Optimal control of Hamilton-Jacobi

Transcription

Introduction
Main results
Proofs
Further results
Optimal control of Hamilton-Jacobi-Bellman
equations
P. Jameson Graber
Commands (ENSTA ParisTech, INRIA Saclay)
17 January, 2014
P. Jameson Graber
Optimal control of Hamilton-Jacobi-Bellman equations
Introduction
Main results
Proofs
Further results
Motivation
Background
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
Existence of minimizers
Characterization of minimizers
3
Proofs
Duality
Equivalence of original and relaxed problems
A priori bounds
Passing to the limit
Existence of weak solutions
Uniqueness of weak solutions
4
Further results
Well-posedness of weak solutions
Long time average behavior
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Objective
We consider an optimal control problem:
Problem
(1) For a given cost functional K and finite measure dm0 , find
(Z Z
)
Z
T
inf
K (t, x, f (t, x))dxdt − u(0, x)dm0 (x)
0
over functions u, f such that f ∈ C ([0, T ] × TN ) and u solves
−∂t u(t, x) + H(x, Du(t, x)) = f (t, x),
u(T , x) = uT (x).
(2) Find a PDE characterization of (weak) minimizers.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Motivation
Purpose: to steer an evolving set.
Model: given a set A of admissible controls and a given region
ΩT ⊂ RN , let Ωt be the backward reachable set at time t defined by
Ωt := {y (t) : ẏ = c(y , α), α ∈ A, y (T ) ∈ ΩT }.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Model
Strategy: use obstacles.
Obstacle construction: Let f ≥ 0 be continuous. Define Kτ , τ ∈ [0, T ] by
Kτ := {x : f (τ, x) > 0}.
Consider the modified “blocked” evolving set
Ω̃t = {y (t) : ẏ = c(y , α), α ∈ A, y (T ) ∈ ΩT , y (τ ) ∈
/ Kτ ∀ τ ∈ [t, T ]}.
Following [Bokanowski, Forcadel, Zidani 2010], we have a
characterization: let uT ≥ 0 (continuous) be such that uT = 0 precisely
on ΩT .
Z T
Ω̃t = {y (t) : uT (y (T )) +
f (s, y (s)) ≤ 0, ẏ = c(y , α), α ∈ A}.
t
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Model
Ω̃t = {u(t, ·) ≤ 0} characterized by an optimal control problem:
(
)
Z T
u(t, x) := inf uT (y (T )) +
f (s, y (s))ds : ẏ = c(y , α), α ∈ A, y (t) = x .
t
u solves a Hamilton-Jacobi-Bellman equation:
−∂t u(t, x) + H(x, Du(t, x)) = f (t, x),
u(T , x) = uT (x).
Conclusion
Steering the front by obstacles is thus related to optimizing solutions to
HJB equations.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Related model
A related open problem:
Optimal control via velocity
Given a cost functional C, find
Z
inf C(c) − u(T , x)dm0 (x) ,
where the infimum is taken over u, c satisfying
ut + c(t, x)|Du| = 0, u(0, x) = u0 (x).
Find a characterization of the minimizers.
Very little appears to be known.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Mass transportation
In 2000, Benamou and Brenier considered a fluid mechanics formulation
of the Monge-Kantorovich problem (MKP).
For ρ0 (x) ≥ 0 and ρT (x) ≥ 0 probability densities on Rd , we say
M : Rd → Rd transfers ρ0 to ρT if ∀A ⊂ Rd bounded,
Z
Z
ρT (x)dx =
ρ0 (x)dx.
x∈A
M(x)∈A
The Lp MKP is to find
Z
p
dp (ρ0 , ρT ) = inf
M
|M(x) − x|p ρ0 (x)dx.
dp is called the Lp Kantorovich (or Wasserstein) distance.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
PDE characterization of MKP
Theorem (Benamou-Brenier, 2000)
The square of the L2 Kantorovich distance is equal to the infimum of
Z
T
Z
ρ(t, x)|v (t, x)|2 dxdt,
T
Rd
0
among all (ρ, v ) satisfying
∂t ρ + ∇ · (ρv ) = 0
for 0 < t < T and x ∈ Rd , and the initial-final condition
ρ(0, ·) = ρ0 , ρ(T , ·) = ρT .
The optimality conditions are given by v (t, x) = ∇φ(t, x), where
∂t φ +
1
|∇φ|2 = 0.
2
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Dolbeault-Nazaret-Savaré distances
In 2009, Dolbeault, Nazaret, and Savaré, building on Benamou-Brenier,
introduced a new class of transport distances between measures, given by
minimizing
Z 1Z
|v (t, x)|2 m(ρ(t, x))dxdt
0
Rd
subject to
∂t + ∇ · (m(ρ)v ) = 0, ρ(0, ·) = ρ0 , ρ(1, ·) = ρ1 .
In 2013, Cardaliaguet, Carlier, and Nazaret studied geodesics for this
class of distances when m(ρ) = ρα , getting optimality conditions

