We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space . For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, if the choice is to use only “regular strategies” or to use also “singular strategies”. We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential equations method “à la Lions–Papanicolaou–Varadhan” with suitable adaptations, the first case has to be treated by control methods (dynamic programming).
In order to describe the homogenization problems we address in this article, we consider a partition of as where , are open subsets of , and . We assume that the ’s are -periodic, i.e. for all and .
The homogenization problems can be written from the partial differential equations (pde in short) point of view as
where ε is a small positive parameter which is devoted to tend to 0, the actualization factor λ is positive and , are classical Hamiltonians of deterministic control problems, which are of the form ()
Precise assumptions will be given later on but we already mention that the functions and satisfy the most classical regularity and boundedness assumptions and and are -periodic in y, for any x and .
Of course, Eq. (1.1) have to be completed by suitable conditions on the hypersurface 1
Our assumptions below actually imply that is a -hypersurface.
and this was the purpose of [9,10] to see what kind of conditions have to be imposed. Unfortunately the classical Ishii inequalities
and
are not sufficient to have a well-posed problem and (1.1)–(1.3)–(1.4) has a maximal solution denoted by and a minimal solution denoted by which can both be described in terms of control. We refer the reader to Section 3 for a complete description of the control problems for and but we just mention that, while is built by using so-called “regular strategies”, is built by using all kind of strategies and in particular “singular strategies” which are excluded in the case of .
We want to describe the asymptotic behavior as of the maximal solution and the minimal solution . The results in [9,10] imply that can be characterized through pdes, by adding a suitable subsolution condition on , while this is not the case anymore for which is just the maximal subsolution of (1.1)–(1.3)–(1.4). The consequence for our study is immediate: while for we can follow and adapt the classical pde arguments of Lions, Papanicolaou and Varadhan [24] and Alvarez and Bardi [2,3], this is not the case anymore for where even if we follow closely the pde ideas, we have to perform all the argument on the control formulas. In that way we are close to some of the arguments of the weak KAM theory (see Fathi [14–16]).
For the convergence of , some specific technical difficulties appear which are solved by an approximation of the cell problem: such ideas, in a slightly different context, are already used in Barles, Da Lio, Lions and Souganidis [11] (see also [7]).
We point out that Forcadel and Rao [18] studied such homogenization problems in a multi-domain framework: they are able to treat cases where the boundaries of the are not smooth but for problems set in and only in the case. As related works we mention the study of Hamilton–Jacobi equations on networks [1,19,20].
The article is organized as follows: in Section 2, we recall basic facts and stability results for the pde approach of (1.1)–(1.3)–(1.4). In Section 3, we describe the control problems for and : we give precise definitions of “regular strategies” and “singular strategies”. The next two sections are devoted to the study of the homogenization problems for and , respectively: we follow a (rather) classical double-scale approach by first studying the cell problems and then we use the solutions of the cell problems to deduce the convergence. In Section 6 we give an explicit example in dimension 1 where the effective Hamiltonians describing the asymptotic behavior of and , are different. Finally in the Appendix we provide several technical results which are useful for the convergence proofs.
Different notions of viscosity solutions for multi-domains problems
This section is devoted to the description of the precise definition of viscosity solutions for problems like (1.1)–(1.3)–(1.4). For the introduction and all the details on these definitions we refer to [9,10] and the reference therein.
Let us remember that in this paper we are considering a partition of as where , are open subsets of , and is a regular hypersurface (). For we denote by , the tangent space to at y and is the scalar product in this tangent space.
We consider the general function that can be differently defined on , and by
We define also
Note that to be consistent with the homogenization problem we will always assume that each is not only defined in but in all .
First of all we recall the classical H. Ishii definition of discontinuous viscosity solution for a discontinuous Hamiltonian G (see [21] and also [8]). Given a real number and a function a viscosity solution of problem
is defined as follows.
We say that a bounded usc function u is a subsolution of (2.1) if it verifies the following inequalities in the viscosity sense
We say that a lsc function v is a supersolution of (2.1) if it verifies the following inequalities in the viscosity sense
We say that a bounded continuous function w is a solution of (2.1) it is both a sub and a supersolution.
We give now a new definition of solution taking in account a tangential equation on of the following problem:
The main difference is the following definition of viscosity subsolution for the tangential Hamiltonian on .
An usc function is a viscosity subsolution of
if, for any and any maximum point y of in , one has
where means the gradient of the restriction of ϕ to at point y (which belongs to ).2
Note that, if is a unitary normal vector to at y, then .
We define the viscosity solutions for multi-domain problems as Ishii classical solutions adding for subsolutions the previous condition, more precisely.
