Control and inverse problems for degenerate parabolic operators - - PowerPoint PPT Presentation

Control and inverse problems for degenerate parabolic operators

Piermarco Cannarsa

University of Rome “Tor Vergata”

Inverse Problems, Control and Shape Optimization April 2 – 4, 2012 Ecole Polytechnique, Palaiseau, France

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 1 / 46

Outline

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 2 / 46

Outline

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 2 / 46

Outline

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 2 / 46

Outline

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 2 / 46

motivations

diffusion processes and Kolmogorov operator

X(·, x) unique solution

• dX(t) = b(X(t))dt + σ(X(t)) dW(t)

t ≥ 0 X(0) = x ∈ Rn b : Rn → Rn , σ : Rn → L(Rn; Rm) Lipschitz W(t) m-dimensional Brownian transition semigroup Ptϕ(x) := E

• ϕ
• X(t, x)
• u(t, x) = Ptϕ(x) solution of Kolmogorov equation

       ut = 1 2 Tr [A(x)∇2u] + b(x), ∇u

• Lu

in (0, ∞) × Rn u(0, x) = ϕ(x) x ∈ Rn where A(x) = σ(x)σ∗(x) ≥ 0 may be degenerate

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 3 / 46

motivations

invariant sets for stochastic flows

K ⊂ Rn invariant x ∈ K = ⇒ X(t, x) ∈ K P − a.s. ∀t ≥ 0

• r x

✬ ✫ ✩ ✪

K Ω ⊂ Rn

• pen set

∂Ω = Γ Problem find (necessary and sufficient) conditions for invariance of Ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 4 / 46

motivations

invariant sets for stochastic flows

K ⊂ Rn invariant x ∈ K = ⇒ X(t, x) ∈ K P − a.s. ∀t ≥ 0

• r x

✬ ✫ ✩ ✪

K Ω ⊂ Rn

• pen set

∂Ω = Γ Problem find (necessary and sufficient) conditions for invariance of Ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 4 / 46

motivations

invariant sets for stochastic flows

K ⊂ Rn invariant x ∈ K = ⇒ X(t, x) ∈ K P − a.s. ∀t ≥ 0

• r x

✬ ✫ ✩ ✪

K Ω ⊂ Rn

• pen set

∂Ω = Γ Problem find (necessary and sufficient) conditions for invariance of Ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 4 / 46

motivations

conditions for invariance

Ω ⊂ Rn

• pen set

∂Ω = Γ

• riented distance from Γ

dΓ(x) =

• dist(x, Γ)

if x ∈ Ω −dist(x, Γ) if x ∈ Ωc Ω Γ dΓ

r r ❅ ❅ ❅ ❅

• Lu = 1

2 Tr [A(x)∇2u] + b(x), ∇u

Theorem Ω invariant ⇐ ⇒ ∀x ∈ Γ      LdΓ(x) ≥ 0 A(x)∇dΓ(x), ∇dΓ(x) = 0 Friedman and Pinsky (1975), Da Prato & Frankowska (2004). . . Ω invariant = ⇒ L degenerate on Γ in normal direction

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 5 / 46

motivations

conditions for invariance

Ω ⊂ Rn

• pen set

∂Ω = Γ

• riented distance from Γ

dΓ(x) =

• dist(x, Γ)

if x ∈ Ω −dist(x, Γ) if x ∈ Ωc Ω Γ dΓ

r r ❅ ❅ ❅ ❅

• Lu = 1

2 Tr [A(x)∇2u] + b(x), ∇u

Theorem Ω invariant ⇐ ⇒ ∀x ∈ Γ      LdΓ(x) ≥ 0 A(x)∇dΓ(x), ∇dΓ(x) = 0 Friedman and Pinsky (1975), Da Prato & Frankowska (2004). . . Ω invariant = ⇒ L degenerate on Γ in normal direction

