difficult. An exceptional achievement was
made with the introduction of viscosity solutions, a weak notion of solution pro-
posed by Crandall, Evans and Lions in the 80s, and the successive refinements (for
a whole presentation of this subject referring to the monographs [2,5]). A special
benefit of this approach is to move the attention from the study of the trajectories
of the problem to some geometric properties of the value function, opening in this
way a wide range of techniques that, in the following, have shown a good efficiency.
In this paper we will consider a problem related to this. It consists in the de-
tection of a collection of subsets, contained in the domain of the problem, where
the value function restricted to each sub-domain is independent on the value of any
other subset. This knowledge is useful for several reasons: it is related to stabi-
lization problems, reachability sets reconstruction, or, as we will show later in this
paper, in parallel procedures for a fast numerical resolution. This point is of special
interest. In fact it is well known as the greatest limitation to the use of the Bellman
approach in optimal control is due to the complexity of solving the Hamilton Jacobi
equation associated; especially in high dimensional context.
2010 Mathematics Subject Classification. Primary: 49L25, 65N55; Secondary: 49M27.
Key words and phrases. Hamilton Jacobi Equations, Viscosity Solutions, Numerical Approxi-
mation, Parallel computing, Domain Decomposition.
1
arXiv:1405.3521v2 [math.NA] 10 Nov 2014
2 ADRIANO FESTA
The “curse of dimensionality”, a meaningful definition proposed by the same
R. Bellman, has been attacked in the last twenty years from several directions
and various tools have been developed. Some examples of acceleration techniques
are Fast Marching [21,25] and Fast Sweeping methods [28]. The strategy of these
methods is to focus on the implicit order driven by the characteristics of the problem
to compute each node only once, reaching convergence to the discrete solution in
finite time. Despite the high efficiency of these techniques, born in the context
of the resolution of the Eikonal case (where the organisation of the nodes is more
evident), a generalization to more general equations is not trivial and still under
investigation. Some proposals can be found (Fast Matching methods) in [10,22,6]
and (generalized Fast Sweeping methods) [24].
Another possible strategy is to decompose the domain in a collection of subsets,
chosen in a number which is sufficient to lower the quantity of nodes to process
in every sub-problem. Meanwhile to solve in parallel on every sub-domain. The
difficulty in this idea is about conditions to impose on the interface between two
different subsets, or equivalently if some regions of overlapping are introduced, about
the manner to handle the computation. Moreover, a technique of that kind requires
an iterative process performed on every subdomain, with a consequent growth of the
total complexity. For a whole dissertation on the subject of domain decomposition
techniques (DD) we refer to the monograph [26], for Hamilton Jacobi equations
[8,23] and about a parallel version of the Fast Sweeping Methods [29,11].
An alternative approach was proposed in [30] where the authors, passing to a
quasi variational inequality formulation which is shown to be equivalent of the
original problem, can handle a decomposition of the domain.
A new direction of research was opened by Ancona and Bressan in [1] where they
introduced the original concept of patchy feedback with the intention of studying an
asymptotic stability problem. Navasca and Krener in [19] used these ideas to develop
a technique of reconstruction for the feedback solution in some special polynomial
cases (patchy solutions). Again, these elements inspired the work of Cacace et al.
[7]. They propose, in a special class of Hamilton-Jacobi equations, a preliminary
procedure called patchy decomposition. This preliminary computation supplies a
partition of the domain in sets which could be computed separately without any
exchange of information between the interfaces. The result is archived using the
multi-grid idea of pre-computing the problem on a coarse grid, solving the synthesis
problem to have an optimal feedback control, and using it to detect a decomposition
of the domain, accordingly with an approximation of the characteristics of the
problem. They show in practice that the error added in this procedure is sufficiently
small in some cases of interest. Our paper can be considered a development of this
idea. By using some recent results in decomposition techniques we will state in a
rigorous way the concept of independent sub-domain (different concept from patchy
subset) and we will show an easy property which permit us an implicit way to
reconstruct them without the delicate step about the feedback control (we recall
that, in general, there is no guarantee of convergence of an approximated optimal
feedback to the continuous control). We will be able to prove the convergence of the
technique and provide some error estimates. With this background we will enlarge
significantly, with respect to the tests performed in [7], the class of equations where
the decomposition is appropriate. Indeed, through the paper there will be discussed
the analogies and the differences with the patchy decomposition.
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 3
The paper is organized as it follows: in Section 2 we will introduce, functionally
to our purposes, the minimum property which is useful for the decomposition and
the concept of independent sub-domains. In Section 3 we propose an algorithm
for their location. Main result of this part is the necessary condition contained in
Proposition 3.1 which characterizes the points of the grid belonging to a certain
independent domain. Finally in Section 4 there will be discussed the application
of the previous results to propose a parallel algorithm for the approximation of the
solution. The main benefit of our proposal will be claimed in Proposition 4.1 where
the convergence of the technique and a bound for the error will be proved. Through
some test examples we will show the good features of the proposal.
2. Formulation and a decomposition property. In this section there are pre-
sented some basic definitions and features which are necessary for the comprehen-
sion of the paper. In particular the decomposition property presented in details in
[18,17] will be shortly recalled.
Let us first of all introduce the classical framework of an exit problem. We will
refer to the general structure of a differential game. A generic optimal control
problem can be viewed as sub-case.
Let the dynamics be given by
˙y(t) = f(y(t), a(t), b(t)), a.e.
y(0) = x,
where x∈Ω is an open subset of Rn,a∈ A := {a:R+→A, measureable}, and
b∈ B := {b:R+→B, measureable}with A, B compact sets of Rm. We take f
Lipschitz continuous with respect to the first variable and continuous with respect
to (x, a, b); this is enough to guarantee the existence of a solution yx(t, a(t), b(t))
which will be called trajectory.
The goal is to find the optimum (a sup −inf optimum) over A,Bof the functional
Jx(a, b) := Zτx(a,b)
0
l(yx(s, a(s), b(s)), a(s), b(s))e−λsds
+e−λτx(a,b)g(yx(τx(a, b))), λ ≥0,
where τis the time of the first exit from the set Ω defined as
τx(a, b) := min {t∈[0,+∞)|yx(t, a(t), b(t)) /∈Ω},
for the continuity of the trajectories yx(τx, a(τx), b(τx)) ∈Ω.
