Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks
We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.
35F21, 49L25, 35B51.
Hamilton-Jacobi equations, networks, quasi-convex Hamiltonians, discontinuous Hamiltonians, viscosity solutions, flux-limited solutions, comparison principle, vertex test function, homogenization, optimal control, discontinuous running cost.
- 1 Introduction
- 2 Relaxed and flux-limited solutions
- 3 Comparison principle on a junction
- 4 Construction of the vertex test function
- 5 Extension to networks
- 6 First application: link with optimal control theory
- 7 Second application: study of Ishii solutions
- 8 Third application: a homogenization result for a network
- A Appendix: proofs of some technical results
- B Appendix: stationary results for networks
This paper is concerned with Hamilton-Jacobi (HJ) equations on networks associated with Hamiltonians that are quasi-convex and coercive in the gradient variable and possibly discontinuous at the vertices of the network in the space variable.
Space discontinuous Hamiltonians have been identified as both important/relevant and difficult to handle; in particular, a few theories/approaches (see below) were developed to study the associated HJ equations. In this paper, we show that if they are assumed to be quasi-convex and coercive in the gradient variable, then not only uniquess can be proved for very general conditions at discontinuities (referred to as junction conditions), but such conditions can even be classified: imposing a general junction condition reduce to impose a junction condition of optimal control type, referred to as a flux-limited junction condition. As far as uniqueness is concerned, a comparison principle is proved. We show that the doubling variable approach can be adapted to the discontinuous setting if we go beyond the classical test function by using a vertex test function instead. This vertex test function can be used to do much more, like dealing with second order terms  or getting error estimates for monotone schemes .
We point out that the present article is written in the one-dimensional setting for pedagogical reasons but our theory extends readily to higher dimensions .
1.1 The junction framework
We focus in this introduction and in most of the article on the simplest network, referred to as a junction, and on Hamiltonians which are constant with respect to the space variable on each edge. Indeed, this simple framework leads us to the main difficulties to be overcome and allows us to present the main contributions. We will see in Section 5 that the case of a general network with -dependent Hamiltonians is only an extension of this special case.
A junction is a network made of one vertex and a finite number of infinite edges. It is endowed with a flat metric on each edge. It can be viewed as the set of distinct copies () of the half-line which are glued at the origin. For , each branch is assumed to be isometric to and
where the origin is called the junction point. For points , denotes the geodesic distance on defined as
For a smooth real-valued function defined on , denotes the (spatial) derivative of at and the “gradient” of is defined as follows,
With such a notation in hand, we consider the following Hamilton-Jacobi equation on the junction
subject to the initial condition
The second equation in (1.3) is referred to as the junction condition. In general, minimal assumptions are required in order to get a good notion of weak (i.e. viscosity) solutions. We shed some light on the fact that Equation (1.3) can be thought as a system of Hamilton-Jacobi equations associated with coupled through a “dynamical” boundary condition involving . This point of view can be useful, see Subsection 1.5. As far as junction functions are concerned, we will construct below some special ones (denoted by ) from the Hamiltonians () and a real parameter .
We consider the important case of Hamiltonians satisfying the following structure condition:
We recall that is quasi-convex if its sub-level sets are convex. In particular, since is also assumed to be coercive, there exist numbers such that
1.2 First main new idea: classification of junction conditions
In the present paper, two notions of viscosity solutions are introduced: relaxed (viscosity) solutions (see Definition 2.1), which can be used to deal with all junction conditions, and flux-limited (viscosity) solutions (see Definition 2.2) which are associated with flux-limited junction conditions. Relaxed solutions are used to prove existence and ensure stability. Flux-limited solutions satisfy the junction condition in a stronger sense and are used in order to prove uniqueness. Our first main result states that relaxed solutions for general junction conditions are in fact flux-limited solutions for some junction conditions of optimal-control type.