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 5 / 46

motivations

conditions for invariance

Ω ⊂ Rn

• pen set

∂Ω = Γ

• riented distance from Γ

dΓ(x) =

• dist(x, Γ)

if x ∈ Ω −dist(x, Γ) if x ∈ Ωc Ω Γ dΓ

r r ❅ ❅ ❅ ❅

• Lu = 1

2 Tr [A(x)∇2u] + b(x), ∇u

Theorem Ω invariant ⇐ ⇒ ∀x ∈ Γ      LdΓ(x) ≥ 0 A(x)∇dΓ(x), ∇dΓ(x) = 0 Friedman and Pinsky (1975), Da Prato & Frankowska (2004). . . Ω invariant = ⇒ L degenerate on Γ in normal direction

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 5 / 46

motivations

conditions for invariance

Ω ⊂ Rn

• pen set

∂Ω = Γ

• riented distance from Γ

dΓ(x) =

• dist(x, Γ)

if x ∈ Ω −dist(x, Γ) if x ∈ Ωc Ω Γ dΓ

r r ❅ ❅ ❅ ❅

• Lu = 1

2 Tr [A(x)∇2u] + b(x), ∇u

Theorem Ω invariant ⇐ ⇒ ∀x ∈ Γ      LdΓ(x) ≥ 0 A(x)∇dΓ(x), ∇dΓ(x) = 0 Friedman and Pinsky (1975), Da Prato & Frankowska (2004). . . Ω invariant = ⇒ L degenerate on Γ in normal direction

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 5 / 46

motivations

• perators with boundary degeneracy

Ω ⊂ Rn bounded      ut − Lu = f in Ω × (0, T) u(0, x) = u0(x) x ∈ Ω + b. c.

• n

Γ × (0, T) where Lu =      div(A(x)∇u) + lower order tms

• r

Tr [A(x)∇2u] + lower order tms with A(x) > 0 in Ω but A(·)∇dΓ(·) = 0

• n

Γ0 ⊂ Γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 6 / 46

motivations

• perators with interior degeneracy

Crocco-type equation ut + ux −

• a(y)uy
• y = χω(x, y)f(t, x, y)

Martinez – Raymond – Vancostenoble (2003) (a ≡ 1) C – Martinez – Vancostenoble (2005, 2008) (a = (1 − y)θ)

Kolmogorov-type equation ut + yux − uyy = χω(x, y)f(t, x, y) Beauchard – Zuazua (2009) Grushin-type equation ut − uxx − |x|2γuyy = χω(x, y)f(t, x, y) Beauchard – C – Guglielmi (2012)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 7 / 46

motivations

• perators with interior degeneracy

Crocco-type equation ut + ux −

• a(y)uy
• y = χω(x, y)f(t, x, y)

Martinez – Raymond – Vancostenoble (2003) (a ≡ 1) C – Martinez – Vancostenoble (2005, 2008) (a = (1 − y)θ)

Kolmogorov-type equation ut + yux − uyy = χω(x, y)f(t, x, y) Beauchard – Zuazua (2009) Grushin-type equation ut − uxx − |x|2γuyy = χω(x, y)f(t, x, y) Beauchard – C – Guglielmi (2012)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 7 / 46

motivations

• perators with interior degeneracy

Crocco-type equation ut + ux −

• a(y)uy
• y = χω(x, y)f(t, x, y)

Martinez – Raymond – Vancostenoble (2003) (a ≡ 1) C – Martinez – Vancostenoble (2005, 2008) (a = (1 − y)θ)

Kolmogorov-type equation ut + yux − uyy = χω(x, y)f(t, x, y) Beauchard – Zuazua (2009) Grushin-type equation ut − uxx − |x|2γuyy = χω(x, y)f(t, x, y) Beauchard – C – Guglielmi (2012)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 7 / 46

motivations

• perators with interior degeneracy

Crocco-type equation ut + ux −

• a(y)uy
• y = χω(x, y)f(t, x, y)

Martinez – Raymond – Vancostenoble (2003) (a ≡ 1) C – Martinez – Vancostenoble (2005, 2008) (a = (1 − y)θ)