Typical hypothesis on the data, are stated: calling lthe running cost, and gthe
exit cost:
f: (Ω, A, B)→R,continuous function,
Lipschitz continuous in the first variable
l: (Ω, A, B)→(ρ, +∞],is a strictly positive continuous function,
Lipschitz continuous in the first variable,
g:¯
Ω→Ris a continuous function.
(H0)
Using the Elliot-Kalton’s notion [13] of non anticipating strategies, we define the
value function of this problem as
v(x) := sup
ϕ∈Φ
inf
a∈A Jx(a, ϕ(a)),(1)
4 ADRIANO FESTA
where
Φ := {ϕ:A→B:t > 0, a(s) = ˜a(s) for all s≤t
implies ϕ[a](s) = ϕ[˜a](s) for all s≤t}.
For a simpler presentation, we will assume the Isaacs’ conditions verified, then the
value function of the problem exists, is unique and coincides with v. It is well known
that such function is a viscosity solution of the problem
λv(x) + H(x, Dv (x)) = 0 x∈Ω
v(x) = g(x)x∈Γ(2)
where the Hamiltonian is defined as H(x, p) := minb∈B maxa∈A{−f(x, a, b)·p−
l(x, a, b)}, and the n−1 dimensional set Γ ∈∂Ω. To avoid a large number of
technicalities and focus on our purposes, we will state as hypothesis:
the problem (2) has an unique Lipschitz continuous viscosity solution v(x).(H1)
This assumption will be essential in the following; conditions to ensure such reg-
ularity of the solution have been largely discussed in literature (just to cite some
monographs [2,5,9]).
A key property of the value function that we will use in the following is the
possibility to solve a collection of Hamilton-Jacobi equations obtaining the original
solution as the point-wise minimum of such family. This property was discussed in
the work [18]; here we report the result and the main points of the proof for the
benefit of the reader.
Consider a decomposition of the set Γ as a union of a collection of subsets, i.e.
Γ := Si∈I Γi, with I:= {1, ...m} ⊂ N. We call vi:¯
Ω→Ra Lipschitz continuous
viscosity solution of the problem
λvi(x) + H(x, Dvi(x)) = 0 x∈Ω
vi(x) = gi(x)x∈Γ(3)
where gi: Γ →Ris a regular function such that
gi(x) = g(x),if x∈Γi,
gi(x)> g(x),otherwise.(4)
Also in this case, we will ask the existence of a Lipschitz continuous solution of
every equation (3).
The limiting superdifferential ∂Lv(x) of the continuous function v(·) at xis de-
fined as:
∂Lv(x) := {p| ∃ sequences pi→pand xi→xs.t. pi∈D+v(xi) for each i},
where D+v(x) is the usual Fr´echet superdifferential.
The active indexes set is stated as
I(x) = {j∈ {1, . . . , m} | vj(x) = min
i∈I vi(x)},for each x∈Ω.
We are now ready to recall the decomposition result:
Theorem 2.1. Let be verified (H0)-(H1) and the Isaacs’ conditions. Define the set
Υ⊂Ω as Υ := {x∈Ω|Card(I(x)) >1}(where Card(A) is the cardinality of the
set A) and the function ¯v: Ω →Ras
v(x) := min
i∈I vi(x).
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 5
Under the hypothesis
λ¯v(x) + H(x, X
i∈I(x)
αipi)≤0,(H2)
where pi∈∂Lvi(x) for each i∈I(x), x∈Υ, and any convex combination {αi|i∈
I(x)}, we have that ¯vis the unique viscosity solution of the problem (2).
Proof. We know that valways verifies the boundary conditions from the definition
of value function and (4). We show now that vis both subsolution and supersolution
in Ω of the problem (2). For uniqueness the thesis follows.
The proof of ¯v(x) supersolution is classic in literature. The property of subsolu-
tion is less trivial. If x /∈Υ, i.e. I(x) contains a single index value j, the property is
directly verified; so x∈Υ. Now, vj(·) is Lipschitz continuous on a neighbourhood
of xfor each j∈I(x). Since p∈D+¯v(x), it is certainly the case that p∈∂L¯v(x).
Using the property that ¯v(x0) coincides with max{vj(x0)|j∈I(x0)}for x0in some
neighbourhood of x, we deduce from the max rule for limiting subdifferentials of
Lipschitz continuous functions (see, e.g. [27]) applied to −¯v(·) the following repre-
sentation for p:
p=X
j∈I(x)
αjpj,
for some convex combination {αj|j∈I(x)}and vectors pj∈∂Lvj(x), j∈I(x).
But then, by (H2),
λ¯v(x) + H(x, p) = λ¯v(x) + H
x, X
j∈I(x)
αjpj
≤0.
This shows that u(x) is a subsolution and concludes the proof.
Remark 2.1. It is quite direct to show that the request (H2) is always verified
with the presence of a convex Hamiltonian. As consequence any optimal control
problem is included in our framework (in a optimal control problem the Hamiltonian
associated is always convex). To pass to this special case, it is sufficient to restrict
the set Bto a singleton.
Let us now define the concept of independent sub-domains.
Definition 2.1. A subset Σ ⊆Ω is an independent sub-domain of the problem (1)
if, given a point x∈Σ and an optimal control (a(t),¯ϕ(a(t)) (i.e. Jx(a, ¯ϕ(¯a)) ≤
Jx(a, ¯ϕ(a)) for every choice of a∈ A, and Jx(a, ¯ϕ(¯a)) ≥Jx(¯a, ϕ(¯a)) for any ϕ∈Φ),
the trajectory yx(¯a(t),¯ϕ(¯a(t))) ∈Σ for t∈[0, τx(¯a, ¯ϕ(¯a))].
It is possible to establish a link between the decomposition result and the concept
of independent sub-domain. In particular we show that Theorem 2.1 provides a
constructive way to build a independent sub-domains decomposition of Ω.
Proposition 2.1. Let be verified (H0),(H1),(H2)and the Isaacs’ conditions.