We now introduce the notion of flux-limited junction condition. Given a flux limiter , the -limited flux through the junction point is defined for as
where is the nonincreasing part of defined by
We now consider the following important special case of (1.3),
We point out that the flux functions associated with coincide if one chooses
As announced above, general junction conditions are proved to be equivalent to those flux-limited junction conditions. Let us be more precise: a junction function is an satisfying
Theorem 1.1 (General junction conditions reduce to flux-limited ones).
The special case of convex Hamiltonians.
In the special case of convex Hamiltonians with different minimum values, Problem (1.7) can be viewed as the Hamilton-Jacobi-Bellman equation satisfied by the value function of an optimal control problem; see for instance  when . In this case, existence and uniqueness of viscosity solutions for (1.7)-(1.4) (with ) have been established either with a very rigid method  based on an explicit Oleinik-Lax formula which does not extend easily to networks, or in cases reducing to for all if Hamiltonians do not depend on the space variable [40, 1]. In such an optimal control framework, trajectories can stay for a while at the junction point. In this case, the running cost at the junction point equals . In this special case, the parameter consists in replacing the previous running cost at the junction point by . In Section 6, the link between our results and optimal control theory will be investigated.
1.3 Second main new idea: the vertex test function
The second main contribution of this paper is to provide the reader with a general yet handy and flexible method to prove a comparison principle, allowing in particular to deal with Hamiltonians that are quasi-convex and coercive with respect to the gradient variable and are possibly discontinuous with respect to the space variable at the vertices.
It is known that the core of the theory for HJ equations lies in the proof of a strong uniqueness result, i.e. of a comparison principle. It is also known that it is difficult to get uniqueness results for discontinuous Hamiltonians. Indeed, the standard proof of the comparison principle in the Euclidian setting is based on the so-called doubling variable technique; and such a method, even in the monodimensional case, generally fails for piecewise constant (in ) Hamiltonians at discontinuities (see the last paragraph of Subsection 1.5). Since the network setting contains the previous one, the classical doubling variable technique is known to fail at vertices [40, 1, 30].
Before discussing the method we develop to prove it, we state the comparison principle.
Theorem 1.5 (Comparison principle on a junction).
Theorem 1.6 (Existence and uniqueness on a junction).
As we previously mentioned it, we prove Theorem 1.5 by remarking that the doubling variable approach can still be used if a suitable vertex test function at each vertex is introduced. Roughly speaking, such a test function will allow the edges of the network to exchange the necessary information. More precisely, the usual penalization term, with , is replaced with . For a general HJ equation
the vertex test function has to (almost) satisfy,
(at least close to the vertex ). This key inequality compensates for the lack of compatibility between Hamiltonians333Compatibility conditions are assumed in [40, 1] for instance.. The construction of a (vertex) test function satisfying such a condition allows us to circumvent the discontinuity of at the junction point.
As explained above, this method consists in combining the doubling variable technique with the construction of a vertex test function . We took our inspiration for the construction of this function from papers like [26, 7] dealing with scalar conservation laws with discontinuous flux functions. In such papers, authors stick to the case .
A natural family of explicit solutions of (1.7) is given by
for in the germ defined as follows,
In the special case of convex Hamiltonians satisfying the vertex test function is a regularized version444Such a function should indeed be regularized since it is not on the diagonal of . of the function , where is defined as follows: for ,
In particular, we have .
1.4 The network setting
We will extend our results to the case of networks and quasi-convex Hamiltonians depending on time and space and to flux limiters depending on time and vertex, see Section 5. Noticeably, a localization procedure allows us to use the vertex test function constructed for a single junction.
In order to state the results in the network setting, we need to make precise the assumptions satisfied by the Hamiltonians associated with each edge and the flux limiters associated with each vertex. This results in a rather long list of assumptions. Still, when reading the proof of the comparison principle in this setting, the reader may check that the main structure properties used in the proof are gathered in the technical Lemma 5.2.
