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

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Submitted on 14 Apr 2011

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

Olivier Ley, Fabio Camilli, Paola Loreti, Vinh Duc Nguyen

To cite this version:

Olivier Ley, Fabio Camilli, Paola Loreti, Vinh Duc Nguyen. Large time behavior of systems of fi

rst-order Hamilton-Jacobi equations. SADCO Kick off, Mar 2011, Paris, France. �inria-00585692�

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Large time behavior of systems of Hamilton- Jacobi equations Olivier Ley SADCO Mar 2011

1. Scalar case 2. Control 3. Result 4. Proof

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

Olivier Ley

Institut de Recherche Math´ematique de Rennes INSA de Rennes, France

Joint work with F. Camilli (Roma), P. Loreti (Roma) and V.

Nguyen (Rennes)

SADCO Meeting, Paris, March 3-4 2011

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Time-dependent systems of HJ equations







∂u

_i

∂t + F

_i

(x, Du

_i

) +

Xm

j=1

d

_ij

(x)u

_j

= f

_i

(x)

T^N

× (0, +∞) u

_i

(x, 0) = u

_0,i

(x)

T^N

for i = 1, . . . , m.

➪

Periodic setting

➪

Linear coupling

➪

More precise assumptions later

Aim :

Study the behavior of u(x, t) = (u

₁

(x, t), . . . , u

_m

(x, t))

when t → +∞

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Plan of the talk

1

Recall of the scalar case

2

Motivations from control

3

Assumptions and a result

4

Sketch of the proof

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The scalar periodic case

(

∂u

∂t + F (x, Du) = f (x)

T^N

× (0, +∞) u (x, 0) = u

₀

(x )

T^N

A lot of works : Lions 82, Fathi 98, Namah-Roquejoffre 99,

Barles-Souganidis 00, Davini-Siconolfi 06, Ishii-Mitake 06,07,...

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Namah-Roquejoffre Thm for

^∂u_∂t

+ F ( x

, Du ) = f ( x )

Theorem.

[Namah-Roquejoffre 99]

Periodicity : F (·, p), f , u

₀

are 1-periodic continuous convexity and coercivity of F (x, ·)

F (x, p) ≥ F (x, 0) = 0

f (x) ≥ 0, A = {x ∈

T^N

: f (x) = 0} 6= ∅

regularity : |F (x, p) − F (y, p)| ≤ ω((1 + |p|)|x − y|) Then, for every u

₀

∈

Lip(T^N

), there exists

(c , v ) ∈

R

×

Lip(T^N

) such that

u(x, t) − ct → v(x) as t → +∞ uniformly in

T^N

v is solution of F (x, Dv ) = f + c in

T^N

c is the ergodic constant, unique,

c = −

min_TN

f =

lim_t→+∞

−

^u(x,t)_t

= 0

.

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Goal

Is this theorem true in the case of systems ?

The Aubry set A := {x ∈

T^N

: f (x) = 0} plays a particular

role. What is the equivalent for systems ?

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Control interpretation (scalar case)

Dynamics :

X ˙ (s) = b(X (s ),

α(s))

s ≥ 0 X (0) = x

α(s)

control, takes its value in a compact space

K

. Value function :

V (x, t) =

inf

α(·)

{

Z t

0

f (X (s ))ds + u

₀

(X (t))}

Then V is the unique viscosity solution of







∂u

∂t +

sup

α∈K

{−b(x,

α)

· Du} = f (x) u(x, 0) = u

₀

(x)

Optimal trajectories are attracted by A =

argmin

f and V (x, t)

t ∼

t→+∞

−c =

min

f .

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Control for systems : piecewise deterministic trajectories with random jumps (1)

Dynamics :

X ˙ (s) = b

_ν(t)

(X (s ), α(s )) s ≥ 0 X (0) = x

solution : (X (s),

ν(s)) with ν(s)

a Markov process with values in

{1,2, . . . ,m}

Transition probabilities :

P

(ν(t

+h)

=j |

ν(t)

= i, X (t) = x) = γ

ij

(x)h + o (h) for j 6= i.

Value function : V

i

(x, t) =

inf

α(·)

E

x,i

{

Z t

0

f

_ν(s)

(X (s ))ds + u

_0,ν(t)

(X (t))}

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Control for systems : piecewise deterministic trajectories with random jumps (2)

Then V = (V

₁

, ..., V

_m

) is the unique viscosity solution of the system







∂u

_i

∂t +

sup

α∈K

{−b

i

(x, α) · Du

_i

} +

Xm

j=1

γ

_ij

(x)(u

_i

− u

_j

) = f

_i

(x ) u

_i

(x, 0) = u

_0,i

(x)

for i = 1, . . . , m.