Given a collection of n−1dimensional subsets {Γi}i=1,...,m such that Γ = ∪m
i=1Γi,
the sets defined as
Σi:= x∈Ω|vi(x) = v(x), i = 1, ..., m, (5)
where viand vare defined accordingly to Theorem (2.1), are independent sub-
domains of the problem (1).
6 ADRIANO FESTA
Proof. The proof obtained for contradiction, is made using the Dynamical Program-
ming Principle (cf. [2]).
For a fixed iconsider a point x∈Σi. Let us then assume the trajectory, for an
optimal control (¯a, ¯ϕ(¯a)) for the original problem, yx(a(t),¯ϕ(a)) = x /∈Σifor a
certain t∈[0, τx(a(t),¯ϕ(a))]. If t= 0 the contradiction comes directly from the
definition of Σi. If t > 0 we recall (Dynamical Programming Principle)
v(x) = sup
ϕ∈Φ
inf
a∈A (Zt
0
l(yx(a(s), ϕ(a(s))), a(s), ϕ(a(s)))e−λsds
+e−λtvyx(a(t), ϕ(a(¯
t)))o,
an analogue formula is obviously valid also for vi(x). Recalling vi(x)> v(x),
vi(x) =
sup
ϕ∈Φ
inf
a∈A (Zt
0
l(yx(a(s), ϕ(a(s))), a(s), ϕ(a(s)))e−λsds +e−λt viyx(a(t), ϕ(a(¯
t)))
>inf
a∈A (Zt
0
l(yx(a(s),¯ϕ(a(s))), a(s),¯ϕ(a(s)))e−λs ds +e−λtvi(yx(a(s),¯ϕ(a(s))))
=Zt
0
l(yx(¯a(s),¯ϕ(¯a(s))),¯a(s),¯ϕ(¯a(s)))e−λsds +e−λt vi(¯x)
≥(Zt
0
l(yx(¯a(s),¯ϕ(¯a(s))),¯a(s))e−λs ds +e−λt v(¯x))=v(x),
then vi(x)> v(x), which contradicts again the definition (5).
That property of the trajectories will play an important role in the following; it
will guarantee the absence of crossing information through the boundary of every
independent sub-domain, or using different words, the solution of the problem (2)
in each sub-domain will not depend on the solution in other sub-domains.
A feature easy to derive from Proposition 2.1 is the connexion of the sets:
Corollary 2.1. Let be verified (H0),(H1)and (H2)and the Isaacs’ conditions. If
Γiis connected, the respective set Σidefined in (5)is also connected.
Proof. If Γiis connected we can always join two points x,yof the set using the
respective optimal trajectories, which, for Proposition 2.1 are contained in the set.
Let us to give a simple example of an independent sub-domains decomposition:
Example 2.1. The equation considered is, with Ω := (−1,1) ×(−1,1),
maxa∈B(0,1){a·Dv(x)}= 1 x∈Ω
v(x)=0 x1∈∂Ω.
Stated ∂Ω = Γ := ∪iΓi=∪[(±1,±1),(±1,±1)], we associate to the Γ1:= [(−1,−1),(−1,1)]
the function g1:∂Ω→Rdefined as
g1(x) := 0 x∈Γ1
g1(x) := γ(1 + x2)x∈Γ\Γ1,
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 7
Figure 1. Example 2.1, the auxiliary solution u1and the inde-
pendent sub-domains decomposition.
for a chosen γ∈R+. The other gis will be defined in a symmetric manner. It is
possible to verify that the unique viscosity solution of such a problem is
v1(x) = (1 + γ)−max(|x1−γ|,|x2|).
Finally the original value function v(x)=1−max(|x1|,|x2|) is recovered as v(x) =
min
i=1,...,4vi(x). The decomposition in independent sub-domains obtained are shown
in Figure 2.
3. Independent sub-domains reconstruction. In this section we will introduce
a numerical technique for the approximation of the independent sub-domains based
on the results of the previous section. The technique is not related to a special
numerical scheme, but it needs an a priori bound for the approximation; this is
necessary to have an important property of inclusion of the sets which will play
a role in the successive error analysis. As example of numerical scheme we will
make reference to a semiLagrangian solver, but the procedure can be easily ex-
tended to finite difference schemes, finite volumes etc. For further details about
semiLagrangian techniques we refer to the monograph by Falcone and Ferretti [16].
Let us consider a structured grid of Ω made by a family of simplices Sj, such
that Ω ∈ ∪jSj. Name xi,i= 1, ..., N the nodes of the triangulation,
∆x:= max
jdiam(Sj) (6)
the size of the mesh (diam(S) is the diameter of the set S). Let be Gthe set of the
internal nodes of the grid and, consequently, ∂G is the set of its boundary points;
in the case of a bounded Ω we call also Φ the nodes corresponding to the set Rn\Ω,
those nodes typically act as ghost nodes. We remark that this discretization space
includes the classical case of regular meshes.
We map all the values at the nodes in V= (V(1), ..., V (N)). By a standard
semiLagrangian discretization [3,16] of (2), it is possible to obtain the following
scheme in fixed point form
V=T(V),(7)
8 ADRIANO FESTA
where F:RN→RNis defined component-wise by
[T(V)]i=
max
b∈Bmin
a∈An1
1+λh I[V](xi−hf(xi, a, b)) −hl(xi, a, b)oxi∈G,
g(xi)xi∈∂G,
+∞xi∈Φ.
The discrete value function Vis extended on the whole space Ω by a linear
n−dimensional interpolation, represented by the operator I, as described in [15,4].
The variable hcorresponds to a fictitious time discretization with the purpose to
imitate the behaviour of the characteristics of the problem. The minimum over A
and the maximum over Bis evaluated by direct comparison using a discrete version
of the control space A, B. Generally the fixed point of the equation (7) is found
through the iterative map Vn+1 := T(Vn) which is shown to be a contraction.
It is important to recall the following result of convergence for the semiLagrangian
scheme. The proof can be found in [20,4] for the case of differential games and in
[12,14] for optimal control problems.
Theorem 3.1. Let vand Vbe the solutions of, respectively, equation (2)and (7).
Assume verified (H0)and (H1)then
||v−V||∞≤C(∆x)q,
where Cis a positive constant independent from ∆x,q∈R+depending on the
regularity of the problem.