As an application of the comparison principle, we consider a model case for homogenization on a network. The network whose vertices are is naturally embedded in . We consider for all edges a Hamiltonian only depending on the gradient variable but which is “repeated -periodically with respect to edges”. We prove that when , the solution of the “oscillating” Hamilton-Jacobi equation posed in converges toward the unique solution of an “effective” Hamilton-Jacobi equation posed in .
A first general comment about the main results.
Our proofs do not rely on optimal control interpretation (there is no representation formula of solutions for instance) but on PDE methods. We believe that the construction of a vertex test function is flexible and opens many perspectives. It also sheds light on the fact that the framework of quasi-convex Hamiltonians, which is slightly more general than the one of convex ones (at least in the evolution case), deserves special attention.
1.5 Comparison with known results
Hamilton-Jacobi equations on networks.
There is a growing interest in the study of Hamilton-Jacobi equations on networks. The first results were obtained in  for eikonal equations. Several years after this first contribution, the three papers [1, 30, 41] were published more or less simultaneously. In these three papers, the Hamiltonians are always convex with respect to the gradient variables and optimal control plays in important role (at least in [1, 30]). Still, frameworks are significantly different.
In , the authors study an optimal control problem in and impose a state constraint: the trajectories of the controlled system have to stay in the embedded network. From this point of view,  is related to [21, 22] where trajectories in are constrained to stay in a closed set which can have an empty interior. But as pointed out in , the framework from [21, 22] implies some restricting conditions on the geometry of the embedded networks. Our approach can now handle the general case for networks.
In , the comparison principle was a consequence of a super-optimality principle (in the spirit of  or [42, 43]) and the comparison of sub-solutions with the value function of the optimal control problem. Still, the idea of using the “fundamental solution” to prove a comparison principle originates in the proof of the comparison of sub-solutions and the value function. Moreover, as explained in Subsection 3.3, the comparison principle obtained in this paper could also be proved, for and under more restrictive assumptions on the Hamiltonians, by using this fundamental solution.
Networks, regional optimal control and stratified spaces.
We already pointed out that the Hamilton-Jacobi equation on a network can be regarded as a system of Hamilton-Jacobi equations coupled through vertices. In this perspective, our work can be compared with studies of Hamilton-Jacobi equations posed on, say, two domains separated by a frontier where some transmission conditions should be imposed. Contributions to such problems are [9, 10, 38, 37, 2]. This can be even more general by considering equations in stratified spaces [12, 11].
We first point out that the framework of these works is genuinely multi-dimensional while in this paper we stick to a monodimensional setting; still, our method generalizes to a higher dimensional setting . Another difference between their approach and the one presented in the present work and in papers like [1, 41, 30] is that these authors write a Hamilton-Jacobi equation on the frontier (which is lower-dimensional). Another difference is that techniques from dynamical systems play also an important role. We mention that the techniques from  can be applied to treat the cases considered in our work.
Still, results can be compared. Precisely, considering a framework were both results can be applied, that is to say the monodimensional one, we will prove in Section 7 that the value function from  coincides with the solution of (1.7) for some constant that is determined. And we prove more (in the monodimensional setting; see also extensions below): we prove that the value function from  coincides with the solution of (1.7) for some (distinct) constant which is also computed.
Hamilton-Jacobi equations with discontinuous source terms.
There are numerous papers about Hamilton-Jacobi equations with discontinuous Hamiltonians. The first contribution is due to Dupuis ; see also [18, 25, 16, 17]. The recent paper  considers a Hamilton-Jacobi equation where specific solutions are expected. In the one-dimensional space, it can be proved that these solutions are in fact flux-limited solutions in the sense of the present paper with where is a constant appearing in the HJ equation at stake in . The introduction of  contains a rather long list of results for HJ equations with discontinuous Hamiltonians; the reader is referred to it for further details.
Contributions of the paper.