 ∂t ρ + ∇ · ( 21 ρα ∇φ) = 0,
ρ > 0 ⇒ ∂t φ + α4 ρα−1 |∇φ|2 = 0,

ρ ≥ 0, ∂t φ ≤ 0, ρ(0, ·) = ρ0 , ρ(1, ·) = ρ1 .
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Motivation
Background
Mean Field Games
In 2007, Lasry and Lions give an overview of the following system of
equations:
∂u
∂t
∂m
∂t
− ν∆u + H(x,
∇u) = V [m]
in Q × (0, T )
+ ν∆m + ∇ · ∂H
∂p (x, ∇u)m = 0 in Q × (0, T )
u|t=0 = V0 [m(0)] on Q, m|t=T = m0 on Q.
There are two characterizations of the system:
1
asymptotic behavior of differential games with large number of
players,
2
optimal control of Fokker-Planck equation, or of Hamilton-Jacobi
equation (by duality).
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Assumptions
Hamiltonian structure. H = H(x, p) is Lipschitz in the first variable:
|H(x, p) − H(y , p)| ≤ L|x − y ||p|.
H(x, ·) is subadditive and positively homogeneous, hence convex, with
linear bounds
c0 |p| ≤ H(x, p) ≤ c1 |p|.
H(x, p) := sup{−c(x, a) · p : a ∈ A}
Cost function. k = k(t, x, m) continuous and strictly increasing with
1
|m|q−1 − C ≤ k(t, x, m) ≤ C |m|q−1 + C .
C
K ∗ (t, x, ·) a primitive of k, K (t, x, ·) its Fenchel conjugate. We will
assume that K (t, x, 0) = 0 and that K (t, x, f ) is increasing in f for f ≥ 0.
Note that
1 p
|f | − C ≤ K (t, x, f ) ≤ C |f |p + C .
C
Final-initial conditions. uT : TN → R is Lipschitz, m0 ∈ L∞ (TN )
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Comparison with the literature
1
This is the first study with linearly bounded Hamiltonian.
2
In [Cardaliaguet, 2013] the Hamiltonian is supposed to satisfy
1
C
|p|r − C2 ≤ H(x, p) ≤ |p|r + C
rC
r
r
for some r > N(q − 1) ∨ 1 and, for some θ ∈ [0, N+1
),
|H(x, p) − H(y , p)| ≤ C |x − y |(|p| ∨ 1)θ .
The Hamiltonian H(x, p) = c(x)|p|r is forbidden by this restriction.
By contrast, H(x, p) = c(x)|p| is permitted under our assumptions.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Existence theorem
Relaxed set
The set K̃ will be defined as the set of all pairs (u, f ) ∈ BV × Lp such
that u ∈ L∞ , u(T , ·) = uT in the sense of traces, and
−∂t u + H(x, Du) ≤ f in the sense of measures, by which we mean that
f − H(x, Du) + ∂t u is a non-negative Radon measure.
Relaxed problem: Find inf (u,f )∈K̃ A(u, f ).
Lemma
The relaxed problem has the same infimum as the original.
Theorem (PJG)
There exists a minimizer to the relaxed problem.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Heuristically, use Lagrange multipliers to get system (MFG)
−∂t u + H(x, Du) = k(t, x, m(t, x))
u(T , x) = uT (x)
∂t m − div (m∂p H(x, Du)) = 0
m(0, x) = m0 (x)
(state equation)
(adjoint equation)
which characterizes minimizers (u, f ) of the relaxed problem (the optimal
choice is f = k(t, x, m)).
Problems:
H not differentiable,
weak solutions otherwise hard to define.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Definition of weak solutions
A pair (u, m) ∈ BV ((0, T ) × TN ) × Lq ((0, T ) × TN ) is called a weak