For differential games with a Lipschitz continuous solution, a possible estimate
is
kv−Vk∞≤Ch 1
2 1 + ∆x
h2!.
If the quantity ∆x
h= 1, we have the relation described in Theorem 3.1 with q= 1/2.
The constant Cdepends on the data of the problem and can be estimated.
In the case of an optimal control problem with λ > 0, a possible convergence
bound is the following
kv−Vk∞≤2(Mv+Mvh)h1
2+Ll
λ(λ−Lf)
∆x
h
with Mv, Mvhmaxima of the absolute value of the continuous and semidiscrete
solution and Ll, LfLipschitz constants of dynamic and running cost. Then, in this
case, for h2= ∆x3,C= 2(Mv+Mvh) + Ll
λ(λ−Lf)and q=1
3.
Other examples of error estimates can be found in literature, even of high order
(i.e. q > 1) in some smooth cases [16].
Using the numerical scheme described above we can obtain an approximation of
the solution of every decomposed problem (3); these discrete solutions are called,
in analogy with the continuous case, Vifor i∈ I.
A simple observation brings us to the following Lemma:
Lemma 3.1. Let be verified (H0),(H1),(H2)and the Isaacs’ conditions. If a
node xj∈Σithen there exists a C > 0independent from ∆xand a q∈R+s.t.
|Vi(j)−v(xj)| ≤ C(∆x)q. The parameters Cand qare the same than in Theorem
3.1.
Proof. It is sufficient observe that |Vi(j)−v(xj)| ≤ |Vi(j)−vi(xj)|+|vi(xj)−v(xj)|.
Proposition 2.1 and Theorem 3.1 give the estimate.
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 9
We can establish a necessary condition for the nodes of Gto belong to a fixed
independent sub-domain Σi. Let be B(x, ρ) the n−dimensional ball centred in x
and of radius ρ.
Proposition 3.1. Assume (H0),(H1),(H2)and the Isaacs’ conditions. Let be
xj∈Gsuch that, taken an ∈[0,∆x)and a direction d∈B(0,1), the point
x=xj−d ∈Ωverifies vi(x) = v(x)for a certain i∈ I. Then the following
estimate holds
|Vi(xj)−V(xj)| ≤ 2(C(∆x)q+M∆x) (8)
Cas in the previous statement and M:= max{Lvi, i ∈ I } where Lviis the Lipschitz
constant of the function vi.
Proof. It is sufficient to observe that
|Vi(xj)−V(xj)| ≤ |Vi(xj)−vi(xj)|+|vi(xj)−vi(x)|+|vi(x)−v(x)|
+|v(x)−v(xj)|+|v(xj)−V(xj)| ≤ kVi(xj)−vi(xj)k∞+kv(xj)−V(xj)k∞
+|vi(xj)−vi(x)|+|vi(x)−v(x)|+|v(x)−v(xj)| ≤ 2C(∆x)q+2M∆x+|vi(x)−v(x)|
since vand viare Lipschitz continuous with Lipschitz constant bounded by M.
From vi(x) = v(x), (8) follows.
The previous condition gives us a necessary condition for the nodes to belong to
a independent subset Ωi. Note that such condition is verified by the nodes lying in
the interior of such set but also by a neighbourhood of the boundary, of thickness
depending by the parameters Cand M. This criteria will be used in the invariant
sub-domains reconstruction algorithm; the list of the nodes of Gbelonging to the
approximation of the independent sub-domain Σiwill be denoted Σi. Consequently,
the relative approximated set will be the region delimited by Σi(concave hull).
Let us define union(X1, X2) the vector composed by all the elements present in
X1and X2and let us call XΓthe vector containing all the indices of the nodes in
Gcorresponding to Γ. We call also V(j) the j−component of the vector V.
Indipendent Sub-Domains Reconstruction Algorithm (RA).
- Given a grid Gof discretization step ∆xand a collection of vectors such that
union(Xi, i = 1, ..., m) = XΓ.
1) (Resolution of auxiliary problems)
for i= 1...m solve iteratively the problem
Vi=Ti(Vi) with Tidefined as (3) with ∂G := Xi.
end
2) (Check and reconstruction of the value function)
If necessary, check numerically (H2),
then obtain Vas V:= mini=1...m{Vi}.
3) (Reconstruction of the sub-domains)
for i=1...m
initialize Σi=Xi
for j=1...N
if |Vi(j)−V(j)| ≤ 2(C(∆x)q+M∆x)
then add xjto vector Σi
end
the i−subset is ¯
Σi.
10 ADRIANO FESTA
end
Let us underline that, from the computational point of view, the difficult step is
only the first one; successive points are faster and with a negligible complexity. In
addition, point 1) is easily performed in parallel, since it consists in a collection of
independent problems, reducing the difficulty of resolution.
Remark 3.1. A delicate phase of the algorithm is the choice of the parameters
Cand M. An error in this point would produce two opposite behaviours. Bigger
parameters will bring to round up the independent sub-domains that could even
include the whole domain, making worthless the technique. A wrong choice of Cor
M, smaller than the correct one, would nullify our following error analysis. In the
test section (Section 4.1) we will show as even a not so tight choice of the parameters
will produce acceptable approximations of the desired sets in most of situations of
interest.
Remark 3.2. It is worth to stress an issue about the stopping criterion used in
the iterative resolution (7) contained in step 1). It is clear that, in general, the
exact discrete solution will not be reached, then the stopping criterion used should
be compatible with our requests of accuracy. For the case of the semiLagrangian
approximation, for a λ > 0, the classical estimate kVn−Vn+1k∞≤1
1+λh kVn−1−
Vn−2k∞brings us a link between the two successive iterations and the distance (in
the L∞norm) from the discrete solution as
kVn−Vk∞≤
∞
X
t=n1
1 + λ∆xt
kVn−Vn+1k∞
then a possible stopping criterion compatible is
kVn+1 −Vnk∞≤, = 2λ∆x(1 + λ∆x)n−1(C(∆x)q+M∆x).
