Null controllability properties of some degenerate parabolic equations.

Carleman estimates and applications for a class
of degenerate parabolic equations in space
dimension 2
P. Martinez
Institut de Mathématiques de Toulouse
Université Paul Sabatier, Toulouse III
IFAC Workshop - CDPS 2009 - Toulouse, july 20-24, 2009
1/26

Joint work with
- Piermarco Cannarsa, Univ. Tor Vergata, Roma 2,
- Judith Vancostenoble, Univ. Toulouse 3,
2/26

Presentation of the problem
Ω bounded, smooth domain of R 2 (R n ) ; ω nonempty open subset
of Ω.
but non uniformly positive :
A : Ω → S 2 (R), A(x) ≥ 0,
∀x ∈ ∂Ω, det(A(x)) = 0.
Null controllability and inverse problems properties of
⎧
⎪⎨ u t − div (A(x)∇u) = h(x, t)χ ω ,
boundary conditions,
⎪⎩
initial condition
(First step to exact controllability to trajectories...)
Uniformly positive case : heat equation Lebeau-Robbiano (95),
general case : Fursikov-Imanuvilov (95,96)
3/26

Some examples of such equations in 1D
◮ aeronautics : the Crocco type equation (boundary layer
model) :
u t + a(y)u x − (b(y)u y ) y = localized control, x ∈ (0, L), y(0, 1)
with a(1) = 0 = b(1) ;
◮ climatology : the Budyko-Sellers model :
RT t − ((1 − x 2 )T x ) x − QS(1 − α) = −I (T ), x ∈ (−1, 1);
◮ economics : the Black-Scholes equation of the type :
u t − x 2 u xx + · · · = · · · , x ∈ (0, L).
4/26

An example in N-D
◮ biology : the Fleming Viot model :
u t − Tr (C(x)D 2 u) · · · = f ,
where
C(x) = (c ij (x)) i,j , c ij (x) = x i (δ ij − x j ),
and
x ∈ {x i ∈ [0, 1], ∑ i
x i ≤ 1};
example : N=2 : degenerate along the sides of the triangle.
5/26

Main results on the 1D degenerate problems : the simplest
problem
The simplest problem in divergence form (Cannarsa, Martinez,
Vancostenoble (2008)) :
u t − (x α u x ) x = h(x, t)χ (a,b) , x ∈ (0, 1), t > 0 :
◮ α ∈ [0, 1[ : well-posed with the Dirichlet boundary condition
(u(0, t) = 0 = u(t, 1), and null controllable ;
◮ α ∈ [1, 2[ : well-posed with the Neumann boundary condition
(x α u x )(0, t) = 0 = u(1, t), and null controllable ;
Main tools : Carleman estimates associated to the degenerate
problem, and Hardy type inequalities ;
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◮ α ≥ 2 : well-posed with the Neumann boundary condition
(x α u x )(0, t) = 0 = u(1, t), and not null controllable ;
Main tools : application of a result of Escauriaza, Seregin,
Sverak (2004) related to the backward uniqueness properties
of the heat equation in half space.
Remark : Strong connection between degenerate problems in
bounded domains and nondegenerate problems in unbounded
domains
7/26