solution to the MFG system if it satisfies the following conditions.
1
u satisfies
Z
Z
Z tZ
u(0)m0 dx =
u(t)m(t)dx +
k(s, x, m)mdxds,
TN
TN
Z
Z
u(t)m(t)dx =
TN
2
Z
0
T
TN
Z
uT m(T )dx +
TN
k(s, x, m)mdxds,
t
TN
for almost all t ∈ [0, T ]. We also have
−∂t u + H(x, Du) ≤ k(t, x, m) in the sense of measures.
Moreover, u(T , ·) = uT in the sense of traces
m satisfies the continuity equation
∂t m + div (mv) = 0
in (0, T ) × TN ,
m(0) = m0
in the sense of distributions, where v ∈ L∞ ((0, T ) × TN ; RN ) is a
vector field such that v(t, x) ∈ c(x, A) a.e.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
Well-posedness
Theorem (PJG)
(i) There exists a weak solution (u, m) of (MFG). Moreover, if u ∈ W 1,p ,
then the Hamilton-Jacobi equation
−∂t u − v · Du = f
holds pointwise almost everywhere in {m > 0}, where v is the bounded
vector field appearing in the definition.
(ii) If (u, m) and (u 0 , m0 ) are both weak solutions (MFG), then m = m0
almost everywhere while u = u 0 almost everywhere in the set {m > 0}.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Smooth problem
Denote by K0 the set of maps u ∈ C 1 ([0, T ] × TN ) such that
u(T , x) = uT (x). The “smooth problem” is to compute the infimum
over K0 of the functional
Z TZ
Z
A(u) =
K (t, x, −∂t u + H(x, Du))dxdt −
u(0, x)dm0 (x).
0
TN
TN
Lemma
The “smooth problem” is equivalent to the original problem, i.e.
inf A(u) =
u∈K0
inf
f ∈C ([0,T ]×TN )
J (f ).
Proof: uses the superposition principle.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Dual problem
Define K1 to be the set of all pairs
(m, w) ∈ L1 ((0, T ) × TN ) × L1 ((0, T ) × TN ; RN ) such that m ≥ 0 almost
everywhere, w(t, x) ∈ m(t, x)c(x, A) almost everywhere, and
∂t m + div (w) = 0
m(0, ·) = m0 (·)
in the sense of distributions. The “dual problem” is to compute the
infimum over K1 of
Z
Z TZ
B(m, w) =
uT (x)m(T , x)dx +
K ∗ (t, x, m(t, x))dxdt.
TN
0
P. Jameson Graber
TN
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Proof of duality
Lemma
The smooth and dual problems really are in duality, i.e.
inf A(u) = −
u∈K0
min
B(m, w).
(m,w)∈K1
Moreover, the minimum on the right hand side is achieved by a pair
(m, w) ∈ K1 , of which m is unique, and which must satisfy the following:
(t, x) 7→ K ∗ (t, x, m(t, x)) ∈ L1 ((0, T ) × TN ) and m ∈ Lq ((0, T ) × TN ).
Proof is an application of the Fenchel-Rockafellar duality Theorem. Let
X := C 1 ([0, T ] × TN ; R),
Y := C ([0, T ] × TN ; R) × C ([0, T ] × TN ; RN ).
Let Λ : X → Y be given by Λ(u) = (∂t u, Du).
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Proof of duality (cont)
Lemma
inf u∈K0 A(u) = − min(m,w)∈K1 B(m, w).
Then define the functionals F : X → R and G : Y → R by
R
− TN u(0, x)dm0 (x)
if
u(T , ·) = uT (·)
F (u) =
∞
otherwise
RT R
G (a, b) = 0 TN K (t, x, −a + H(x, b))dxdt
One can show the lemma is equivalent to
max −F ∗ (Λ∗ (m, w)) − G ∗ (−(m, w))
(m,w)∈Y 0
=−
min
(m,w)∈Y 0
F ∗ (Λ∗ (m, w)) + G ∗ (−(m, w)).