The RA builds an approximation of the independent sub-domains. It is guaran-
teed that such approximation is performed exceeding the desired set, in the sense
that Σi⊆Σi. Another important property coming from the Proposition 3.1 is that,
for two discretization steps ∆x1and ∆x2such that ∆x1≥∆x2, the approximate
independent sub-domains of a same decomposition of Γ have the feature
Σ∆x1
i⊇Σ∆x2
i(9)
where with Σ∆x
iwe intend the discrete independent set obtained performing the
RA with discretization space step ∆x.
A point to discuss is the relation with the decomposition technique proposed in
[7]. Despite the analogies, in particular the idea of finding a decomposition in subdo-
mains which preserves a certain mutual independence, in general the decomposition
obtained can be slightly different. Let us show it with an example.
Example 3.1. Let us consider the domain Ω := (−1,1) ×Rthe dynamics
f(x, a) := a1, λ := δ, a =B2(0,1),
running cost l(x, a)≡1, and the set Γ := ∪2
i=1Γi, with Γ1:= {x1= 1}, Γ2:= {x1=
−1}. Let us impose g: Γ →Rnull in Γ1and Γ2. It is possible to check that the
solution is
v(x) = (1−eδx1
eδfor x1≤0
−1 + eδ x1
eδotherwise.
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 11
Figure 2. Example 3.1, the flatness of the central region depends
on δ, in tones of grey the incorrect division coming from numerical
incertitude.
We can notice how for every choice of δthe invariant domains relative to the two
subtargets are respectively {x∈Ω|x1≥0}and {x∈Ω|x1≤0}.
For δsufficiently small the numerics, although reconstructing correctly according
to a certain error bound the approximate value function, will not be able to solve
appropriately the synthesis problem. The assignment of the approximated optimal
control of the region “too flat” will depend on the priority chosen in the comput-
ing. So using the patchy decomposition, the sub-domains reconstructed will be, for
example {x∈Ω|x1+r
δ≥0}and {x∈Ω|x1+r
δ≤0},r∈(0, c), for a fixed cde-
pending on the computing parameters, which are arbitrarily different (for a generic
δ) from the correct division. Differently, our implicit reconstruction will produce
{x∈Ω|x1+c
δ≥0}and {x∈Ω|x1−c
δ≤0}which are larger sets containing the
correct decomposition.
4. Application to parallel computing. In this section we show as the results
about the reconstruction of the independent sub-domains can be used to compute in
parallel the correct solution of the discrete problem (7). We prove the convergence
of the technique, and provide a bound for the numerical error. Roughly speaking
the proposal is based on the reconstruction of a collection of independent subsets,
computed in parallel on a coarse grid, and successively the computation of the
solution on every sub-domain on a fine grid, recovering at the end the result, using
the minimum property on the regions of overlapping.
Let us state more rigorously the technique.
Consider two families of simplices: a coarse grid Kof discretization step ∆xKand
afine grid Gof step ∆xGwhich both cover the domain Ω, (i.e. Ω ⊆ ∪jSj⊆ ∪jKj),
call zk,k= 1, ..., N1, the nodes of the first triangulation and xk,k= 1, ..., N2the
nodes of the second triangulation. Triangulations are chosen such that N1N2.
The set of the nodes of Kcorresponding to Γ will be called ZΓ.
The parallel invariant sub-domains based algorithm is the following:
Independent-Sets Algorithm (ISA).
- Given a grid Kand a collection of vectors Zisuch that
union(Zi, i = 1, ..., M ) = ZΓ
12 ADRIANO FESTA
1) (Reconstruction of the approximated independent sub-domains)
Using RA get a collection Σ∆xk
i,i= 1, ..., M subsets of the grid K.
2) (Projection on the fine grid)
Project Σ∆xK
ion the grid Ggetting ΣG
ifor i= 1, ..., M ,
3) (Resolution on the fine grid)
for i= 1, ..., M solve iteratively the problem on ΣG
i
Vi=T(Vi) with Tdefined as (3)
end
4) (Assembly of the final Solution)
for j= 1, ..., N2
V(j) = min{Vi(j)|xj∈ΣG
i}
end
Some observations about the algorithm described above:
•computing of the RA at point 1) it is not more expansive than a single com-
putation on the coarse grid. RA is an algorithm which can work in parallel,
and the number of threads that it needs, are the same requested at point 4).
•The projection is very easy if the grid are chosen to be partially superimposed
i.e. every point zi∈Kis also a point of the fine grid G; in every case the
condition to impose is
xj∈ΣG
i⇐⇒ xj∈Con({zj|j∈Σ∆xK
i});
where Con(·) is the concave hull of the set, i.e. the union of the simplexes
with vertexes in the set. The computational cost of this passage is negligible.
•It is evident from definitions and from (9) that
Σi⊆Σ∆xG
i⊆ΣG
i≡Σ∆xK
i.(10)
This last observation will guarantee a delicate point about the convergence of the
method, as we show in the following proposition:
Proposition 4.1. Assume (H0),(H1),(H2)and the Isaacs’ conditions. Called V
the exact discrete solution of the ISA algorithm (i.e. all Vi=Ti(Vi)are verified
exactly) and Vthe exact solution of (7)(i.e. V=T(V)is exactly verified), then
there exists a C > 0and a q∈R+independent from ∆Gsuch that
kV−vk∞≤C(∆xG)q
holds. The parameters Cand qare the same than in Theorem 3.1.
Proof. For the observation (10) we know that the independent sub-domains even-
tually obtained on the fine grid, should be subsets of the sub-domains we used in
the algorithm.
Let us take a xj∈G, through Proposition 3.1 and (10) it is assured that there
exists at least one index i∈ {1, ..., M }such that v(xj) = vi(xj) (solution of (3)),
and |V(xj)−Vi(xj)|= 0. Then, using Lemma 3.1
|V(xj)−v(xj)|≤|V(xj)−Vi(xj)|+|Vi(xj)−v(xj)|≤kVi−vik∞≤C(∆xG)q
for the arbitrariness of the choice of xjwe have the thesis.
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 13
Figure 3. Distance function: exact decomposition and two (of the
four) approximated independent subsets found with a coarse grid of
152points (the third tone of grey in the centre is the superposition
area between the two sets).