Kolmogorov-type equation ut + yux − uyy = χω(x, y)f(t, x, y) Beauchard – Zuazua (2009) Grushin-type equation ut − uxx − |x|2γuyy = χω(x, y)f(t, x, y) Beauchard – C – Guglielmi (2012)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 7 / 46

existence, uniqueness, regularity

• ne space dimension

a ∈ C([0, 1]) ∩ C1(]0, 1]) and a > 0

• n

]0, 1]

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) u(t, 1) = 0 + b.c. at x = 0 u0 ∈ L2(0, 1) , f ∈ L2(QT) Campiti, Metafune, Pallara (1998) weakly degenerate case: 1/a ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(0) = 0 = u(1)

• strongly degenerate case:

1/a / ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(1) = 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 8 / 46

existence, uniqueness, regularity

• ne space dimension

a ∈ C([0, 1]) ∩ C1(]0, 1]) and a > 0

• n

]0, 1]

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) u(t, 1) = 0 + b.c. at x = 0 u0 ∈ L2(0, 1) , f ∈ L2(QT) Campiti, Metafune, Pallara (1998) weakly degenerate case: 1/a ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(0) = 0 = u(1)

• strongly degenerate case:

1/a / ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(1) = 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 8 / 46

existence, uniqueness, regularity

• ne space dimension

a ∈ C([0, 1]) ∩ C1(]0, 1]) and a > 0

• n

]0, 1]

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) u(t, 1) = 0 + b.c. at x = 0 u0 ∈ L2(0, 1) , f ∈ L2(QT) Campiti, Metafune, Pallara (1998) weakly degenerate case: 1/a ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(0) = 0 = u(1)

• strongly degenerate case:

1/a / ∈ L1(0, 1) H1

a(0, 1) =

• u ∈ L2(0, 1)
• 1

au2

x dx < ∞ & u(1) = 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 8 / 46

existence, uniqueness, regularity

well-posedness

• D(A) =
• u ∈ H1

a(0, 1)

• aux ∈ H1(0, 1)
• Au =
• aux
• x

generates analytic semigroup in L2(0, 1) unique solution u ∈ C(0, T; L2(0, 1)) ∩ L2(0, T; H1

a(0, 1))

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) maximal regularity u0 ∈ H1

a(0, 1)

= ⇒ u ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T; D(A)) (needed to justify integration by parts) strongly degenerate case incorporates b.c. u ∈ D(A) = ⇒ aux

(x→0)

− → 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity

well-posedness

• D(A) =
• u ∈ H1

a(0, 1)

• aux ∈ H1(0, 1)
• Au =
• aux
• x

generates analytic semigroup in L2(0, 1) unique solution u ∈ C(0, T; L2(0, 1)) ∩ L2(0, T; H1

a(0, 1))

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) maximal regularity u0 ∈ H1

a(0, 1)

= ⇒ u ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T; D(A)) (needed to justify integration by parts) strongly degenerate case incorporates b.c. u ∈ D(A) = ⇒ aux

(x→0)

− → 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity

well-posedness

• D(A) =
• u ∈ H1

a(0, 1)

• aux ∈ H1(0, 1)
• Au =
• aux
• x

generates analytic semigroup in L2(0, 1) unique solution u ∈ C(0, T; L2(0, 1)) ∩ L2(0, T; H1

a(0, 1))

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) maximal regularity u0 ∈ H1

a(0, 1)

= ⇒ u ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T; D(A)) (needed to justify integration by parts) strongly degenerate case incorporates b.c. u ∈ D(A) = ⇒ aux

(x→0)

− → 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity

well-posedness

• D(A) =
• u ∈ H1

a(0, 1)

• aux ∈ H1(0, 1)
• Au =
• aux
• x

generates analytic semigroup in L2(0, 1) unique solution u ∈ C(0, T; L2(0, 1)) ∩ L2(0, T; H1

a(0, 1))

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) maximal regularity u0 ∈ H1

a(0, 1)

= ⇒ u ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T; D(A)) (needed to justify integration by parts) strongly degenerate case incorporates b.c. u ∈ D(A) = ⇒ aux

(x→0)

− → 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity

well-posedness

• D(A) =
• u ∈ H1

a(0, 1)