We say that a bounded usc function u is a subsolution of (2.6) if it verifies (2.2), (2.3) and
We say that a lsc function v is a supersolution of (2.6) if it verifies (2.4) and (2.5).
We say that a bounded continuous function w is a solution of (2.6) it is both a sub and a supersolution.
Note that an analogous definition can be given for a solution u depending on the x-variable instead of the y-variable, so we will not detail it.
Moreover, let us remark that the function does not need to be defined for any but only in the tangent space .
Now we focus our attention on the Bellman equations when we have Hamiltonians of the form (1.2). We state our main assumptions and define the different type of dynamics and Hamiltonians on the interfaces.
For , is a compact metric space and is a continuous bounded function. More precisely, there exists , such that for any , and , ,
For any and , the function is -periodic.
Moreover, there exists such that, for any , and
and there exists such that, for any , and
For , the function is a continuous, bounded function. More precisely, there exists , such that for any , and , ,
Moreover, for any , , the function is -periodic.
There exists a modulus such that, for any , and
and there exists such that, for any , and
Let us recall that a modulus is a function such that where ω is an increasing function such that .
For each and the sets , are closed and convex. There is a such that for any , , ,
where .
We set
For , , we denote by
and
The set of tangential controls is given by:
where is the unitary normal exterior vector to in y, and the subset of of “regular” tangential controls is given by
the tangential Hamiltonians:
where and has been identified with its orthogonal projection on and the Hamiltonians associated with on , on (defined in (1.2)) and respectively and on :
We are interested in the following equations:
where and is associated with on , on and on , is associated with on , on and on , is associated with on , on (in the following we delete the index ε in these notations for the sake of simplicity).
Setting the optimal control problem at ε-fixed
The aim of this section is to give the precise definition of infinite horizon control problems whose value functions are “solutions” of the Hamilton–Jacobi–Bellman equations (2.14) or (2.13).
Note that, assumptions (H0), (H1), are the classical hypotheses used in infinite horizon control problems. We have strengthened them in (H2) in order to have uniformly Lipschitz continuous value functions. Let us remark also that the first part of assumption (H2) avoids the use of relaxed controls.
In order to define the optimal control problems in all , we have to define the dynamics and therefore we are led to consider an ordinary differential equation with discontinuous right-hand side. This kind of ode has been treated for the first time in the pioneering work of Filippov [17]. We are going to define the trajectories of our optimal control problem by using the approach through differential inclusions which is rather convenient here. This approach has been introduced in [26] (see also [4]) and has become now classical. To do so in a more general setting, and since the controllability condition (2.8) plays no role in the definition of the dynamic, we are going to use assumption (H2) which is (H2) without (2.8).
Fix . Our trajectories are Lipschitz continuous functions which are solutions of the following differential inclusion
where, for any ,
the notation referring to the convex closure of the set and we recall that are defined in (H2).
For each, there exists a Lipschitz functionwhich is a solution of the differential inclusion (3.1).
For each solutionof (3.1), there exists a controlsuch that(wherestands for the indicator function of the set I, and for the sake of simplicity the ε-dependence of the controlis not written).
Moreover,
It is worth remarking that, in Theorem 3.1, a solution can be associated to several controls . Fix , to set properly the control problems, we introduce the set of admissible controlled trajectories starting from the initial datum
and we set (the ε and dependence is not explicitly written)
We finally define the set of regular controlled trajectories
Recall that a regular dynamics on with is such that (where is the unitary normal exterior vector to in ) while a singular a dynamic is such that .
The cost functional. Our aim is to minimize an infinite horizon cost functional such that we respectively pay if the trajectory is in , and if it is on .
More precisely, the cost associated to is
where the Lagrangian is given by ( is defined in (2.10))
For the sake of simplicity we also set ( is defined in (2.9))
The value functions. For each initial data , we define the following two value functions
The most important consequence of the controllability assumption (H2) is that both value functions and are uniformly Lipschitz continuous.
Assume (H0),(H1) and (H2). Then, the value functionsandare bounded, Lipschitz continuous functions frominto. Theirnorm is also uniformly bounded with respect to ε.
For the details of the proof see Theorem 2.3 in [9]. Here we only recall that: if and δ are given in (H2) and (H3) then the Lipschitz constant is , hence it does not depend on ε. □
The first key result is the Dynamic Programming Principle.
Assume (H0),(H1) and (H2). Let,be the value functions defined in (3.3) and (3.4), respectively. For each initial data, and each time, we have
The proof is classical so we will omit it. □
As a consequence of the DPP we obtain that both the value functions and are viscosity solutions of the Hamilton–Jacobi–Bellman equation (2.15), while they fulfill different inequalities on the hyperplane .