4.1. Some examples. In this section we will give some examples of problems
solved with ISA. The aim of this section is purely descriptive, and our intention
is to give an experimental confirmation to the results stated previously. We will
show practically that the procedure of the independent domain approximation RA
is computational cheap and does not add an excessive number of nodes, even when
the coarse grid Kconsists of a small number of control points. Together we will
verify that our technique does not add a numerical error with respect to the solution
found on the whole domain and we will briefly compare the performances of our
proposal to the literature. For a rigorous and more detailed comparison we postpone
to future works more computationally oriented.
Let us first of all recall the discrete analogue of the L∞,L1norms for a vector
Xof Nelements:
kXk∆∞:= max
j=1,...,N |X(j)|,kXk∆1:= 1
N
N
X
j=1
|X(j)|.
Example 4.1 (Distance function).Let us to start with the very easy case shown
in Example 2.1, which will be useful to observe some general features. Therefore
we consider the Eikonal equation on the set Ω := (−1,1)2with the boundary value
fixed to zero on Γ := ∂Ω. This equation models the distance from the boundary of
such set. The case considered here will be with λ= 1, it will correspond to a non
linear monotone scaling of the solution (this relation is classically shown through
the Kruzkov transform see [2]), so the correct viscosity solution is the function
v(x)=1−min{e|x1|, e|x2|}
e,
solving the equation
v(x) + max
a∈B(0,1){a·Dv(x)}= 1.
In the following we will refer to a uniform decomposition of the set Γ, for example
in a 2-treads decomposition Γ1:= [−1,1] × {−1} ∪ −1×[−1,1], Γ2:= [−1,1] ×
14 ADRIANO FESTA
Table 1. Distance function: comparison of the accuracy of the
decomposition with various discretization steps
N. of variables ∆xKTime elapsed maxi|Σi|/|Ω|maxi|Σi|/|Ω|
520.4 1 ·10−350%
720.28 2 ·10−343%
1020.2 4 ·10−338%
1520.133 2 ·10−235% 25%
2020.1 5 ·10−233%
3020.06 1.01 30%
4020.05 3 29%
5020.04 11 28.3%
Table 2. Distance function: comparison between the efficiency of
the various methods (ND no decomposition, DD domain decompo-
sition, ISA Independent set decomposition)
N. of variables ∆xKTime ND Time (it) DD Time ISA
2520.08 0.13 0.06(2) 0.035
5020.04 7.02 1.2(2) 0.68
7520.026 57.5 12.2(3) 4.8
10020.02 1.5·10365.3(3) 16.6
20020.01 1.9·1051.2·104(5) 3 ·103
30020.006 >1061.8·105(11) 4.6·104
Table 3. Distance function: approximation error Error in norm
∆∞(and ∆1)
50210022002
original 1.2·10−2(1.1·10−2) 6.5·10−3(3.6·10−3) 2.5·10−3(1.6·10−3)
2-subsets 1.2·10−2(7.2·10−3) 6.5·10−3(3.7·10−3) 2.5·10−3(1.4·10−3)
4-subsets 9·10−3(7.2·10−3) 4.6·10−3(3.6·10−3) 1.4·10−3(1.3·10−3)
8-subsets 9·10−3(7.2·10−3) 4.6·10−3(3.6·10−3) 1.4·10−3(1.3·10−3)
{1}∪{1} × [−1,1], in a 4-treads Γ1:= [−1,1] × {−1}, Γ2:= {1} × [−1,1], Γ3:=
[−1,1] × {1}, Γ2:= {−1} × [−1,1]; etc.
In this case it is easy also to give an estimation of the constants introduced above,
M= 1, C= 1 and = 10−3,q= 1/2. It is evident the fact that the precision of the
independent subdomains reconstruction will be affected by the discretization step
used in the procedure. In Table 1there is a comparison, in the case of a 4-subsets
decomposition, of such accuracy. The percentage reported is the maximal extension
of an approximated subset Σion the total area of Ω. Evidently in every case, the
exact decomposition will be contained in the approximated one. It is worth to
underline how, even a very coarse grid (with 102or 152elements) the technique
is able to provide a sufficiently accurate estimate, giving a good reduction of the
dimension of the sub problems with a cost of the precomputing section absolutely
negligible. In Figure 3is reported the exact decomposition and two approximation
sets Σ1, Σ3with ∆x= 0.2.
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 15
After the decomposition, the problem can be solved separately, and possibly at
the same time on each Σi. It is consequential a large gain in term of computational
cost, compared to the resolution on the whole domain. We show, in Table 2, the time
of computation of the resolution in the whole domain ND, compared with a standard
domain decomposition methods DD (4 equal sub-domains, no superposition, as
described in [8]) where we show also the number of iterations between the sub-
domains necessary to reach the solution and our algorithm ISA.
As expected our performances are comparable with a single iteration of the DD,
which consists in a resolution on a part of the grid containing the 25% of the
nodes of Ω. The ISA is performed on a approximated independent sub-domains
decomposition obtained using RA on a 202grid; accordingly as shown in Table 1
the dimension of such decomposed domain will be approximately 33% of the original
one.
The most important peculiarity of this proposal consists in having a bound for
the convergence of the method. Table 3shows the experimental error in various fine
grids, in the case of the original problem (solved on the whole domain Ω) or in pres-
ence of various decompositions in independent domains. It is important to remark
again the fact that the error introduced in not affected by the discretization step
of the coarse grid for the reconstruction of the subdomains. This is not shown in a
table because it would be simply constant through the various choices of ∆xK. The
experimental evidence shows us an error even smaller than the normal resolution,
or at least equal. This is due to the fact that in presence of a “favourable” decom-
position of the problem, where non differentiability points of the solution lie on a
regular point of every decomposed solution, the value in that point is reconstructed
as a minimum of a collection of smooth functions. Numerically, this decreases the
local numerical error due to the smoothing effect of the interpolation in (7).
Example 4.2 (Van Der Pol oscillator).Let us pass to a more difficult case. A
well known example in the field is the Van Der Pol oscillator, here formulated
as target problem. We consider Γ := ∂ B(0, ρ) (in this case ρ= 0.2) and Ω :=
(−1,1)2\¯
B(0, ρ). Also in this case the example is an optimal control problem, then
the second control will not be present in the dynamics. The nonlinear system will
be:
f(x, a) = x2
(1 −x2
1)x2−x1+a.