For instance Fleming-Zhang 98

Xm

j=1

γ

ij

(x)(u

i

− u

j

) =

Xm

j=1

d

ij

(x)u

j

with d

ii

=

X

j6=i

γ

ij

≥ 0 and d

ij

= −γ

ij

≤ 0 for i 6= j

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Assumptions on the Hamiltonian and initial conditions

The same as in Namah-Roquejoffre Theorem.

For all i = 1, ...m :

Periodicity : F

_i

(·, p), f

_i

, u

_0,i

are 1-periodic continuous convexity and coercivity of F

_i

(x, ·)

F

_i

(x, p) ≥ F

_i

(x, 0) = 0 f

i

(x) ≥ 0

regularity : |F

i

(x, p) − F

_i

(y, p)| ≤ ω((1 + |p|)|x − y |)

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Assumptions on the coupling matrix D ( x ) = ( d

_ij

( x ))

1≤i,j≤m

For all x ∈

T^N

:

d

_ii

≥ 0, d

_ij

≤ 0 for j 6= i ,

Xm

j=1

d

_ij

≥ 0.

➪

D is a M -matrix

Classical assumptions to have a monotone system

⇒ maximum principle for the evolution problem d

_ij

are periodic in x

D(x) has non zero coefficients or : is an irreducible matrix in

T^N

equivalent definitions :

(i) ∀I

(

{1, ..., m}, ∃i ∈ I and j ∈ I / s.t. d

_ij

6= 0 (ii) ∀i, j ∈ {1, ..., m}, there exists

a “chain” i = i

₀

, i

₁

, ..., i

n

= j s.t. d

i_l−1i_l

6= 0.

➪

means that the coupling is nontrivial

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Introduction of a “Aubry set” and assumptions

Let F =

\

1≤i≤m

argmin

f

_i

D =

\

1≤i≤m

{x ∈

T^N

:

Xm

j=1

d

ij

(x) =0}

We define

A = F ∩ D.

Assume that

A is non empty

all the f

_i

’s have the same minimum f (

➪

to simplify we take f = 0)

D =

T^N

(

➪

to simplify)

With these assumptions, since f

_i

≥ 0, A = F = {x ∈

T^N

:

Xm

i=1

f

_i

(x) = 0}

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A Lemma on the coupling matrix

Lemma.

Suppose that :

D(x) is an irreducible M -matrix on

T^N Xm

j=1

d

ij

= 0 (i.e., D =

T^N

) Then, for all x ∈

T^N

, :

D(x) is degenerate of rank m − 1

the kernel of D(x) is spanned with (1, ..., 1)

there exists a positive function Λ :

T^N

→

R^m

such that D(x)

^T

Λ(x) = 0 (i.e.,

Xm

i=1

Λ

i

(x)d

ij

= 0) [Proof : Perron-Froebenius+continuous dependence]

For simplicity, we suppose that Λ(x) = (1, ..., 1)

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A result

Theorem.

Under the previous assumptions,

for every u

₀

= (u

_0,1

, ..., u

_0,m

) ∈

Lip(T^N

), there exists ((c

₁

, ..., c

m

), (v

₁

, ..., v

m

)) ∈

R^m

×

Lip(T^N

) such that

u(x, t) − ct → v(x) as t → +∞ uniformly in

T^N

v is solution of the stationary system

F

_i

(x, Dv

_i

) +

Xm

j=1

d

_ij

(x)v

_j

= f

_i

+ c

_i in T^N

, i = 1, . . . , m c is in the kernel of D(x) so c = (c

1

, ..., c

₁

), with c

₁

= −f =

lim_t→+∞

−

^uⁱ^(x,t)_t

= 0 for all i .

v

i

= v

j

on A.

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Comparison result for the stationary system

F

_i

(x, Dv

_i

) +

Xm

j=1

d

_ij

(x)v

_j

= f

_i inT^N

, i = 1, . . . , m

Theorem.

Let u be a bounded subsolution and v a bounded supersolution.

(Classical case) If, for all i ,

Xm

j=1

d

_ij

> 0 (D = ∅) then u ≤ v on

T^N

.

(Degenerate case) If

Xm

j=1

u

_j

≤

Xm

j=1

v

_j

on A then u ≤ v

➪

A may be empty

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Existence & uniqueness with prescribed datas on A

Theorem.

(Classical case) there exists a unique viscosity solution to the stationary system.

(Degenerate case) For all g continuous on A with compatibility conditions (see Fathi-Siconolfi 05,

Ishii-Mitake 07), there exists a unique viscosity v solution such that v = g on A.

➪

A is a uniqueness set

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Idea of the proof of the comparison theorem (1)

Let 0 < µ < 1 M :=

sup

1≤i≤m

sup

T^N

{µu

i

− v

_i

} is achieved at x.

Let I = {i : max is achieved for index i}

➪Case 1 : I ={1, ...,m} andx∈ A.