Assume (H0),(H1) and (H2). The value functionsandare both viscosity solutions of the Hamilton–Jacobi–Bellman equation (2.15).
Moreover,is a solution of (2.14) andis a solution of (2.13).
The proof is given in [9, Theorem 2.5] (see also [10, Theorem 3.3]). □
We end this section by stating two comparison results we will need for (2.14). The first one is a strong comparison result in , while the second one is a local comparison result we will need in the proof of the convergence result Theorem 4.5. Moreover, we prove that and are the minimal supersolution and the maximal subsolution of (2.15), respectively.
Assume (H0),(H1) and (H2).
is the minimal supersolution and solution of (2.15).
is the maximal subsolution and solution of (2.15).
Let u be a bounded, Lipschitz continuous subsolution of (2.14) and v be a bounded, lsc supersolution of (2.14). Thenin.
Fixand. Let u be a bounded, Lipschitz continuous subsolution and v a lsc supersolution of (2.14) for all.
Fixand. Let u be a bounded, Lipschitz continuous subsolution of (2.15) for all.
In order to prove (i), (ii) and (iii) we remark that the proof of [9, Theorem 4.1] (or [10, Theorem 4.4]) is local, therefore it can be adapted easily to this case, so we will omit it. The local comparison (iv) follows directly from [10, Theorem 4.1] while (v) can be easily proved adapting the proof of [10, Theorem 4.4(ii)]. In particular, note that, since all the argument are local we can use Theorem 3.5 and Theorem 3.7 in [10] to adapt the proof of [10, Lemma 6.1]. (The same idea is detailed in [9, Theorem 4.1].) □
We recall here that when we are dealing with the two equations (2.13) and (2.15) are equivalent. Indeed in [10, Theorem 3.7] is it proved that any subsolution u of (2.15) fulfills in the sense of Definition 2.2. However, it is worth emphasizing the fact that plays a central role in the proof of Theorem 3.5(ii) and (v), this is why we keep it here.
The homogenization result for
The cell problem and the definition of the effective Hamiltonian
Le us first study the cell problem.
Assume (H0),(H1) and (H2). For any, there exist a unique constantsuch that the following cell problem has a Lipschitz continuous,-periodic viscosity solution
To prove Theorem 4.1 we introduce the classical ρ-problem: let , for any and (x and p are the “frozen” variables), we denote by the solution of
As in Section 3 (see also the results in [10]) it is possible to prove that there is one and only one solution of (4.2). Moreover, is characterized by being the value function of an optimal control problem. For the sake of clarity we describe all these results in Section 4.2.
The proof is classical. Thanks to the characterization (4.7), is a Lipschitz continuous, -periodic function. Moreover, as a consequence of the uniform controllability condition in (H2), the Lipschitz constant of can be chosen independent of ρ (cf. the proof of Theorem 3.2).
Let us define and . By easy and classical estimates, the are bounded and since the are equi-Lipschitz continuous and periodic, they are also equi-bounded. Therefore, we can apply Ascoli–Arzelà theorem and, up to extractions of subsequences, we may assume that converges uniformly to a Lipschitz -periodic function and converges to a constant .
Next we can use the stability result, Theorem A.4 and deduce that and satisfy (4.1).
To prove the uniqueness of the constant , we argue by contradiction, assuming that and are two different solutions of (4.1). We suppose, for instance that and, without loss of generality, we can assume that in by adding a suitable large constant to v.
Let us fix ρ small enough to have for all . Since is a solution of (4.1) we have that v is a solution of
while w is a solution of
Thus, by the comparison result, we have that in all which is a contradiction and the claim is proved. □
Study of the ρ-control problem
In this section, we describe the control problem associated to the Bellman equation (4.2) which is used in the proof of Theorem 4.1.
As in Section 3 in order to define the dynamic, we have to consider the solutions of the differential inclusion
We recall that there exist controls , , such that, for almost every , one has
with (the dependence on x and is not explicitly written)
Finally we set
Following [10], one can prove that there is one and only one solution of (4.2) which is given by
where
and
Assume (H0),(H1) and (H2). Fix. Forletbe a bounded continuous function and,be the solution of the following equation:
The result follows from the following characterization of the solutions :
Properties of
We complement this result by proving that the effective Hamiltonian fulfills the properties needed to obtain a comparison result for the limit problem.
Assume (H0),(H1) and (H2). Letbe defined in Theorem4.1.
There exists a modulussuch that for any,,
There exists a constant M such that for any, for any,
For anyand for anywe have
The proofs are classical. Proofs of (i) and (ii) are based on the comparison principle for the ρ-problem defining .