Apply Fenchel-Rockafellar to finish.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Original and relaxed problems equivalent
Below we prove the following:
Proposition
A(u, f ) = −
inf
min
B(m, w).
(m,w)∈K1
(u,f )∈K̃
Equivalently,
inf
A(u, f ) = inf A(u).
(u,f )∈K̃
u∈K0
First part of the proof:
Since for u ∈ K0 we have that (u, −ut + H(x, Du)) ∈ K̃, it follows that
inf (u,f )∈K̃ A(u, f ) ≤ inf u∈K0 A(u).
It remains to show other inequality.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
A technical lemma
Lemma
For u ∈ BV and f an integrable function, the following are equivalent:
−∂t u + H(x, Du) ≤ f in the sense of measures, and
for every continuous vector field ṽ = ṽ(t, x) ∈ c(x, A) we have
Z
T
Z
Z
T
Z
φ∂t u + φṽ · Du ≤
−
0
TN
f φdtdx.
0
TN
for every continuous function φ : [0, T ] × TN → [0, ∞).
Remark: the crucial thing is that we only need to consider ṽ continuous.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Proof of technical lemma
Lemma
ZZ
−∂t u + H(x, Du) ≤ f ⇔ −
ZZ
φ(∂t u + ṽ · Du) ≤
f φ ∀ṽ cont.
One direction is obvious. For the other direction, let > 0. By Lusin’s
Theorem, there is a compact set K with |Du|([0, T ] × TN \ K ) ≤ and
Du/|Du| continuous on K . By continuity and convexity of c(·, A) we
construct ṽ(t, x) ∈ c(x, A) by partition of unity so that
−ṽ · Du/|Du| ≥ H(x, Du/|Du|) − . Use this to obtain the estimate
Z
T
Z
Z
T
Z
φH(x, Du) ≤ C +
0
TN
φṽ · Du.
0
TN
Then let → 0.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Key lemma
Lemma
Suppose u ∈ BV ∩ L∞ satisfies −∂t u + H(x, Du) ≤ f in the sense of
measures. Then for any m ∈ Lq satisfying the continuity equation
∂t m + div(mv) = 0 for some vector field v with v(t, x) ∈ c(x, A) a.e., we
have
Z
Z
Z tZ
u(0)m(0)dx ≤
u(t)m(t)dx +
fmdxds,
TN
TN
Z
Z
u(t)m(t)dx ≤
TN
Z
0
T
TN
Z
uT m(T )dx +
TN
fmdxds,
t
TN
for almost every t ∈ (0, T ).
Proof: obtain smooth approximations of m and mv through time scaling
and convolution, apply previous lemma and integration by parts.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Proof of equivalence
Second half of the proof: Let (m, w) be a minimizer of the dual problem.
It suffices to show that for (u, f ) ∈ K̃, we have A(u, f ) ≥ −B(m, w).
This essentially follows from the last lemma. Indeed, we have
Z
T
Z
Z
A(u, f ) =
K (t, x, f (t, x))dxdt −
Z
u(0, x)m0 (x)dx
TN
0
T
TN
Z
Z
∗
≥
fm − K (t, x, m)dxdt −
0
Z
TN
T Z
≥−
0
u(0, x)m0 (x)dx
TN
K ∗ (t, x, m)dxdt −
TN
Z
uT (x)m(T , x)dx = −B(m, w).
TN
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Upper bound lemma
Lemma
Suppose u is a continuous viscosity solution to the Hamilton-Jacobi
equation
−∂t u + H(x, Du) = f ≥ 0
in a region [0, T ] × U for some open set U in Euclidean space. Then for
any 0 ≤ t < s ≤ T and for any 0 < β < 1, we have the following
estimate:
u(t, x) − u(s, y ) ≤ C (1 − β 2 )−N/2p kf kp |t − s|α ,
∀|x − y | ≤ βc0 (s − t),
where α = 1 − (N + 1)/p and the constant C depends on p, N, and c0−1 .