X

i

u

_i

(x) ≤

X

i

v

_i

(x) and (µu

_i

− v

_i

)(x) = M for all i

⇒ mM =

X

i

(µu

_i

− v

_i

)(x) ≤ (1 − µ)m|u|

∞

⇒ M ≤ 0 when µ → 1

(19)

Idea of the proof of the comparison theorem (2)

➪Case 2 : I ={1, ...,m} andx6∈ A.

⇒ ∃i s.t. f

i

(x) > 0

Subsolution and supersolutions inequalities for equation

i

: µF

_i

(x,

^Dµu_µⁱ^(x)

) +

P

j

d

_ij

µu

_j

(x) ≤ µf

_i

(x) F

i

(x, Dv

i

(x)) +

P

j

d

ij

v

j

(x) ≥ f

i

(x)

Therefore µF

i

(x, Dµu

i

(x)

µ )−F

i

(x, Dv

_i

(x))

| {z }

≥0 (convexity and max.point)

+

X

j

d

_ij

(µ u

_j

−v

j

)(x)

| {z }

=(P

jdij)M

≤ (µ−1) f

_i

(x)

| {z }

<0

(20)

Idea of the proof of the comparison theorem (3)

➪Case 3 : I 6={1, ...,m}

D(x ) irreducible ⇒ ∃i ∈ I,

k

6∈ I s.t. d

_ik

(x) < 0 Equation for

i

(as in case 2) leads to

P

j

d

_ij

(µu

_j

− v

_j

)(x) ≤ (µ − 1)f

_i

(x)

But

k

6∈ I ⇒ (µu

_k

− v

_k

)(x) ≤ M −

δ, δ

> 0 Therefore (

P

j

d

_ij

)M −

δd_ik

(x) ≤ (µ − 1)f

_i

≤ 0

contradiction

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Ergodic problem (1)

λv_i^λ

+ F

_i

(x, Dv

_i^λ

) +

Xm

j=1

d

_ij

(x)v

_j^λ

= f

_i inT^N

, i = 1, . . . , m

Theorem.

There exists a unique solution which is Lipschitz continuous with constant L independent of λ.

Up to extract, as λ → 0,

λv^λ

→ −(c

1

, ..., c

_m

) = −(c

1

, ..., c

₁

) ∈

ker

D v

^λ

− v

^λ

(x

^∗

) → v ∈

Lip(T^N

)

and (c, v) is solution of F

_i

(x, Dv

_i

) +

Xm

j=1

d

_ij

(x)v

j

= f

_i

+

c₁ inT^N

, i = 1, . . . , m

➪

In fact −c

1

=

min

f

_i

= f = 0 here

(22)

Ergodic problem (2)

In a more general context

X

i

min

f

_i

≤ −mc

1

≤

min X

i

f

_i

(23)

Proof of the convergence theorem u ( x

, t ) − ct → v ( x ) as t → +∞ (1)

Recall that −c

1

= −min f

_i

= 0

Comparison for the evolution problem : ∃C ≥ |u

0

|

∞

s.t.

v(x) − C ≤ u (x, t) ≤ v (x) + C

The function u

^ε

(x, t) = u(x,

^t_ε

) is solution to the system





 ε

∂u

i

∂t + F

i

(x, Du

i

) +

Xm

j=1

d

ij

(x)u

j

= f

i

(x)

T^N

× (0, +∞) u

_i

(x, 0) = u

_0,i

(x)

T^N

for i = 1, . . . , m.

Then, by stability, the half-relaxed limits u(x) =

lim sup

ε→0

∗

u

^ε

(x, t)

and

u(x) =

lim inf

ε→0 ∗

u

^ε

(x, t)

are respectively sub and supersolutions to the stationary

system.

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Proof of the convergence theorem u ( x

, t ) − ct → v ( x ) as t → +∞ (2)

Summing the evolution equations for 1 ≤ i ≤ m,

∂

∂t (

X

i

u

_i

) +

X

i

F

_i

(x , Du

_i

)

| {z }

≥0

+

X

i

Xm

j=1

d

_ij

(x)u

_j

| {z }

=P

jujPm

i=1dij(x)=0

=

X

i

f

_i

(x)

But

P

i

f

_i

(x) = 0

onA

Therefore

∂

∂t

(

P

i

u

_i

) ≤ 0 ⇒

X

i

u

_i

(·, t) →

t→+∞φ

uniformly

on A

(25)

Proof of the convergence theorem u ( x

, t ) − ct → v ( x ) as t → +∞ (3)

Lemma.

u

i

= u

_i

= u

j

= u

_j onA

Therefore

X

i

u

_i

=

X

i

u

_i

=

lim

t→+∞

X

i

u

_i

(·, t) =

φ on A

Comparison theorem for the subsolution u and the supersolution u implies

u = u

on T^N

.