• aux ∈ H1(0, 1)
• Au =
• aux
• x

generates analytic semigroup in L2(0, 1) unique solution u ∈ C(0, T; L2(0, 1)) ∩ L2(0, T; H1

a(0, 1))

• ut −
• a(x)ux
• x = f

in QT =]0, 1[×]0, T[ u(x, 0) = u0(x) maximal regularity u0 ∈ H1

a(0, 1)

= ⇒ u ∈ H1(0, T; L2(0, 1)) ∩ L2(0, T; D(A)) (needed to justify integration by parts) strongly degenerate case incorporates b.c. u ∈ D(A) = ⇒ aux

(x→0)

− → 0

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 9 / 46

existence, uniqueness, regularity

Budyko-Sellers model

• ut −
• (1 − x2)ux
• x = f(x) g(u) − h(u)

x ∈ (−1, 1) (1 − x2)ux|x=±1 = 0

r ✫✪ ✬✩

• x = sin α

α u(t, x)= sea-level zonally averaged temperature f(x)= solar input g(u)= co-albedo h(u)= outgoing infrared radiation

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 10 / 46

existence, uniqueness, regularity

Budyko-Sellers model

• ut −
• (1 − x2)ux
• x = f(x) g(u) − h(u)

x ∈ (−1, 1) (1 − x2)ux|x=±1 = 0

r ✫✪ ✬✩

• x = sin α

α u(t, x)= sea-level zonally averaged temperature f(x)= solar input g(u)= co-albedo h(u)= outgoing infrared radiation

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 10 / 46

existence, uniqueness, regularity

the simplest problem in 2d

similar theory n = 2      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ σ(A(x)) = {λ1(x), λ2(x)} eigenvectors ε1(x), ε2(x)

• λ1(x) ∼ dΓ(x)α ,

ε1(x) ∼ −DdΓ(x) = νΓ(ΠΓ(x)) near Γ λ2(x) ≥ m > 0 ∀x ∈ Ω

• ✠

❅ ❅ ■ q q

ΠΓ(x)

✬ ✫ ✩ ✪

Ω Γ x ε1(x) ε2(x) C–, Rocchetti & Vancostenoble (2008)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 11 / 46

boundary degeneracy

• verview

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 12 / 46

boundary degeneracy

• verview

controlled parabolic equations

ω ⊂⊂ Ω T > 0 uf →      ut − div(A(x)∇u) = χω(x)f(t, x) in QT := Ω×]0, T[ u(x, 0) = u0(x) x ∈ Ω + b. c. f control χω characteristic function of ω A(x) =

• aij(x)

n

i,j=1

aij = aji ∈ C(Ω) ∩ C1(Ω) positive definite in Ω (not in Ω) ✒✑ ✓✏

ω

✬ ✫ ✩ ✪

Ω also of interest boundary control Γ1 ⊂ Γ u(t, x) = g(t, x) (t, x) ∈ (0, T) × Γ1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 13 / 46

boundary degeneracy

• verview

controlled parabolic equations

ω ⊂⊂ Ω T > 0 uf →      ut − div(A(x)∇u) = χω(x)f(t, x) in QT := Ω×]0, T[ u(x, 0) = u0(x) x ∈ Ω + b. c. f control χω characteristic function of ω A(x) =

• aij(x)

n

i,j=1

aij = aji ∈ C(Ω) ∩ C1(Ω) positive definite in Ω (not in Ω) ✒✑ ✓✏

ω

✬ ✫ ✩ ✪

Ω also of interest boundary control Γ1 ⊂ Γ u(t, x) = g(t, x) (t, x) ∈ (0, T) × Γ1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 13 / 46

boundary degeneracy

• verview

controlled parabolic equations

ω ⊂⊂ Ω T > 0 uf →      ut − div(A(x)∇u) = χω(x)f(t, x) in QT := Ω×]0, T[ u(x, 0) = u0(x) x ∈ Ω + b. c. f control χω characteristic function of ω A(x) =