For instance, let us prove (i):
For , for any , and , we denote by the solution of (4.2) and by the solution of
Our aim is to prove that is a supersolution of (4.2). Indeed, if this is true, using the comparison principle (Theorem 3.5(iii)) with (subsolution of (4.2) satisfying ) we deduce that
Multiplying by ρ and letting , we have
and, reversing the roles of x and z, we conclude the proof.
Since the argument is completely similar we detail only the case .
Since is a solution of (4.13), by coerciveness of there exists three constants , , (depending only on the constant defined in (H0)–(H2)) such that
and
We deduce that there exists a constant K such that
Moreover, there exist a modulus w such that for any , , ,
Therefore, in the viscosity sense:
Thanks to (4.14) we have that there exist a modulus w (note that this is not exactly the same as before) such that satisfies
in the viscosity sense and (i) holds.
The coerciveness property (iii) follows from
letting . Indeed, thanks to periodicity and continuity, has a maximum point and a minimum point called respectively and .
Then .
Using only definition (2.1) of sub and supersolution, there exist and such that taking as test function in the equations satisfied by we have
this implies
and thanks to the controllability condition (H2) and the boundedness of in (H1) it is easy to remark that
The convergence result
Before proving the convergence result we state here a comparison result for the limiting problem.
Assume (H0),(H1) and (H2). Letbe defined in Theorem4.1. Let u and v be respectively a bounded usc subsolution and a bounded lsc supersolution ofThenin.
Thanks to Proposition 4.3 we are able to apply the classical comparison results for Hamilton–Jacobi–Bellman equation in (see for instance [5,6,8,22] or [23]). □
We are finally ready to prove the convergence result. More precisely.
Assume (H0),(H1) and (H2). Letbe defined in Theorem4.1. The sequenceconverges locally uniformly into a functionwhich is the unique solution of
We first remark that, in view of Theorem 3.2, the functions are bounded and Lipschitz continuous uniformly with respect to ε. Therefore, thanks to Ascoli–Arzelà’s theorem, we may assume, up to the extraction of a subsequence, that the sequence converges locally uniformly in to a bounded, Lipschitz continuous function . Because of Theorem 4.4, in order to conclude, we only need to prove that is a sub and a supersolution of
Since both proofs use the same technique we will detail only the proof of being a supersolution.
Let ϕ be a -function in and be a local strict minimum point of : we may assume without loss of generality that
Our aim is to prove that
We argue by contradiction, assuming that
Let be defined as in Theorem 4.1 and be the related unique solution of system (4.1), i.e.
There exist,,such that the functionis a viscosity subsolution of (2.14) for all, for anyand, if.
Fix and a point . Since is a viscosity solution of (4.22) we have that
in the viscosity sense. We remark now that,
thanks to the Lipschitz properties of and for and Proposition A.2 for . Moreover, since is bounded and ϕ is regular we have . Therefore, by assumption (4.21) we can deduce the following inequality in the viscosity sense
if, say, , for r and ε small enough, depending only on η. The proof of the lemma is then completed. □
Now we note that, by (4.19), if , then there exists , such that
Since the limit is uniform and is bounded on , for ε small enough, we have
Choosing now γ, r small enough ( and r small enough), we can have, at the same time,
Therefore, applying the comparison result (cf. Theorem 3.5), we obtain
and letting we have
Taking we get which is in contradiction with (4.19) and the proof is complete. □
The homogenization result for
As remarked in the Introduction, the function can only be characterized by being the maximal solution of (1.1)–(1.3)–(1.4). Therefore, we are going to prove the homogenization result by performing all the arguments on the control formulas. However, since we are closely following the ideas of pde argument’s we start again by studying the cell problem.
The cell problem
The following theorem plays the same role as Theorem 4.1 for .
Assume (H0),(H1) and (H2). For any, there exists a unique constantsuch that there exists a Lipschitz continuous, periodic functionsatisfying, for anyandwhereis defined in (4.9). Moreover,is a viscosity subsolution of
Note that, thanks to Remark 3.6 we only have to prove that is a subsolution of .
Finally, for allwe have
Note that, since
formula (5.3) can be rewritten as
Moreover, by taking the infimum on the set of all the trajectories (instead of the set of regular trajectories ) we can obtain the same characterization for .
We introduce the classical ρ-problem. Let , for any fixed and
where and the cost function is defined as in (4.8) but here the infimum is taken considering only the regular trajectories
(Recall that a control a is regular if , .)