The others parameters of the system are:
Ω=(−1,1)2, A = [−1,1], λ = 1, l(x, y, a) = (x2
1+x2
2)1
2, g(x)≡0.
For this problem we do not have an analytic formula for the solution, then we will
consider, (in the error estimation), a numerical solution computed on a very fine
grid of 4002elements.
We consider a division of the target in “slices of a cake”, meaning that a 2-parts
division will be Γ1:= {x∈B(0,0.2)|x2≥0}, Γ2:= {x∈B(0,0.2)|x2≤0}and a
4-parts, Γ1:= {x∈B(0,0.2)|x1≥0, x2≥0}, Γ2:= {x∈B(0,0.2)|x1≥0, x2≤0},
Γ3:= {x∈B(0,0.2)|x1≤0, x2≤0}, Γ4:= {x∈B(0,0.2)|x1≤0, x2≥0}, etc.
The estimation of the constants will be in this case less elementary: we will
choose in this example C= 1, M= 1, = 10−3,q=3
4.
In Figure 4is shown a comparison between the exact division in sub-domains
and two approximated sets (Σ1, Σ3). An additional difficulty in this case consists
16 ADRIANO FESTA
Figure 4. Van Der Pol: exact decomposition and two (of the four)
approximated independent subsets found with a coarse grind of 152
points.
Table 4. Van Der Pol: comparison of the accuracy of the decom-
position with various discretization steps
N. of variables ∆xKTime elapsed maxi|Σi|/|Ω|maxi|Σi|/|Ω|
520.4 1.4·10−362%
1020.2 0.011 55%
2020.1 0.103 47% 42.2%
3020.06 1.47 45%
4020.05 5.6 44.6%
5020.04 16.3 44.1%
Table 5. Van Der Pol: approximation error Error in norm ∆∞
(and ∆1)
50210022002
original 0.09(0.07) 0.03(0.01) 0.01(6 ·10−3)
2-subsets 0.09(0.07) 0.03(0.01) 0.01(6 ·10−3)
4-subsets 0.09(0.07) 0.03(0.01) 0.01(6 ·10−3)
8-subsets 0.09(0.07) 0.03(0.01) 0.01(6 ·10−3)
in the fact that there are some points of the domain not reachable by the trajecto-
ries without exiting from the computational domain (in white in the l.h.s. figure).
These points are automatically included in each approximation but, although they
contribute to enlarge every subset, they do not create serious problems to the pro-
cedure.
In Table 4is shown the accuracy of the 4-independent subset reconstruction with
various discretization steps. In this case it is possible to see an innate limitation
of the efficacy of such decomposition for parallel computing: the exact division in
independent subset is not balanced, then the reduction of dimension in the greater
subset will be less considerable. In some cases this problem could even nullify the
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 17
Figure 5. A pursuit evasion game: approximated value function
of the differential game presented in Example 4.3
Figure 6. A pursuit evasion game: exact decomposition and two
(of the four) approximated independent subsets found with a coarse
grind of 402points.
efficacy of the method (we can obtain a decomposition in some empty sets and the
whole Ω), we will discuss this point in the conclusions section.
As already remarked this decomposition guarantees a rate of convergence to the
solution equal to the resolution on the whole domain. The experimental evidence
in Table 6shows something more, with an error (both in ∆∞than in ∆2norm)
constant, in its significant figures, in the various cases of decomposition. This is the
typical situation: simply the decomposition does not affect the convergence of the
numerical method.
Example 4.3 (A pursuit-evasion game).Let us now to pass to a decomposable
differential game. We will consider a pursuit evasion game, where two agents have
the opposite goal to reduce/postpone the time of capture. The dynamics considered
are the following:
f(x, a, b) := f1(x)(a1−b1)
f2(x)(a2−b2)
18 ADRIANO FESTA
Table 6. A pursuit evasion game: comparison of the accuracy of
the decomposition with various discretization steps
N. of variables ∆xKTime elapsed maxi|Σi|/|Ω|maxi|Σi|/|Ω|
520.4 10−360%
1020.2 0.008 46%
3020.06 1.38 38% 25%
5020.04 15.9 36.1%
where the functions f1,f2are f1(x) := x2+ 1 and f2(x) := 1. The running cost
l(x, a, b) := x1+ 0.1. This is a modification of the classical pursuit evasion game on
a plane to put in evidence another aspect of the technique, as will be clear in the
following. The controls are taken in the unit ball for the pursuer A=B(0,1) and
B=B(0,1/2) for the evader. The capture happens when the trajectory is driven
to touch the small ball B(0, ρ), (ρ= 0.2, in this case), then the set Γ will be, as in
the previous example Γ := ∂B(0,0.2).
It is possible to show that the Hamilton-Jacobi equation associated to this prob-
lem verifies the decomposability condition (H2); the easiest way to do it is to con-
sider the norm (it is possible because |fi(x)|>0 for i= 1,2)
kpk∗:= max
a∈B(0,1) f1(x)
f2(x)aT·p,
the Hamiltonian associated is equivalent to
H(x, p) := kpk∗−kpk∗
2−(x2
1+ 0.1)
evidently convex everywhere with respect to p; then (H2) is automatically verified.
The value function of the game is shown in Figure 5. The main point is that the
function is very flat along the axis x2= 0, this will produce a critical effect in the
sets approximation, shown in Figure 6and in Table 6, in this test, the parameter
are fixed as C= 1, M= 3, q= 1/2. The convergence to the exact division in
independent subsets will be very slow.
5. Conclusions. We have shown a constructive manner to obtain a decomposition
of the domain of a Hamilton Jacobi equation verifying condition (H2) in independent
subsets which have the property of being computed independently from each others.
The procedure resumes some general ideas already presented in [7], clarifying the
theoretical background, enlarging the class of the Hamilton-Jacobi equations where
the technique is relevant, proving the convergence of the parallel algorithm derived
ISA and producing some estimates for the error.
A detailed evaluation of the performances of ISA are still an open question post-
poned to a forthcoming work. Despite that, we can expect results similar to [7], since
as shown in the Section 4.1, our pre-computing step gives a division in sub-domains
sufficiently close to a partition. Anyway, the main advantage of our proposal is in
the guarantee to converge to the correct solution with an error of the order O(∆xG).