• aij(x)

n

i,j=1

aij = aji ∈ C(Ω) ∩ C1(Ω) positive definite in Ω (not in Ω) ✒✑ ✓✏

ω

✬ ✫ ✩ ✪

Ω also of interest boundary control Γ1 ⊂ Γ u(t, x) = g(t, x) (t, x) ∈ (0, T) × Γ1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 13 / 46

boundary degeneracy

• verview

the null controllability problem

want to study null-controllability in time T > 0 ∀u0 ∈ L2(Ω) ∃f ∈ L2(QT) :    uf(·, T) ≡ 0

• QT |f|2 ≤ CT
• Ω |u0|2

uniformly parabolic equations: ∃ m > 0 : A(x) ≥ m In = ⇒ null-controllability ∀ T > 0 Fattorini and Russell (1971), Russell (1978) Lebeau and Robbiano (1995) Fursikov and Emanouilov (1996) Tataru (1997)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 14 / 46

boundary degeneracy

• verview

the null controllability problem

want to study null-controllability in time T > 0 ∀u0 ∈ L2(Ω) ∃f ∈ L2(QT) :    uf(·, T) ≡ 0

• QT |f|2 ≤ CT
• Ω |u0|2

uniformly parabolic equations: ∃ m > 0 : A(x) ≥ m In = ⇒ null-controllability ∀ T > 0 Fattorini and Russell (1971), Russell (1978) Lebeau and Robbiano (1995) Fursikov and Emanouilov (1996) Tataru (1997)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 14 / 46

boundary degeneracy

• verview

the null controllability problem

want to study null-controllability in time T > 0 ∀u0 ∈ L2(Ω) ∃f ∈ L2(QT) :    uf(·, T) ≡ 0

• QT |f|2 ≤ CT
• Ω |u0|2

uniformly parabolic equations: ∃ m > 0 : A(x) ≥ m In = ⇒ null-controllability ∀ T > 0 Fattorini and Russell (1971), Russell (1978) Lebeau and Robbiano (1995) Fursikov and Emanouilov (1996) Tataru (1997)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 14 / 46

boundary degeneracy

• verview

the roadmap to null controllability

show equivalence with observability inequality adjoint problem

• vt+div(A(x)∇v) = 0

in QT v = 0

• n Γ×]0, T[

= ⇒

• Ω

v 2(x, 0) dx ≤ CT T

• ω

v 2(x, t) dxdt prove observability by Carleman estimates τ >> 1

• QT

τ 3θ3(t)v 2

• +τθ(t)|Dv|2+···

e2sφ(x,t) dxdt ≤ C T

• ω

v 2 dxdt φ(x, t) = θ(t)

• erψ(x) − e2rψ∞

any Dψ(x) = 0 in Ω \ ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 15 / 46

boundary degeneracy

• verview

the roadmap to null controllability

show equivalence with observability inequality adjoint problem

• vt+div(A(x)∇v) = 0

in QT v = 0

• n Γ×]0, T[

= ⇒

• Ω

v 2(x, 0) dx ≤ CT T

• ω

v 2(x, t) dxdt prove observability by Carleman estimates τ >> 1

• QT

τ 3θ3(t)v 2

• +τθ(t)|Dv|2+···

e2sφ(x,t) dxdt ≤ C T

• ω

v 2 dxdt φ(x, t) = θ(t)

• erψ(x) − e2rψ∞

any Dψ(x) = 0 in Ω \ ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 15 / 46

boundary degeneracy

• verview

the roadmap to null controllability

show equivalence with observability inequality adjoint problem

• vt+div(A(x)∇v) = 0

in QT v = 0

• n Γ×]0, T[

= ⇒

• Ω

v 2(x, 0) dx ≤ CT T

• ω

v 2(x, t) dxdt prove observability by Carleman estimates τ >> 1

• QT

τ 3θ3(t)v 2

• +τθ(t)|Dv|2+···

e2sφ(x,t) dxdt ≤ C T

• ω

v 2 dxdt φ(x, t) = θ(t)