By Theorems 3.4 and 3.5 we know that is the maximal subsolution of
Thanks to definition (5.5), is a -periodic function and since the , () are bounded, then an easy estimate proves that is bounded, uniformly in ρ. Moreover, thanks to the uniform controllability condition in (H2), as proved in Theorem 3.2, is Lipschitz continuous and its Lipschitz constant is independent of ρ.
Let us define and . Up to an extraction of a subsequence, thanks to Ascoli–Arzelà theorem, we may assume that converges uniformly in to a Lipschitz -periodic function and converges to a constant that we will denote by .
Using the stability property (cf. Theorem A.4), we have that is a subsolution of (5.2). Moreover, by the Dynamic Programming Principle for , we have, for each and each time ,
Next we use that
which yields, by subtracting
Using the uniform convergence of the sequence , it is easy to pass to the limit in this equality and to get
It is worth pointing out that here τ is arbitrary thus satisfies this property for any .
Now we want to prove the uniqueness of .
Let us suppose now that there exist two constants and such that there exist respectively two continuous periodic functions and satisfying (5.1), then using that , we obtain
Thanks to the properties of periodicity and continuity of and there exist a such that
and we can use the preceding equality with .
This leads to
and for this inequality to hold for , this clearly implies that . Exchanging the roles of and , we obtain the opposite inequality, i.e. .
We end the proof by remarking that we can deduce (5.3) by (5.1) thanks to the boundness of . □
We remark that, not only the above proof just requires that there exists a positive τ such that (5.1) holds to obtain the uniqueness of but one can also use this proof to obtain further results. For example, if is a subsolution of (5.2) associated to , one can prove that it satisfies a suboptimality principle, i.e.
Moreover, since is bounded the above argument with and leads to . This means that , where the infimum is taken on the set of subsolutions of (5.2).
Now we provide additional properties for to state that the effective Hamiltonian fulfills the properties needed to obtain a comparison result for the limiting problem as in Theorem 4.3 for .
Assume (H0),(H1) and (H2).
There exists a modulussuch that for any,,
For any, for any,
For anyand for anywe have
The proof is similar to that of Theorem 4.3 but we cannot use a comparison principle and we replace it by the use of maximal subsolutions.
For instance, let us prove (i):
Let us remember that , that here we will denote by , is the maximal subsolution of (5.6):
while is the maximal subsolution of
We want to prove that there exists a modulus w such that the function
is a subsolution of (5.13) (in the sense of Definition 2.1). If this is true , this implies that
letting we have and the proof is completed.
Let us suppose that (the other cases are analogous and even simpler), there exists (or 2) such that in the sense of viscosity:
Thanks to the coerciveness properties of there exist a constant and a constant such that and using the definition of there exists a constant such that
then there exists a constant depending only on the constants defined in (H0), (H1) and (H2) such that
This implies, using the regularity properties of and (5.14) that there exists a modulus w such that
therefore the function verify in the viscosity sense
The convergence result
We are now ready to prove the convergence result. More precisely.
Assume (H0),(H1) and (H2). Letbe defined as in Theorem5.1andas in (3.4). Then sequenceconverges locally uniformly into a continuous function, which is the unique viscosity solution of
We first remark that, in view of Theorem 3.2, the functions are equi-bounded and equi-Lipschitz continuous. Therefore, by Ascoli–Arzelà theorem, we can extract a subsequence, still denoted by , which converges locally uniformly on to a function . Our aim is then to prove that is a solution of (5.15). If this is the case, since by Theorem 5.4, (5.15) has a unique viscosity solution (because satisfies the classical assumption of classical comparison results) then the whole sequence will converge to .
Since some parts of the proof are rather technical, we split it into three steps. The first step, concerning the supersolution property for is rather similar to the analogous proof for , the principal tool is the local comparison principle stated in Theorem 3.5(v), unfortunately to prove the subsolution property for will be more difficult because a comparison principle concerning supersolutions and does not hold, in the second step we prove the subsolution property in the case when b and l do not depend on the first variable and in the last step the subsolution property is proved in the general case using a sequence of approximating problems.
( is a supersolution of (5.15)).
In this step, we follow readily the pde arguments which are already used for .
Let ϕ be a -function in and be a local strict minimum point of : we may assume without loss of generality that
Our aim is to prove that
where is defined as in Theorem 5.1. We argue by contradiction, assuming that
Let be the subsolution related to fulfilling (5.2), i.e.
Let us first prove the following lemma.
There existssmall enough such that the functionis a subsolution of (2.13) for all, ifis small enough.