Cf. [Cardaliaguet 2013]. This lemma gives us upper bounds.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Let (un , fn ) be a minimizing sequence. WLOG, un and fn are Lipschitz,
hence differentiable a.e., and fn ≥ 0. We have:
fn bounded in Lp by estimates on K , so fn → f weakly.
This implies un pointwise bounded.
The above implies H(x, Dun ) is bounded in L1 , so Dun is as well.
The above implies ∂t un is bounded in L1 , so un is bounded in BV .
(∂t un , Dun ) → (∂t u, Du) in the weak measure sense, while un → u
in L1 .
Conclusion: (u, f ) is the minimizer.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Duality
A priori bounds
Introduction
Main results
Proofs
Further results
Minimizers are weak solutions
Suppose (u, f ) ∈ K̃ is a minimizer of relaxed problem and (m, w) ∈ K1 is
a minimizer of dual problem.
We know that −∂t u + H(x, Du) ≤ f in the sense of measures, so
Z tZ
Z
u(t)m(t) − u(0)m0 ≥ 0,
fm +
Z
0
T Z
Z
fm +
uT m(T ) − u(t)m(t) ≥ 0.
t
RT R
R
The main point is to get 0 fm + uT m(T ) − u(0)m0 = 0, so
that the above inequalities become equality.
Everything else comes directly from the definition of K̃ and K1 and
the properties of minimizers already proved.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Minimizers are weak solutions, part 2
(u, f ) ∈ K̃ is a minimizer of relaxed problem and (m, w) ∈ K1 is a
minimizer of dual problem.
By the duality theorem,
Z
T
Z
0=
K (t, x, f ) + K ∗ (t, x, m) +
Z
uT m(T ) − u(0)m0
0
Z
≥
T
Z
Z
fm +
uT m(T ) − u(0)m0 ≥ 0,
0
which implies the desired result.
It also implies K (t, x, f (t, x)) + K ∗ (t, x, m(t, x)) = f (t, x)m(t, x)a.e. so
f (t, x) = k(t, x, m).
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Weak solutions are also minimizers
Suppose (u, m) is a weak solution. We set w = mv and f = k(·, ·, m).
Then (u, f ) is a minimizer of the relaxed problem and (m, w) a minimizer
of the dual problem. For example:
ZZ
Z
A(u 0 , f 0 ) =
K (t, x, f 0 ) − u 0 (0)m0
ZZ
Z
≥
K (t, x, f ) + ∂f K (t, x, f )(f 0 − f ) − u 0 (0)m0
ZZ
Z
0
=
K (t, x, f ) + m(f − f ) − u 0 (0)m0
ZZ
Z
=
K (t, x, f ) + mf 0 + m(T )uT − m0 u(0) − u 0 (0)m0
ZZ
Z
≥
K (t, x, f ) − m0 u(0) = A(u, f ).
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Duality
A priori bounds
Weak solutions are unique
Suppose (u1 , m1 ) and (u2 , m2 ) are both weak solutions.
1
m1 and m2 are both minimizers of dual problem, so m1 = m2 =: m.
2
f := k(·, ·, m), then both (u1 , f ) and (u2 , f ) are minimizers of
relaxed problem.
3
u := max{u1 , u2 }; we show −∂t u + H(x, Du) ≤ f in the sense of
measures.
4
Then (u, f ) is a minimizer of relaxed problem, so
Z
ZZ
Z
ZZ
K (t, x, f ) − ui (0)m0 =
K (t, x, f ) − u(0)m0 .
5
Combine with previous inequalities and use u ≥ ui to get u = ui in
{m > 0}.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Table of Contents
1
Introduction
Motivation
Background
2
Main results
Assumptions
3
Proofs
Duality
A priori bounds
4
Further results
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Recent work
The following is joint