The problem in 2D : simplest assumptions on the
degeneracy
A(x) ∼
(
λ1 (x) 0
0 λ 2 (x)
)
with 0 ≤ λ 1 (x) ≤ λ 2 (x).
ε 1 (x) denotes the eigenvector associated to λ 1 (x), and ε 2 (x) the
eigenvector associated to λ 2 (x).
Simplest assumptions (H s (A)) : Ω is of class C 4 , and there exists
some α ≥ 0, and some neighborhood of the boundary Γ such that :
◮ λ 1 (x) = d(x, Γ) α for all x ∈ V(Γ),
◮ ε 1 (x) = ν(p Γ (x)) for all x ∈ V(Γ), where p Γ (x) is the
projection of x on the boundary Γ,
◮ 0 < m ≤ λ 2 (x) ≤ M for all x ∈ Ω.
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The control problem
We consider
⎧
⎪⎨ u t − div (A(x)∇u) = h(x, t)χ ω ,
boundary conditions,
⎪⎩
initial condition,
under (H s (A)) .
◮ Question 1 : well-posedness
yes, in natural weighted Sobolev spaces.
◮ Question 2 : null controllability : given T > 0, u 0 ∈ L 2 (Ω),
does there exist h ∈ such that u(T ) = 0
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Null controllability result when α ∈ [0, 1)
Theorem
Assume that A satisfies Hypothesis (H s (A)) with α ∈ [0, 1). Let
T > 0 be given and consider ω any non-empty open subset of Ω.
Then for all u 0 ∈ L 2 (Ω), there exists h ∈ L 2 (Ω T ) such that the
solution u of
⎧
⎪⎨ u t − div (A(x)∇u) = h(x, t)χ ω , x ∈ Ω, t > 0
u(t, x) = 0 on Γ,
⎪⎩
u(0, x) = u 0 (x)
satisfies u(T , ·) = 0 in L 2 (Ω). Moreover h is bounded with respect
to α ∈ [0, 1) :
‖h‖ L 2 (Ω×(0,T )) ≤ C(Ω, ω, T )‖u 0 ‖ L 2 (Ω).
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Associated observability estimate
The null controllability result derives from the following
observability estimate :
Theorem
Under (H s (A)), there is some C(Ω, ω, T ) independent of α ∈ [0, 1)
such that, given v T ∈ L 2 (Ω), the solution v of the adjoint problem
satisfies :
∫
Ω
⎧
⎪⎨ v t + div (A(x)∇v) = 0, x ∈ Ω, t > 0
v(t, x) = 0 on Γ,
⎪⎩
v(T , x) = v T (x)
∫ T ∫
v(0, x) 2 dx ≤ C(Ω, ω, T ) v(t, x) 2 dx dt.
0 ω
Hence : usual observability estimate, but in the degenerate context.
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Null controllability result when α ∈ [1, 2)
Theorem
Assume that A satisfies Hypothesis (H s (A)) with α ∈ [1, 2). Let
T > 0 be given and consider ω any non-empty open subset of Ω.
Then for all u 0 ∈ L 2 (Ω), there exists h ∈ L 2 (Ω T ) such that the
solution u of
⎧
⎪⎨ u t − div (A(x)∇u) = h(x, t)χ ω , x ∈ Ω, t > 0
A∇u(t, x) · ν = 0 on Γ (generalized Neumann condition),
⎪⎩
u(0, x) = u 0 (x)
satisfies u(T , ·) = 0 in L 2 (Ω). Moreover there is C(Ω, , T )
independent of α ∈ [1, 2) such that
‖h‖ L 2 (Ω×(0,T )) ≤ e eC(Ω,ω,T )/(2−α)2 ‖u 0 ‖ L 2 (Ω).
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Associated observability estimate
The null controllability result derives from the following
observability estimate :
Theorem
Under (H s (A)), there is some C(Ω, ω, T ) independent of α ∈ [1, 2)
such that, given v T ∈ L 2 (Ω), the solution v of the adjoint problem
satisfies :
∫
Ω
⎧
⎪⎨ v t + div (A(x)∇v) = 0, x ∈ Ω, t > 0
A∇v(t, x) · ν = 0 on Γ,
⎪⎩
v(T , x) = v T (x)
∫
v(0, x) 2 T ∫
dx ≤ e eC(Ω,ω,T )/(2−α)2 v(t, x) 2 dx dt.
0
ω
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Comments
◮ estimate on the cost of observability and controllability with
respect to the degeneracy parameter α (new even in 1-D,
thanks to improved Hardy type inequalities) ;
◮ the bound on the control and the observability cost explode as
α → 2 − ;
◮ indeed : negative result in the case α ≥ 2 : counter-example :
Ω = disc of radius 1 ; explicit matrix A(x) whose eigenvalues
are λ 1 (x) = (1 − r) α , λ 2 (x) = 1 ;
transformation to write the problem in polar coordinates : the
associated function v(r, θ) is solution of a degenerate parabolic
equation in 2D ;
the means w(r) = ∫ 2π
v(r, θ) dθ is solution of a degenerate
0
parabolic equation in 1D, for which we already know that there
is no null controllability ;
return to the initial problem : no null controllability ;
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Extensions
◮ more general degenerate parabolic problem
u t − div (A(x)∇u) + b(x) · ∇u + c(x)u = h(x, t)χ ω ,
when (b, ε 1 ) = O(d Γ (x) α/2 , (b, ε 2 ) = O(1), c bounded : same
results.
◮ weakened assumptions on the degeneracy : (H g (A)) : there
exists some α ≥ 0 such that :
λ 1 (x) ∼ d(x, Γ) α as x → Γ,
ε 1 (x) − ν(p Γ (x)) → 0 as x → Γ,
0 < m ≤ λ 2 (x) ≤ M for all x ∈ Ω.
(work in progress)
15/26