Fix and a point . Since is a subsolution of (4.22) we have that
in the viscosity sense. We remark now that,
thanks to the Lipschitz properties of and for and Remark A.3 for . Moreover, since is bounded and ϕ is regular we have . Therefore, by assumption (5.18) we can deduce the following inequality in the viscosity sense
if, say, , for r and ε small enough, depending only on η. The proof of the lemma is then completed. □
Now we remark that, by (5.16), if , then there exists , such that
Since this limit is uniform and is bounded on , for ε small enough, we have
Choosing now γ, r small enough, we can have, at the same time, (i) is a subsolution of (2.13) in and (ii) in order that this function is less than on the boundary of this ball.
Therefore, applying the comparison result (cf. Theorem 3.5(v)), we obtain
and letting we have
Taking we get which is in contradiction with (5.16) and the proof of Step 1 is complete. □
( is a subsolution of (5.15) – the case when b and l do not depend on the first variable).
In this step, we write for simplicity , and since there is not dependence on the first variable. Let ϕ be a function and a local (strict) maximum point for such that , i.e.
Our aim is to prove that
where is defined as in Theorem 5.1. We argue by contradiction, assuming that
Let be the subsolution related to fulfilling (5.1), therefore we have for any and
Let . In order to write explicitly we consider, for , any regular trajectory satisfying , with . Setting , we obtain a solution of the differential inclusion
We rewrite (5.23) by using the trajectories under the form . After a scaling in time, and rewriting any control of the form , we arrive at
Our aim is now to prove that the function almost fulfills a super-optimality principle for , more precisely.
For eachandwe havewhere η is given by (5.22) and theis uniform with respect to.
Since we have for each and
choosing in (5.25)
Notice that since is bounded, (note that the is uniform in ε). In order to estimate the two last terms, we use the fact that if g is bounded, then
where the only depends on . Hence, since the trajectory is continuous, b and l are bounded and , we have
which gives the result. □
We consider now the DPP for the function at point and time
and combine it with (5.26) to get
Therefore, using again that is bounded (and can be chosen, say, less than 1), we get
We choose now in order that the trajectory belongs to for all . In particular, since , (5.20) implies that
We then pass to the limit as in (5.29) and get
which is a contradiction for small enough. Hence the proof of Step 2 is complete. □
In the above proof, the key fact was that the change we made on the trajectories lead us to (5.24) which is exactly the dynamic for the control problem which gives . On the contrary, when b depends on the first variable, this change is going to provide a dynamic instead of : if b was continuous, we could handle this difference but here it may change the times when is reached or left by the trajectories and we cannot compare the control problem for and .
In order to overcome this difficulty, we introduce a κ-sequence of problems for where, for κ fixed, the dynamics are constant with respect to the slow variable and the new trajectories can be interpreted as a subset of . For κ fixed we shall use Step 2 which provides a modified corrector , solution of a suitable ergodic problem and a modified ergodic constant . Then we conclude letting , indeed, thanks to the stability properties proved in [10] the ergodic constants are stable.
As in Step 2 let ϕ be a function and a local (strict) maximum point for such that , i.e.
Our aim is to prove (5.21), i.e.
In order to define the approximate corrector, we first introduce, for and , the sets
where
With such definitions, we can build as we built for (3.1) and we are interested in the differential inclusion
Notice that solutions of this differential inclusion are couples , but that does not depend on .
So, despite we are not exactly in the framework of Theorem 3.1, there is no difficulty to solve differential inclusion (5.31), using (H0), (H1), (H2), since the set-valued map is upper semi-continuous with convex compact images. And, as in Theorem 3.1, for each solution of (5.31) and for each function with a.e., there exists a control such that
A key remark here is that, for the associated control problem, the running cost is going to be given by and, if we fix the solution of the differential inclusion, is independent of the choice of the control. However, we have to define the set of regular trajectories as for the original problem and we say that is a regular trajectory if there exists function with a.e. such that is a regular trajectory.
By analogy with (4.9), we replace
in (4.8), for any satisfying the dynamics (5.32)–(5.33) for some controls , and .
Following the ρ-problem construction as in the proof of Theorem 5.1, we obtain a Lipschitz continuous periodic function (the approximate corrector) such that, for any
To do so, it is worth pointing out that, (H2) being uniform with respect to x and y, the trajectory is still controllable.
In order to prove (5.21), we are going to show that
and then we will prove that .
We argue by contradiction assuming that for some .
We introduce the function defined by and we first examine the properties of .
For any solution of (5.31), if we set with , this trajectory solves
Our hypotheses on the dynamics imply that for any κ, there exists such that for all and all , . By the definition of (choosing ), this implies that, for such t
This key inclusion property means that the -trajectories can be seen as particular regular -trajectories on of . Hence, using (5.32)–(5.33) in the definition of and taking the infimum on a bigger set we conclude that, for any
In the previous formula, since the integrand is bounded, we also have
Finally, considering the function , we have
and, taking in account the facts that ϕ is smooth and b is uniformly bounded, we have
We deduce from these properties that, for any and
Finally we use that
In order to conclude, we consider the DPP for the function at point and time t
Since combining with (5.37) we obtain
Arguing now exactly as in Step 2, we have the contradiction and we have proved that, for any , (5.34) holds.