In light of the review we made above, we can emphasize the main contributions of the paper: compared to [40, 41], we deal not only with eikonal equations but with general Hamilton-Jacobi equations. In contrast to , we are able to deal with networks with infinite number of edges, that are not embedded. In constrast to [1, 30, 40, 41], we can deal with quasi-convex (but not necessarily convex) discontinuous Hamilton-Jacobi equations with general junctions conditions. For such equations, flux-limited solutions are introduced and a flexible PDE framework is developed instead of an optimal control approach. Eventhough, the link with optimal control (in the spirit of [1, 9, 10]) and with regional control (in the spirit of [9, 10]) are thoroughly investigated. In particular, a PDE characterization of the two value functions introduced in  is provided, one of the two characterizations being new.
Several applications are also developed: the extension to the network setting and some homogenization results.
More homogenization results were recently obtained in . An example of applications of this result is the case where a periodic Hamiltonian is perturbed by a compactly supported function of the space variable , say. Such a situation is considered in lectures by Lions at Collège de France . Rescaling the solution, the expected effective Hamilton-Jacobi equation is supplemented with a junction condition which keeps memory of the compact perturbation.
1.6 Organization of the article and notation
Organization of the article.
The paper is organized as follows. In Section 2, we introduce the notion of viscosity solution for Hamilton-Jacobi equations on junctions, we prove that they are stable (Proposition 2.4) and we give an existence result (Theorem 2.14). In Section 3, we prove the comparison principle in the junction case (Theorem 1.5). In Section 4, we construct the vertex test function (Theorem 3.2). In Section 6, a general optimal control problem on a junction is considered and the associated value function is proved to be a solution of (1.7) for some computable constant . In Section 7, the two value functions introduced in  are shown to be solutions of (1.7) for two explicit (and distinct) constants . In Section 5, we explain how to generalize the previous results (viscosity solutions, HJ equations, existence, comparison principle) to the case of networks. In Section 8, we present a straightforward application of our results by proving a homogenization result passing from an “oscillating” Hamilton-Jacobi equation posed in a network embedded in an Euclidian space to a Hamilton-Jacobi equation in the whole space. Finally, we prove several technical results in Appendix A and we state results for stationary Hamilton-Jacobi equations in Appendix B.
Notation for a junction.
A junction is denoted by . It is made of a finite number of edges and a junction point. The edges of a junction, () are isometric to . The open edge is denoted by . Given a final time , denotes .
The Hamiltonians on the branches of the junction are denoted by ; they only depend on the gradient variable. The Hamiltonian at the junction point is denoted by and is defined from all and a constant which “limits” the flux of information at the junction.
Given a function , its gradient at is denoted by ; it is a real number if but it is a vector of at . We let denote outside the junction point and at the junction point. If now also depends on the time , denotes the time derivative.
Notation for networks.
A network is denoted by . It is made of vertices and edges . Each edge is either isometric to or to a compact interval whose length is bounded from below; hence a network is naturally endowed with a metric. The associated open (resp. closed) balls are denoted by (resp. ) for and .
In the network case, an Hamiltonian is associated with each edge and is denoted by . It depends on time and space; moreover, the limited flux functions can depend on time and the vertex : .
Given a metric space , denotes the space of continuous real-valued functions defined in . A modulus of continuity is a function which is non-increasing and .
2 Relaxed and flux-limited solutions
This section starts with the introduction of two notions of viscosity solutions in the junction case and of their studies. Relaxed (viscosity) solutions are first introduced; they are defined for general junction conditions. They naturally satisfy good stability properties (see for instance Proposition 2.4). Flux-limited solutions are associated with flux-limited junction conditions. They satisfy the junction condition in a stronger sense (see Proposition 2.5). The main contribution of this section is the proof of Theorem 1.1. It relies on the observation that the set of test functions for flux-limited solutions can be reduced drastically: it is enough to consider test functions with fixed space slopes (Theorem 2.7).
