We show a large time behavior result for class of weakly coupled systems of
first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE
approach to extend the convergence result proved by Namah and Roquejoffre
(1999) in the scalar case. Our proof is based on new comparison, existence and
regularity results for systems. An interpretation of the solution of the system
in terms of an optimal control problem with switching is given

Duc Nguyen

Fabio Camilli

Jacobi Equations

Olivier Ley

Paola Loreti

Vinh

English

arXiv

International audienceWe show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given

Camilli, Fabio

Ley, Olivier

Loreti, Paola

Nguyen, Vinh Duc

HAL-Rennes 1

Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

Name not available

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (Commun. Partial. Differ. Equ. 24(5-6):883-893, 1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given. © 2011 Springer Basel AG

CAMILLI, FABIO

LORETI, Paola

Vinh Duc Nguyen

Archivio della ricerca- Università di Roma La Sapienza

Archive Ouverte a LUniversite Lyon 2

International audience1 Recall of the scalar case 2 Motivations from control 3 Assumptions and a result 4 Sketch of the proo

Large time behavior of systems of fi rst-orderHamilton-Jacobi equationsOlivier Ley, Fabio Camilli, Paola Loreti, Vinh NguyenTo cite this version:Olivier Ley, Fabio Camilli, Paola Loreti, Vinh Nguyen. Large time behavior of systems offi rst-order Hamilton-Jacobi equations. SADCO Kick off, Mar 2011, Paris, France. <inria-00585692>HAL Id: inria-00585692https://hal.inria.fr/inria-00585692Submitted on 14 Apr 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofLarge time behavior of systems offirst-order Hamilton-Jacobi equationsOlivier LeyInstitut de Recherche Mathe´matique de RennesINSA de Rennes, FranceJoint work with F. Camilli (Roma), P. Loreti (Roma) and V.Nguyen (Rennes)SADCO Meeting, Paris, March 3-4 2011Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofTime-dependent systems of HJ equations∂ui∂t+ Fi (x ,Dui ) +m∑j=1dij(x)uj = fi (x) TN × (0,+∞)ui (x , 0) = u0,i (x) TNfor i = 1, . . . ,m.➪ Periodic setting➪ Linear coupling➪ More precise assumptions laterAim : Study the behavior of u(x , t) = (u1(x , t), . . . , um(x , t))when t → +∞Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofPlan of the talk1 Recall of the scalar case2 Motivations from control3 Assumptions and a result4 Sketch of the proofLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofThe scalar periodic case{∂u∂t+ F (x ,Du) = f (x) TN × (0,+∞)u(x , 0) = u0(x) TNA lot of works : Lions 82, Fathi 98, Namah-Roquejoffre 99,Barles-Souganidis 00, Davini-Siconolfi 06, Ishii-Mitake 06,07,...Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofNamah-Roquejoffre Thm for ∂u∂t + F (x ,Du) = f (x)Theorem. [Namah-Roquejoffre 99]Periodicity : F (·, p), f , u0 are 1-periodic continuousconvexity and coercivity of F (x , ·)F (x , p) ≥ F (x , 0) = 0f (x) ≥ 0, A = {x ∈ TN : f (x) = 0} 6= ∅regularity : |F (x , p)− F (y , p)| ≤ ω((1 + |p|)|x − y |)Then, for every u0 ∈ Lip(TN), there exists(c , v) ∈ R× Lip(TN) such thatu(x , t)− ct → v(x) as t → +∞ uniformly in TNv is solution of F (x ,Dv) = f + c in TNc is the ergodic constant, unique,c = −minTN f = limt→+∞−u(x ,t)t= 0.Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofGoalIs this theorem true in the case of systems ?The Aubry set A := {x ∈ TN : f (x) = 0} plays a particularrole. What is the equivalent for systems ?Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofControl interpretation (scalar case)Dynamics :{X˙ (s) = b(X (s), α(s)) s ≥ 0X (0) = xα(s) control, takes its value in a compact space K .Value function :V (x , t) = infα(·){∫ t0f (X (s))ds + u0(X (t))}Then V is the unique viscosity solution of∂u∂t+ supα∈K{−b(x , α) · Du} = f (x)u(x , 0) = u0(x)Optimal trajectories are attracted by A = argmin f andV (x , t)t∼t→+∞−c = min f .Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofControl for systems : piecewise deterministictrajectories with random jumps (1)Dynamics :{X˙ (s) = bν(t)(X (s), α(s)) s ≥ 0X (0) = xsolution : (X (s), ν(s)) with ν(s) a Markov process with valuesin {1, 2, . . . ,m}Transition probabilities :P (ν(t + h)= j | ν(t)= i ,X (t)=x) = γij(x)h + o(h)for j 6= i .Value function :Vi (x , t) = infα(·)Ex ,i{∫ t0fν(s)(X (s))ds + u0,ν(t)(X (t))}Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofControl for systems : piecewise deterministictrajectories with random jumps (2)Then V = (V1, ...,Vm) is the unique viscosity solution of thesystem∂ui∂t+ supα∈K{−bi (x , α) · Dui}+m∑j=1γij(x)(ui − uj) = fi (x)ui (x , 0) = u0,i (x)for i = 1, . . . ,m.For instance Fleming-Zhang 98m∑j=1γij(x)(ui − uj) =m∑j=1dij(x)ujwith dii =∑j 6=iγij ≥ 0 and dij = −γij ≤ 0 for i 6= jLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofAssumptions on the Hamiltonian and initialconditionsThe same as in Namah-Roquejoffre Theorem.For all i = 1, ...m :Periodicity : Fi (·, p), fi , u0,i are 1-periodic continuousconvexity and coercivity of Fi (x , ·)Fi (x , p) ≥ Fi (x , 0) = 0fi (x) ≥ 0regularity : |Fi (x , p)− Fi (y , p)| ≤ ω((1 + |p|)|x − y |)Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofAssumptions on the coupling matrixD(x) = (dij(x))1≤i ,j≤mFor all x ∈ TN :dii ≥ 0, dij ≤ 0 for j 6= i ,m∑j=1dij ≥ 0.➪ D is a M-matrixClassical assumptions to have a monotone system⇒ maximum principle for the evolution problemdij are periodic in xD(x) has non zero coefficients or :is an irreducible matrix in TNequivalent definitions :(i) ∀I ( {1, ...,m}, ∃i ∈ I and j /∈ I s.t. dij 6= 0(ii) ∀i , j ∈ {1, ...,m}, there existsa “chain” i = i0, i1, ..., in = j s.t. dil−1il 6= 0.➪ means that the coupling is nontrivialLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofIntroduction of a “Aubry set” and assumptionsLet F =⋂1≤i≤margmin fiD =⋂1≤i≤m{x ∈ TN :m∑j=1dij(x) =0}We defineA = F ∩ D.Assume thatA is non emptyall the fi ’s have the same minimum f(➪ to simplify we take f = 0)D = TN (➪ to simplify)With these assumptions, since fi ≥ 0,A = F = {x ∈ TN :m∑i=1fi (x) = 0}Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofA Lemma on the coupling matrixLemma. Suppose that :D(x) is an irreducible M-matrix on TNm∑j=1dij = 0 (i.e., D = TN)Then, for all x ∈ TN , :D(x) is degenerate of rank m − 1the kernel of D(x) is spanned with (1, ..., 1)there exists a positive function Λ : TN → Rm such thatD(x)TΛ(x) = 0 (i.e.,m∑i=1Λi (x)dij = 0)[Proof : Perron-Froebenius+continuous dependence]For simplicity, we suppose that Λ(x) = (1, ..., 1)Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofA resultTheorem. Under the previous assumptions,for every u0 = (u0,1, ..., u0,m) ∈ Lip(TN), there exists((c1, ..., cm), (v1, ..., vm)) ∈ Rm × Lip(TN) such thatu(x , t)− ct → v(x) as t → +∞ uniformly in TNv is solution of the stationary systemFi (x ,Dvi ) +m∑j=1dij(x)vj = fi + ci in TN , i = 1, . . . ,mc is in the kernel of D(x) so c = (c1, ..., c1), withc1 = −f = limt→+∞−ui (x ,t)t= 0 for all i .vi = vj on A.Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofComparison result for the stationary systemFi (x ,Dvi ) +m∑j=1dij(x)vj = fi in TN , i = 1, . . . ,mTheorem. Let u be a bounded subsolution and v a boundedsupersolution.(Classical case) If, for all i ,m∑j=1dij > 0 (D = ∅)then u ≤ v on TN .(Degenerate case) Ifm∑j=1uj ≤m∑j=1vj on Athen u ≤ v➪ A may be emptyLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofExistence & uniqueness with prescribed datas on ATheorem.