Main steps in the proof of the positive results (α ∈ [0, 2))
The observability results derives from a Carleman estimate related
to the degenerate problem :
◮ “some” computations,
◮ some useful tools : a special geometrical lemma, and special
Hardy type inequalities,
What does not help :
◮ the operator is degenerate,
◮ the solution is not regular enough (u(t, .) ∈ HA 2 (Ω) that
strictly contains H 2 (Ω)) ;
What helps :
◮ the degeneracy occurs only on the boundary, in a very
specified way ((H s (A)) or (H g (A))),
◮ the solution has some sufficient regularity (proved in the thesis
of D. Rocchetti, Univ Roma 2, 2008).
16/26

Useful 1-D Hardy type inequalities
The simplest one :
Lemma
Given L > 0 and α ∈ [0, 1), then the following inequality holds for
all z ∈ D((0, L])
∫ L
0
x α−2 z(x) 2 dx ≤
∫
4 L
(1 − α) 2 x α z x (x) 2 dx.
0
Comments :
◮ natural extension by density to absolutely continuous
functions z such that
z(x) → 0
x→0 +
and
◮ the constant
4
(1−α) 2 is optimal ;
◮ the result is false for α = 1 ;
◮ similar version for α ∈ (1, 2).
∫ L
0
x α z x (x) 2 dx < ∞;
17/26

Proof : several ones, but probably the most simple one :
0 ≤
∫ L
=
(
0
∫ L
0
x α/2 z x − 1 − α
2
x z) (α−2)/2 dx
2
x α z 2 x +
(1 − α)2
4
∫ L
Then an integration by parts leads to
0 ≤
Hence
∫ L
0
x α z 2 x +
(1 − α)2
4
∫ L
0
0
x α−2 z 2 − 1 − α
2
x α−2 z 2
− 1 − α L α−1 z(L) 2 +
2
(1 − α)
2
∫ L
0
∫ L
0
x α−1 (z 2 ) x .
(α − 1)x α−2 z 2 .
(1 − α) 2
4
∫ L
0
x α−2 z 2 ≤
∫ L
0
x α zx 2 − 1 − α L α−1 z(L) 2
2
≤
∫ L
0
x α z 2 x .
18/26

A useful improved version (J. Vancostenoble) :
Lemma
Given L > 0, α ∈ [0, 1), β > 0, n > 0, there exists
C β,n = C(L, β, n) > 0 and x β,n = x(L, β, n) ∈ (0, L) such that the
following inequality holds : for all z ∈ D((0, L])
(1 − α) 2
4
∫ L
0
x α−2 z(x) 2 dx + n
≤
∫ L
0
∫ L
0
x α−2+β z(x) 2 dx
x α z x (x) 2 dx + C β,n
∫ L
x β,n
z(x) 2 dx. (1)
◮ it allows us to estimate “non critical” terms of the form
∫ L
0 xα−2+β z(x) 2 dx uniformly with respect to α ∈ [0, 1) ;
◮ it remains valid for α ∈ [1, 2), in particular for α = 1 ;
◮ it is the key to obtain the observability cost estimates.
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Proof : the main step is to prove it for z ∈ D((0, L)) :
∫ L
(
0 ≤ x α/2 z x − 1 − α
2
x (α−2)/2 z + x z) (α−2+β)/2 dx
2
0
gives (integrations by parts)
(1 − α) 2
4
∫ L
0
≤
x α−2 z 2 + n
∫ L
0
x α z 2 x +
∫ L
0
∫ L
0
x α−2+β z 2
(
(n + 1)x β/2 − β )
x α−2+β/2 z 2 .
2
Since β > 0, we have
(n + 1)x β/2 − β ( )
2 ≤ 0 for all x ∈ [0, β 2/β
];
2(n + 1)
therefore
(1 − α) 2
4
∫ L
0
x α−2 z 2 + n
≤
∫ L
0
∫ L
0
x α−2+β z 2
x α z 2 x + C 1 (L, α, β, n)
∫ L
x 1
z 2 .
20/26