In order to introduce the two notions of viscosity solution which will be used in the remaining of the paper, we first introduce the class of test functions. For , set . We define the class of test functions on by
We (classically) say that a test function touches a function from below (respectively from above) at if reaches a minimum (respectively maximum) at in a neighborhood of it.
We recall the definition of upper and lower semi-continuous envelopes and of a (locally bounded) function defined on ,
Definition 2.1 (Relaxed solutions).
We say that is a relaxed sub-solution (resp. relaxed super-solution) of (1.3) in if for all test function touching from above (resp. from below) at , we have
if , and
We say that is a relaxed solution if is both a relaxed sub-solution and a relaxed super-solution.
We give a second definition of viscosity solutions in the case of flux-limited junction functions : the junction condition is satisfied “in a classical sense” for test functions touching sub- and super-solutions at the junction point.
Definition 2.2 (Flux-limited solutions).
Assume that the Hamiltonians satisfy (1.5) and let .
We say that is a flux-limited sub-solution (resp. flux-limited super-solution) of (1.7) in if for all test function touching from above (resp. from below) at , we have
We say that is a flux-limited solution if is both a flux-limited sub-solution and a flux-limited super-solution.
2.2 The “weak continuity” condition for sub-solutions
If not only satisfies (1.9) but is also semi-coercive, that is to say if
then any -relaxed sub-solution satisfies a “weak continuity” condition at the junction point. Precisely, the following lemma holds true.
Lemma 2.3 (“Weak continuity” condition at the junction point).
Since is upper semi-continuous, we know that for all and ,
Assume that there exists and such that
Since is upper semi-continuous, we know that we can find arbitrarily close to such that is arbitrarily close to and such that there exists a function (strictly) touching from above at . In particular, we can ensure
In particular, since , there exist and small enough such that
We now consider the test function for . We claim that for and for large enough, reaches its maximum on at . We first remark that . Moreover, for and , (2.4) implies that
For and , we have for large enough
Hence the supremum is reached either for or in the interior of . In the latter case, this yields the viscosity inequality
which cannot hold true for large . We conclude that
We now get
where we have used (2.3) for any negative and any small enough . This implies that
which cannot hold true for very negative because of (2.2). The proof is now complete. ∎
2.3 General junction conditions and stability
The first stability result is concerned with the supremum of relaxed sub-solutions. Such a result is used in the Perron process to construct relaxed solutions. Its proof is standard so we skip it.
Proposition 2.4 (Stability by supremum/infimum).
Assume that the Hamiltonians satisfy (1.5) and that satisfies (1.9). Let be a nonempty set and let be a familly of relaxed sub-solutions (resp. relaxed super-solutions) of (1.3) on . Let us assume that
is locally bounded on . Then is a relaxed sub-solution (resp. relaxed super-solution) of (1.3) on .
In the following proposition, we assert that, for the special junction functions , the junction condition is in fact always satisfied in the classical (viscosity) sense, that is to say in the sense of Definition 2.2 (and not Definition 2.1).
Proposition 2.5 (flux-limited junction conditions are satisfied in the classical sense).
Assume that the Hamiltonians satisfy (1.5) and consider . If , then relaxed super-solutions (resp. relaxed sub-solutions) coincide with flux-limited super-solutions (resp. flux-limited sub-solutions).
Proof of Proposition 2.5.
The proof was done in  for the case , using the monotonicities of the . We follow the same proof and omit details.
The super-solution case. Let be a relaxed super-solution satisfying the junction condition in the viscosity sense and let us assume by contradiction that there exists a test function touching from below at for some , such that
Then we can construct a test function satisfying in a neighborhood of , with equality at such that
Using the fact that at , we deduce a contradiction with (2.5) using the viscosity inequality satisfied by for some .