The final step consists in passing to the limit as , thanks to the following lemma.
When,for any fixed, p.
For the sake of clarity we prove this lemma below and we provide now the conclusion of the proof. Passing to the limit in (5.34) we get . Hence, is a subsolution of the ergodic problem (5.15). Step 3 is then complete and combined with Step 1, the proof of Theorem 5.5 is also complete. □
Since the are equi-Lipschitz (by (H2)) and periodic, after normalizing by they are uniformly bounded. Applying Ascoli–Arzelà theorem, up to the extraction of a subsequence, we may assume that locally uniformly, for some Lipschitz periodic continuous function V (which, a priori, may not be even up to an additive constant). Similarly, we can assume that for any fixed , for some constant .
Since, for any we have
for each there exists a regular κ-trajectory such that this infimum is attained. To pass to the limit, it is clear that we have a subsequence of trajectories which converges uniformly to a trajectory of the limit problem, thanks to (H0)–(H1). However, we need the limit trajectory Y to be regular.
So, in order to use Lemma 5.3 in [10], we first remark that, if is built in the same way as , using instead of , then any solution of the -differential inclusion is a solution of the -differential inclusion since .
Thus, if the subsequence converges to , then solves the -differential inclusion. Moreover, since the trajectories are regular, there exists such that can be interpreted as a regular trajectory in , associated with a partition , being interpreted as part of the control.
The assumptions of [10, Lemma 5.3] are then fulfilled and we deduce that is a regular trajectory. This means that there exists a control (of course, there is no e playing a role at the limit) such that the limit trajectory is regular: , and we have
Hence for any
Now we prove the converse inequality. To do so we notice that the function is a subsolution of the evolution problem
where is defined as , but with the . Arguing as above we can pass to the limit as using the stability result for [10, Theorem 5.1] and we obtain that is a subsolution of
Hence, writing down the maximal (sub-)solution of this problem, we obtain that for any y and τ,
Combining both inequalities for , we get equality and the characterization of in Theorem 5.1 implies that necessarily, . □
Actually, in Step 3 we only need the inequality to conclude that is a subsolution. However, knowing that is of an independent interest.
The 1-D case: An example
In this section we present an 1-D example showing that and can be indeed different. We refer to Lions, Papanicolaou and Varadhan [24], Concordel [12,13] and Namah and Roquejoffre [25] for explicit computations of effective Hamiltonians: despite our aim is not really to compute explicitly , our arguments are inspired by these works.
We choose and . In this context, , are 2-periodic functions and from (5.1) (and its analogue for ), we have
where we have used that solves (4.5). Of course, the admissible controls are different for , .
Since we are interested in the ergodic constant, we can choose and we also consider large t. For the optimal trajectories we have two possible cases.
For some , there exists such that .
We point out that, by the Dynamic Programming Principle, this case is equivalent to for some and clearly this happens, for large t, if is large since the term is playing a more important role in the minimization process than the -one.
Recalling the periodicity of , and we obtain from (6.1)
This gives the behavior of and one can be more precise about the different terms of this equality since , , a solves the control problem
where the infimum is taken on all the trajectories such that .
For any t and , .
In that case, since are bounded and the optimal trajectory too, we divide (6.1) by t and letting t tend to we obtain
Again we insist on the fact that the infimum is taken on different sets of controls for and .
At this point, we consider the following example. Let and , and
If we consider the case then
In the -case, it is clear that the best strategy is to choose and . Indeed, while , this strategy allows to stay at 0 with a zero cost because and . Hence .
But this trajectory is a “singular” one and cannot be used in the -case. If only the “regular” strategies are allowed, then either the best strategy is to have a trajectory which stays in 0 but the controls have to satisfy and so and this would give because , . Or the best regular strategy consists in leaving 0 but in this case, we know from the Dynamic Programming Principle that the optimal trajectory has to be monotone (see [9, Section 5.1, Lemma 5.4] in particular) and we can conclude that .
Indeed, either we are in a situation which is analogue to Case 1 but we remark that, for the trajectory, the cost to go from 0 to 1 and then from 1 to 2 (or from 0 to and then from to ) is strictly positive. Or this trajectory remains bounded and therefore converges to some point in but in this case it is easy to prove that the cost is larger than 1 using that is almost constant for large s and that it is not possible to stay in with a “regular” trajectory.