(Classical case) there exists a unique viscosity solution tothe stationary system.(Degenerate case) For all g continuous on A withcompatibility conditions (see Fathi-Siconolfi 05,Ishii-Mitake 07), there exists a unique viscosity v solutionsuch that v = g on A.➪ A is a uniqueness setLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofIdea of the proof of the comparison theorem (1)Let 0 < µ < 1M := sup1≤i≤msupTN{µui − vi} is achieved at x .Let I = {i : max is achieved for index i}➪ Case 1 : I = {1, ...,m} and x ∈ A.∑iui (x) ≤∑ivi (x) and (µui − vi )(x) = M for all i⇒ mM =∑i(µui − vi )(x) ≤ (1− µ)m|u|∞⇒ M ≤ 0 when µ→ 1Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofIdea of the proof of the comparison theorem (2)➪ Case 2 : I = {1, ...,m} and x 6∈ A.⇒ ∃i s.t. fi (x) > 0Subsolution and supersolutions inequalities for equation i :µFi (x ,Dµui (x)µ) +∑j dijµuj(x) ≤ µfi (x)Fi (x ,Dvi (x)) +∑j dijvj(x) ≥ fi (x)ThereforeµFi (x ,Dµui (x)µ)−Fi (x ,Dvi (x))︸ ︷︷ ︸≥0 (convexity and max.point)+∑jdij(µuj−vj)(x)︸ ︷︷ ︸=(∑j dij )M≤ (µ−1)fi (x)︸ ︷︷ ︸<0Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofIdea of the proof of the comparison theorem (3)➪ Case 3 : I 6= {1, ...,m}D(x) irreducible ⇒ ∃i ∈ I, k 6∈ I s.t. dik(x) < 0Equation for i (as in case 2) leads to∑j di j(µuj − vj)(x) ≤ (µ− 1)fi (x)But k 6∈ I ⇒ (µuk − vk)(x) ≤ M − δ, δ > 0Therefore (∑j di j)M − δdik(x) ≤ (µ− 1)fi ≤ 0contradictionLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofErgodic problem (1)λvλi + Fi (x ,Dvλi ) +m∑j=1dij(x)vλj = fi in TN , i = 1, . . . ,mTheorem. There exists a unique solution which is Lipschitzcontinuous with constant L independent of λ.Up to extract, as λ→ 0,λvλ → −(c1, ..., cm) = −(c1, ..., c1) ∈ kerDvλ − vλ(x∗) → v ∈ Lip(TN)and (c , v) is solution ofFi (x ,Dvi ) +m∑j=1dij(x)vj = fi + c1 in TN , i = 1, . . . ,m➪ In fact −c1 = min fi = f= 0 hereLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofErgodic problem (2)In a more general context∑imin fi ≤ −mc1 ≤ min∑ifiLarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofProof of the convergence theoremu(x , t)− ct → v(x) as t → +∞ (1)Recall that −c1 = −min fi = 0Comparison for the evolution problem : ∃C ≥ |u0|∞ s.t.v(x)− C ≤ u(x , t) ≤ v(x) + CThe function uε(x , t) = u(x , tε) is solution to the systemε∂ui∂t+ Fi (x ,Dui ) +m∑j=1dij(x)uj = fi (x) TN × (0,+∞)ui (x , 0) = u0,i (x) TNfor i = 1, . . . ,m.Then, by stability, the half-relaxed limitsu(x) = lim supε→0∗uε(x , t) and u(x) = lim infε→0 ∗uε(x , t)are respectively sub and supersolutions to the stationarysystem.Large timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofProof of the convergence theoremu(x , t)− ct → v(x) as t → +∞ (2)Summing the evolution equations for 1 ≤ i ≤ m,∂∂t(∑iui ) +∑iFi (x ,Dui )︸ ︷︷ ︸≥0+∑im∑j=1dij(x)uj︸ ︷︷ ︸=∑j uj∑mi=1 dij (x)=0=∑ifi (x)But∑i fi (x) = 0 on ATherefore∂∂t(∑i ui ) ≤ 0 ⇒∑iui (·, t) →t→+∞φ uniformly on ALarge timebehavior ofsystems ofHamilton-JacobiequationsOlivier LeySADCOMar 20111. Scalar case2. Control3. Result4. ProofProof of the convergence theoremu(x , t)− ct → v(x) as t → +∞ (3)Lemma. ui = ui = uj = uj on ATherefore∑iui =∑iui = limt→+∞∑iui (·, t) = φ on AComparison theorem for the subsolution u and thesupersolution u impliesu = u on TN .

Large time behavior of systems of fi rst-order Hamilton-Jacobi equations

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Large time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations

https://link.springer.com/content/pdf/10.1007%2Fs00030-011-0149-7.pdf

Large time behavior of weakly coupled systems of first-order
  Hamilton-Jacobi equations

Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

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