 (i)
(ii)

(iii)
work with Pierre Cardaliaguet. We study
−∂t φ + H(x, Dφ) = f (x, m)
∂t m − div (mDp H(x, Dφ)) = 0
φ(T , x) = φT (x), m(0, x) = m0 (x)
and the corresponding ergodic problem

 (i) λ + H(x, Dφ) = f (x, m(x))
(ii) −div(mD
R p H(x, Dφ)) = 0

(iii) m ≥ 0, Td m = 1
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Assumptions
1
2
(Initial-final conditions) m0 ∈ C (Td ), m0 > 0, φT : Td → R
Lipschitz.
(Conditions on the Hamiltonian) H : Td × Rd → R is continuous in
both variables, convex and differentiable in the second variable, with
Dp H continuous in both variables. Also,
1
C
|p|r − C ≤ H(x, p) ≤ |p|r + C .
rC
r
3
(Conditions on the coupling) Let f be continuous on Td × (0, ∞),
strictly increasing in the second variable, satisfying
1
|m|q−1 − C ≤ f (x, m) ≤ C |m|q−1 + C ∀ m ≥ 1.
C
4
The relation holds between the growth rates of H and of F :
r > max{d(q − 1), 1}.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Optimal control problems
Optimal control of HJ equation: find the infimum of
T
Z
Z
F ∗ (x, −∂t φ + H(x, Dφ))dxdt −
A(φ) =
Td
0
Z
φ(0, x)dm0 (x)
Td
over the set of maps φ ∈ C 1 ([0, T ] × Td ) such that φ(T , x) = φT (x).
Optimal control of transport equation: find the infimum of
Z
B(m, w ) =
Td
L1 ((0, T )
over (m, w ) ∈
a.e. t ∈ (0, T ), and
T
Z
Z
mH ∗ x, −
φT m(T )dx +
Td
0
×
Td )
×
L1 ((0, T )
w
+ F (x, m)dxdt
m
× Td ; Rd ), m ≥ 0 a.e.,
R
Td
m(t, x)dx = 1
∂t m + div (w ) = 0, m(0, ·) = m0 (·)
in the sense of distributions.
Theorem (Cardaliaguet, PJG)
The optimal control problems above are in duality, i.e.
inf A(φ) = −
φ∈K0
P. Jameson Graber
min
B(m, w ).
(m,w )∈K1
Introduction
Main results
Proofs
Further results
Relaxed problem
K will be defined as the set of all pairs (φ, α) ∈ BV × L1 such that
Dφ ∈ Lr ((0, T ) × Td ), φ(T , ·) ≤ φT in the sense of traces,
α+ ∈ Lp ((0, T ) × Td ), φ ∈ L∞ ((t, T ) × Td ) for every t ∈ (0, T ), and
−∂t φ + H(x, Dφ) ≤ α.
in the sense of distribution. Find
Z TZ
Z
inf A(φ, α) = inf
F ∗ (x, α(t, x))dxdt−
(φ,α)∈K
(φ,α)∈K
0
Td
φ(0, x)m0 (x)dx.
Td
The relaxed problem has a minimizer. Moreover,
inf
(φ,α)∈K
A(φ, α) = −
B(m, w ) = inf A(φ).
min
(m,w )∈K1
P. Jameson Graber
φ∈K0
Introduction
Main results
Proofs
Further results
Definition of weak solution
A pair (φ, m) ∈ BV ((0, T ) × Td ) × Lq ((0, T ) × Td ) is called a weak
solution if it satisfies the following conditions.
1
Dφ ∈ Lr and the maps mf (x, m), mH ∗ (x, −Dp H(x, Dφ)) and
mDp H(x, Dφ) are integrable,
2
φ satisfies −∂t φ + H(x, Dφ) ≤ f (x, m) in the sense of distributions,
the boundary condition φ(T , ·) ≤ φT in the sense of trace and the
following equality
ZZ
m (H(x, Dφ) − hDφ, Dp H(x, Dφ)i − f (x, m)) dxdt
Z
= (φT m(T ) − φ(0)m0 )dx
3
m satisfies the continuity equation
in (0, T ) × Td ,
m(0) = m0
in the sense of distributions.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Well-posedness
(i) If (m, w ) ∈ K1 is a minimizer of the dual problem and (φ, α) ∈ K̃ is a
minimizer of the relaxed problem, then (φ, m) is a weak solution and
α(t, x) = f (x, m(t, x)) almost everywhere.
(ii) Conversely, if (φ, m) is a weak solution, then there exist functions
w , α such that (φ, α) ∈ K is a minimizer of the relaxed problem and
(m, w ) ∈ K1 is a minimizer of the dual problem.
(iii) If (φ, m) and (φ0 , m0 ) are both weak solutions, then m = m0 almost
everywhere while φ = φ0 almost everywhere in the set {m > 0}.
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Main result
Let φf ∈ C 1 on Td .
Let (φT , mT ) be a “good” solution of

 −∂t φ + H(x, Dφ) = f (x, m)

φ(T , x) = φf (x), m(0, x) = m0 (x).
and (λ̄, φ̄, m̄) be a solution of the ergodic MFG system. Define
ψ T (s, x) = φT (sT , x),
µT (s, x) = mT (sT , x)
∀(s, x) ∈ (0, 1)×Td .
As T → +∞,
(µT ) converges to m̄ in Lθ ((0, 1) × Td ) for any θ ∈ [1, q),
ψ T /T converges to the map s → λ̄(1 − s) in Lθ ((δ, 1) × Td ) for any
θ ≥ 1 and any δ ∈ (0, 1).
P. Jameson Graber
Introduction
Main results
Proofs
Further results
Thank you!
P. Cardaliaguet and P. Graber, “Mean field games systems of first
order,” submitted to ESAIM: Control, Optimization, and Calculus of
Variations. Available on the arXiv.
P. Graber, “Optimal control of first-order Hamilton-Jacobi equations
with linearly bounded Hamiltonians,” to appear in Applied
Mathematics and Optimization. Also available on the arXiv.
P. Jameson Graber

Optimal control of Hamilton-Jacobi

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