Therefore, we have shown that
Footnotes
Acknowledgements
The authors were partially funded by the ANR project ANR HJnet (ANR-12-BS01-0008-01) and by the EU under the 7th Framework Programme Marie Curie Initial Training Network “FP7-PEOPLE-2010-ITN”, SADCO project, GA number 264735-SADCO.
Regularity results for the tangential Hamiltonian
In this section we can use some weaker hypothesis on b, we can replace (H0) by the following hypothesis.
In order to obtain some regularity result for the tangential Hamiltonian we prove the following useful property.
Let us now prove some regularity properties on the tangential Hamiltonian (see also Lemma 7.2 in [10]).
We are finally ready to prove the stability result.
References
1.
Y.Achdou, F.Camilli, A.Cutri and N.Tchou, Hamilton–Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications20(3) (2012), 413–445. doi:10.1007/s00030-012-0158-1.
2.
O.Alvarez and M.Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim.40(4) (2002), 1159–1188.
3.
O.Alvarez and M.Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal.170(1) (2003), 17–61.
4.
J.-P.Aubin and H.Frankowska, Set-Valued Analysis. Systems & Control: Foundations & Applications, Vol. 2, Birkhäuser, Boston, MA, 1990.
5.
M.Bardi and I.Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA, 1997.
6.
G.Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Springer, Paris, 1994.
7.
G.Barles, A short proof of the -regularity of viscosity subsolutions for superquadratic viscous Hamilton–Jacobi equations and applications, Nonlinear Anal.73(1) (2010), 31–47.
8.
G.Barles, First order Hamilton–Jacobi equations and applications, in: Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Mathematics, Vol. 2074, Springer, 2011, pp. 49–110.
9.
G.Barles, A.Briani and E.Chasseigne, A Bellman approach for two-domains optimal control problems in , ESAIM: Control, Optimisation and Calculus of Variations19(3) (2013), 710–739.
10.
G.Barles, A.Briani and E.Chasseigne, A Bellman approach for regional optimal control problems in , SIAM J. Control Optim.52 (2014), 1712–1744.
11.
G.Barles, F.Da Lio, P.L.Lions and P.E.Souganidis, Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions, Indiana Univ. Math. J.57(5) (2008), 2355–2375.
12.
M.C.Concordel, Periodic homogenization of Hamilton–Jacobi equations: Additive eigenvalues and variational formula, Indiana Univ. Math. J.45(4) (1996), 1095–1117.
13.
M.C.Concordel, Periodic homogenization of Hamilton–Jacobi equations: II: Eikonal equations, Proc. R. Soc. Edinb., Sect. A127(4) (1997), 665–689.
14.
A.Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci., Paris, Sér. I324 (1997), 1043–1046.
15.
A.Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci., Paris, Sér. I, Math.325(6) (1997), 649–652.
16.
A.Fathi, Sur la convergence du semi-groupe de Lax–Oleinik, C. R. Acad. Sci., Paris, Sér. I, Math.327(3) (1998), 267–270.
17.
A.F.Filippov, Differential equations with discontinuous right-hand side, Matematicheskii Sbornik51 (1960), 99–128; English transl. in American Mathematical Society Translations, English Translation Series 2, Vol. 42, 1964, pp. 199–231.
18.
N.Forcadel and Z.Rao, Singular perturbation of optimal control problems on multi-domains, Preprint, HAL: hal-00812846, 2013.
19.
C.Imbert and R.Monneau, Level-set convex Hamilton–Jacobi equations on networks, Preprint, HAL: hal-00832545, 2014.
20.
C.Imbert, R.Monneau and H.Zidani, A Hamilton–Jacobi approach to junction problems and applications to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations19(1) (2013), 129–166.
21.
H.Ishii, Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Faculty Sci. Eng. Chuo Univ.28 (1985), 33–77.
22.
H.Ishii, A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton–Jacobi equations, in: Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Mathematics, Vol. 2074, Springer, 2011, pp. 111–250.
23.
P.L.Lions, Generalized Solutions of Hamilton–Jacobi Equations, Research Notes in Mathematics, Vol. 69, Pitman, Boston, MA, 1982.
24.
P.L.Lions, G.Papanicolaou and S.R.S.Varadhan, Homogenization of Hamilton Jacobi Equation, Unpublished manuscript, circa 1998.
25.
G.Namah and J.-M.Roquejoffre, The “hump” effect in solid propellant combustion, Interfaces Free Bound2 (2000), 449–467.
26.
T.Wasewski, Systèmes de commande et équation au contingent, Bull. Acad. Pol. Sc.9 (1961), 151–155.