The sub-solution case. Let now be a sub-solution satisfying the junction condition in the viscosity sense and let us assume by contradiction that there exists a test function touching from above at for some , such that
Let us define
and for , let be such that
where we have used the fact that . Then we can construct a test function satisfying in a neighborhood of , with equality at , such that
Using the fact that at , we deduce a contradiction with (2.6) using the viscosity inequality for for some . ∎
The last stability result is concerned with sub-solutions of the Hamilton-Jacobi equation away from the junction point and which satisfy the “weak continuity” condition. The following proposition asserts that such a “weak continuity” is stable under upper semi-limit.
Proposition 2.6 (Stability of the “weak continuity” condition).
Consider a family of Hamiltonians satisfying (1.5). We also assume that the coercivity of the Hamiltonians is uniform in . Let be a family of subsolutions of
for all such that, for all ,
If the upper semi-limit is everywhere finite, then it satisfies for all
We argue by contradiction by assuming that there exists and such that
Our goal is first to use a perturbation argument to get a test function touching strictly from above at a time where the previous inequality still hold true. Using the upper semi-continuity of , we can keep away from in a neighborhood of the point corresponding to the boundary of the time interval where and are strictly separed. From the definition of , we also get a sequence of points realizing the value . Considering now for positive and very large, we use the sequence in order to get a contact point of with this test-function away from . This will lead to the desired contradiction since is arbitrarily large.
We now make precise how to use the previous strategy. Since is upper semi-continuous, we know that we can find arbitrarily close to such that is arbitrarily close to and such that there exists a function (strictly) touching from above at . In particular, we can ensure
In particular, since , there exist and such that
Since is the upper relaxed-limit of , this implies in particular that for small enough,
We claim that
Indeed, if the previous inequality is replaced with an equality, this would contradict (2.7). In particular, reducing and if necessary, we can further assume that for ,
Let be such that
By (2.10), we know that for small enough. We also know that there exists such that for small enough (along a subsequence) with . Indeed, if (at least along a subsequence), then
which is in contradiction with (2.8).
We now consider with and we consider the point where the maximum of is reached in . Remark that for and , (2.10) implies that
Analogously, for and , (2.9) implies that
Finally, for and , we have for small and some ,
Since is locally bounded from above (because it is upper semi-continuous), we conclude that we can choose large (depending on and a local bound of from above) such that for and , we have for small and some ,
Finally, the maximum of in satisfies
We conclude that belongs to the interior of which entails
which cannot hold true for very large because of the uniform coercivity of . The proof is now complete. ∎
2.4 Reducing the set of test functions
We show in this subsection, that to check the flux-limited junction condition, it is sufficient to consider very specific test functions. This important property is useful both from a theoretical point of view and from the point of view of applications.
We consider functions satisfying a Hamilton-Jacobi equation in , that is to say, solutions of
for . The non-increasing part of the Hamiltonian is used in the definition of flux-limited junction conditions. In the next theorem, the non-decreasing part is needed. It is defined by
where we recall that is a point realizing the minimum of .
Theorem 2.7 (Reduced set of test functions).
let us fix any time independent test function satisfying
Given a function , the following properties hold true.
If for all , is an upper semi-continuous sub-solution of (2.11) and satisfies
then is a -flux limited sub-solution.
Theorem 2.7 exhibits (necessary and) sufficient conditions for sub- and super-solutions of (2.11) to be flux-limited solutions. After proving Theorem 2.7, we realized that this result shares some similarities with the way of checking the entropy condition at the junction for conservation law equations associated to bell-shaped fluxes. Indeed it is known that it is sufficient to check the entropy condition only with one particular stationary solution of the Riemann solver (see [13, 7, 6]).
The set of test functions can be reduced to a single one for flux-limited sub-solution only if the “weak continuity” condition (2.13) is imposed. Indeed, if this condition is not satisfied, then the conclusion is false. Consider for instance Hamiltonians reaching their minimum at and such that and consider such that and consider