Useful 2-D (N-D) Hardy type inequalities
◮ normal parametrization of the boundary by arclength :
Γ = γ([0, l(Γ)]), γ(0) = γ(l(Γ)), |γ ′ (t)| = 1 ;
◮ this gives a parametrization of the neighborhood C(Γ, η) of Γ :
ψ : [0, l(Γ)] × (0, η) → C(Γ, η), ψ(s, t) = γ(s) − tν(γ(s)) :
ψ is a C 1 -diffeomorphism between (0, l(Γ)) × (0, η) and
C(Γ, η) \ ψ({0} × (0, η)) ;
◮ then use the 1-D hardy type inequality on every normal
segment, and integrate over Γ :
∫∫
C(Γ,η)
∫ ( ∫
d Γ (x) α−2 z 2 η
=
≤ HARDY
∫
Γ
(
Γ
0
∫
4 η
(α − 1) 2
)
“d Γ (x) α−2 z 2 “
0
)
“d Γ (x) α (∇z · ν) 2 “
c
≤
(α − 1)
∫∫C(Γ,η)
2 d Γ (x) α (∇z · ν) 2
21/26

The standard form of the Carleman estimate
We consider the solution of the adjoint problem
⎧
⎪⎨ w t + div (A(x)∇w) = f , x ∈ Ω, t > 0,
w(t, x) = 0 on Γ,
⎪⎩
w(T , x) = w T (x).
A Carleman estimate will be of the type
∫∫
weight 0 w 2 + weight 1 |∇w| 2 + weight 2 |D 2 w| 2 + weight 3 wt
2
Ω T
∫∫
∫∫
≤ C weight 4 f 2 + C weight 5 w 2 ,
Ω T ω T
hence a parabolic regularity type result, but without involving
‖w T ‖.
22/26

How to obtain the Carleman estimate : the method and
the difficulties
◮ description of the method : the weighted function
z(t, x) := e −Rσ w satisfies the differential problem :
and so
P + R z + P− R z = fe−Rσ ,
‖P + R z‖2 + ‖P − R z‖2 + 2〈P + R z, P− R z〉 = ‖fe−Rσ ‖ 2 ,
and the Carleman estimates comes from a suitable lower
bound of the scalar product 〈P + R z, P− R z〉.
◮ the difficulties due to the degeneracy :
adapt all the weights to the degeneracy ; and use suitable
Hardy type inequalities ;
the solution has no enough global regularity : compute on
subdomains Ω δ := {x ∈ Ω, d(x, Γ) > δ}, and pass to the limit
Ω δ → Ω and Γ δ → Γ for the boundary integrals (using
properties of W 1,1 functions).
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The weights that finally appear are :
◮ degenerate in time : classical and natural, since we do not
want to make appear w(T ) ;
◮ degenerate in space : equal to zero on the boundary, because
of the degeneracy of the operator :
∫∫
(2 − α) ρ 3 e −2Rσ d Γ (x) 2−α w 2
Ω T
∫∫
+ (2 − α) ρe −2Rσ A(x)∇w · ∇w
Ω
∫∫ T
∫∫
≤ C e −2Rσ f 2 + C ρ 3 e −2Rσ w 2 .
Ω T ω T
In particular, these estimates require α < 2.
And we have also similar estimates for w t , D 2 w.
24/26

But the Hardy type inequalities allow (once again) to correct this
when α < 2, and to obtain in particular
∫∫
(2 − α) ρ 3/2 e −2Rσ w 2
Ω T
∫∫
≤ C e −2Rσ f 2 + C
Ω T
∫∫
ω T
ρ 3 e −2Rσ w 2 ,
which implies the desired observability inequality, and to estimate
the observability cost with respect to the degeneracy parameter :
∫
Ω
w(0, x) 2 dx ≤ 2 T
∫ 3T /4
T /4
∫
Ω
w(t, x) 2 dx
∫∫
≤ C(Ω, ω, T , α) ρ 3/2 e −2Rσ w 2 ≤ · · ·
Ω T
25/26

Applications of the obtained Carleman estimate
◮ the observability inequalities (α ∈ [0, 1) and α ∈ [1, 2)), and
then the related null controllability results ;
◮ inverse problems results (more appropriate for biological and
climate models) : suitably adapting the methods developped
in particular by O. Imanuvilov, M. Yamamoto, A.
Benabdallah, P. Gaitan, J. Le Rousseau :
determination of a source, of a potential term (in particular :
concerning Budyko-Sellers type problems :
Cannarsa-Tort-Yamamoto, work in progress)
of a diffusion coefficient (work in progress), ...
measuring the solution at some time, and in a part of the
space domain.
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