Some further improvements can be adapted to the technique. The critical occur-
rence shown in Example 4.2, about the balance of the dimension of the subsets can
be solved with a recursive refinement of the division of Γ, producing in some few
steps a more equilibrate division. More unavoidable the case presented in example
INDEPENDENT SUB-DOMAINS RECONSTRUCTION 19
4.3. In that case it is, for the moment, impossible to obtain a satisfactory reduc-
tion of the dimension of the decomposed domains without solving the problem on
a sufficiently fine grid.
Acknowledgements. This work was partially supported by the European Union
under the 7th Framework Programme FP7-PEOPLE-2010-ITN SADCO, Sensitivity
Analysis for Deterministic Controller Design.
REFERENCES
[1] F. Ancona and A. Bressan. Patchy vector fields and asymptotic stabilization. ESAIM Control
Optim. Calc. Var. 4, (1999) 445–471.
[2] M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-
Jacobi-Bellman equations. Systems and Control: Foundations and Applications. Birkh¨auser,
Boston, 1997.
[3] M. Bardi and M. Falcone. An approximation scheme for the minimum time function. SIAM
J. of Contol Optim. 28, (1990) 950–965.
[4] M. Bardi, M. Falcone and P. Soravia. Numerical methods for pursuit-evasion games via vis-
cosity solutions. In Stochastic and differential games, 105–175, Birkhauser, Boston, 1999.
[5] G. Barles. Solutions de viscosit´e des ´equations de Hamilton-Jacobi, volume 17 of Math´ema-
tiques et Applications. Springer, Paris, 1994.
[6] S. Cacace, E. Cristiani and M. Falcone. A local ordered upwind method for hamilton-jacobi
and isaacs equations. In Proceedings of 18th IFAC World Congress, (2011), 6800–6805.
[7] S. Cacace, E. Cristiani, M. Falcone and A. Picarelli. A patchy dynamic programming scheme
for a class of hamilton–jacobi–bellman equations. SIAM J. Sci. Comput. 34(5), (2012), A2625–
A2649.
[8] F. Camilli, M. Falcone, P. Lanucara, and A. Seghini. A domain decomposition method for
bellman equations. In Domain Decomposition methods in Scientific and Engineering Com-
puting, Contemporary Mathematics 180, AMS, 477-483, 1994.
[9] P. Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobi equations, and
optimal control. Progress in Nonlinear Differential Equations and their Applications, 58.
Birkh¨auser, Boston, MA, 2004.
[10] E. Carlini, M. Falcone, N. Forcadel and R. Monneau. Convergence of a generalized fast-
marching method for an eikonal equation with a velocity-changing sign. SIAM J, Numer.
Anal. 46(6), (2008), 2920–2952.
[11] M. Detrixhe, F. Gibou and C. Min. A parallel fast sweeping method for the eikonal equation.
J. Comput. Phys. 237, (2013), 46–55.
[12] I. Capuzzo Dolcetta and H. Ishii. Approximate solutions of the Bellman equation of deter-
ministic control theory. Appl. Math. Optim. 11, (1984), 161–181.
[13] R.J. Elliott and N.J. Kalton. The existence of value in differential games, Bull. Amer. Math.
Soc. 78:3, (1972), 427–431.
[14] M. Falcone. A numerical approach to the infinite horizon problem of deterministic control
theory. Appl. Math. Optim. 15, (1987), 1–13.
[15] M. Falcone and R. Ferretti. Convergence analysis for a class of high-order semi-Lagrangian
advection schemes. SIAM J. Numer. Anal. 35(3), (1998), 909–940.
[16] M. Falcone and R. Ferretti. Semi-Lagrangian Approximation Schemes for Linear and
Hamilton-Jacobi Equations. Applied Mathematics series. SIAM, Philadelphia, 2014.
[17] A. Festa and R.B. Vinter. A decomposition technique for pursuit evasion games with many
pursuers. 52nd IEEE Control and Decision Conference (CDC), (2013), 5797–5802.
[18] A. Festa and R.B. Vinter. Decomposition of differential games. Submitted.
[19] C. Navasca and A.J. Krener. Patchy solutions of Hamilton-Jacobi-Bellman partial differential
equations. in Modeling, Estimation and Control, Lecture Notes in Control and Inform. Sci.
364, A. Chiuso, A. Ferrante, and S. Pinzoni, eds., Springer, Berlin, 2007, 251–270.
[20] P. Soravia. Estimates of convergence of fully discrete schemes for the Isaacs equation of
pursuit-evasion differential games via maximum principle. SIAM J. Cont. Opt. 36.1 (1998),
1–11.
[21] J.A. Sethian. Level sets methods and fast marching methods. Cambridge University Press,
Cambridge, UK, 1999.
20 ADRIANO FESTA
[22] J.A. Sethian and A. Vladimirsky. Ordered upwind methods for static hamilton–jacobi equa-
tions: Theory and algorithms. SIAM J. Numer. Anal. 41(1), (2003), 325–363.
[23] M. Sun. Domain decomposition algorithms for solving hamilton-jacobi-bellman equations.
Numer. Func. Anal. Opt. 14(1-2), (1993), 145–166.
[24] Y.R. Tsai, L.T. Cheng, S. Osher and H. Zhao. Fast sweeping algorithms for a class of
hamilton–jacobi equations. SIAM J. Numer. Anal. 41(2), (2003), 673–694.
[25] J.N. Tsitsiklis. Efficient algorithms for globally optimal trajectories. IEEE T. Automat.
Contr. 40(9), (1995), 1528–1538.
[26] A. Valli and A. Quarteroni. Domain decomposition methods for partial differential equations.
Oxford University Press, Oxford, 1999.
[27] R.B. Vinter. Optimal Control Theory. Birkh¨auser, Boston Heidelberg, 2000.
[28] H. Zhao. A fast sweeping method for eikonal equations. Math. Comput. 74, (2004), 603–627.
[29] H. Zhao. Parallel implementations of the fast sweeping method. J. Comput. Math. 25(4),
(2007), 421–429.
[30] S. Zhou and W. Zhan. A new domain decomposition method for an hjb equation. J. Comput.
Appl. Math. 159(1), (2003), 195–204.
E-mail address:festa@ensta.fr