• erψ(x) − e2rψ∞

any Dψ(x) = 0 in Ω \ ω

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 15 / 46

boundary degeneracy

• verview

difficulties in degenerate case

• bservability (⇒ null controllability) may fail

(for violent degeneracies) φ in Carleman must be adapted to degeneracy Hardy’s inequality essential

• Ω

dα−2

Γ

w2 dx ≤ Cα

• Ω

dα

Γ |∇w|2 dx

(α = 1)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 16 / 46

boundary degeneracy

• verview

difficulties in degenerate case

• bservability (⇒ null controllability) may fail

(for violent degeneracies) φ in Carleman must be adapted to degeneracy Hardy’s inequality essential

• Ω

dα−2

Γ

w2 dx ≤ Cα

• Ω

dα

Γ |∇w|2 dx

(α = 1)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 16 / 46

boundary degeneracy

• verview

difficulties in degenerate case

• bservability (⇒ null controllability) may fail

(for violent degeneracies) φ in Carleman must be adapted to degeneracy Hardy’s inequality essential

• Ω

dα−2

Γ

w2 dx ≤ Cα

• Ω

dα

Γ |∇w|2 dx

(α = 1)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 16 / 46

boundary degeneracy

• verview

difficulties in degenerate case

• bservability (⇒ null controllability) may fail

(for violent degeneracies) φ in Carleman must be adapted to degeneracy Hardy’s inequality essential

• Ω

dα−2

Γ

w2 dx ≤ Cα

• Ω

dα

Γ |∇w|2 dx

(α = 1)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 16 / 46

boundary degeneracy

• ne space dimension

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 17 / 46

boundary degeneracy

• ne space dimension

the simplest example of degeneracy

ω =]a, b[⊂⊂ (0, 1) a(x) = xα (α > 0) ut −

• xαux
• x = χωf ,

u(0, x) = u0(x) Theorem (C – Martinez – Vancostenoble, 2008)

• n. c.

     false α ≥ 2 (→ regional null controllability) true 0 ≤ α < 2

• any b.c.

0 ≤ α < 1 weak Neumann b.c. 1 ≤ α < 2 strong ω 1 T regional

r r r r r r r

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 18 / 46

boundary degeneracy

• ne space dimension

the simplest example of degeneracy

ω =]a, b[⊂⊂ (0, 1) a(x) = xα (α > 0) ut −

• xαux
• x = χωf ,

u(0, x) = u0(x) Theorem (C – Martinez – Vancostenoble, 2008)

• n. c.

     false α ≥ 2 (→ regional null controllability) true 0 ≤ α < 2

• any b.c.

0 ≤ α < 1 weak Neumann b.c. 1 ≤ α < 2 strong ω 1 T regional

r r r r r r r

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 18 / 46

boundary degeneracy

• ne space dimension

Carleman estimate 0 < α < 2

wt +

• xαwx
• x = f

in (0, 1) × (0, T) +

• b. c.

let ϕ(t, x) = θ(t)ψ(x) where θ(t) =

• 1

t(T − t) 4 ψ(x) = x2−α − 2 (2 − α)2 Theorem (C – Martinez – Vancostenoble, 2008) There exists τ0, C > 0 such that ∀τ ≥ τ0

• QT

w2

t

τθ + τθxαw2

x + τ 3θ3x2−αw2

e2τϕ dxdt ≤ C

• QT

|f|2e2τϕ dxdt + C T

• ω

w2(x, t) dxdt

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 19 / 46

boundary degeneracy

• ne space dimension

Carleman estimate 0 < α < 2

wt +

• xαwx
• x = f

in (0, 1) × (0, T) +

• b. c.

let ϕ(t, x) = θ(t)ψ(x) where θ(t) =

• 1

t(T − t) 4 ψ(x) = x2−α − 2 (2 − α)2 Theorem (C – Martinez – Vancostenoble, 2008) There exists τ0, C > 0 such that ∀τ ≥ τ0

• QT

w2

t

τθ + τθxαw2

x + τ 3θ3x2−αw2

e2τϕ dxdt ≤ C

• QT

|f|2e2τϕ dxdt + C T

• ω

w2(x, t) dxdt

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 19 / 46

boundary degeneracy

• ne space dimension

Carleman estimate 0 < α < 2

wt +

• xαwx
• x = f

in (0, 1) × (0, T) +

• b. c.

let ϕ(t, x) = θ(t)ψ(x) where θ(t) =

• 1

t(T − t) 4 ψ(x) = x2−α − 2 (2 − α)2 Theorem (C – Martinez – Vancostenoble, 2008) There exists τ0, C > 0 such that ∀τ ≥ τ0

• QT

w2

t

τθ + τθxαw2

x + τ 3θ3x2−αw2

e2τϕ dxdt ≤ C

• QT

|f|2e2τϕ dxdt + C T

• ω

w2(x, t) dxdt

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 19 / 46

boundary degeneracy

• ne space dimension

more general 1-d problems

divergence form

Martinez – Vancostenoble (2006) ut −

• a(x)ux
• x = χωf

Alabau – C – Fragnelli (2006) ut −

• a(x)ux
• x + g(u) = χωf

Flores – de Teresa (2010) ut −

• xθux
• x + xσb(x, t)ux = χωf

non-divergence form C – Fragnelli – Rocchetti (2007, 2008) ut − a(x)uxx − b(x)ux = χωf degenerate/singular problems

Vancostenoble – Zuazua (2008), Vancostenoble (2009) ut −

• xθux
• x −

λ xσ u = χωf

systems

C – de Teresa (2009) cascade 2 × 2 Maniar et al. (2011) general 2 × 2

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 20 / 46

boundary degeneracy

• ne space dimension

more general 1-d problems

divergence form

Martinez – Vancostenoble (2006) ut −

• a(x)ux
• x = χωf

Alabau – C – Fragnelli (2006) ut −

• a(x)ux
• x + g(u) = χωf

Flores – de Teresa (2010) ut −

• xθux
• x + xσb(x, t)ux = χωf

non-divergence form C – Fragnelli – Rocchetti (2007, 2008) ut − a(x)uxx − b(x)ux = χωf degenerate/singular problems

Vancostenoble – Zuazua (2008), Vancostenoble (2009) ut −

• xθux
• x −

λ xσ u = χωf

systems

C – de Teresa (2009) cascade 2 × 2 Maniar et al. (2011) general 2 × 2

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 20 / 46

boundary degeneracy higher space dimension

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 21 / 46

boundary degeneracy higher space dimension

extension to n ≥ 2

n = 2      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ σ(A(x)) = {λ1(x), λ2(x)} eigenvectors ε1(x), ε2(x)

• λ1(x) ∼ dΓ(x)α ,

ε1(x) ∼ −DdΓ(x) = νΓ(ΠΓ(x)) near Γ λ2(x) ≥ m > 0 ∀x ∈ Ω

• ✠

❅ ❅ ■ q q

ΠΓ(x)

✬ ✫ ✩ ✪

Ω Γ x ε1(x) ε2(x)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 22 / 46

boundary degeneracy higher space dimension

extension to n ≥ 2

n = 2      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ σ(A(x)) = {λ1(x), λ2(x)} eigenvectors ε1(x), ε2(x)

• λ1(x) ∼ dΓ(x)α ,

ε1(x) ∼ −DdΓ(x) = νΓ(ΠΓ(x)) near Γ λ2(x) ≥ m > 0 ∀x ∈ Ω

• ✠

❅ ❅ ■ q q

ΠΓ(x)

✬ ✫ ✩ ✪

Ω Γ x ε1(x) ε2(x)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 22 / 46

boundary degeneracy higher space dimension

extension to n ≥ 2

n = 2      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ σ(A(x)) = {λ1(x), λ2(x)} eigenvectors ε1(x), ε2(x)

• λ1(x) ∼ dΓ(x)α ,

ε1(x) ∼ −DdΓ(x) = νΓ(ΠΓ(x)) near Γ λ2(x) ≥ m > 0 ∀x ∈ Ω

• ✠

❅ ❅ ■ q q

ΠΓ(x)

✬ ✫ ✩ ✪

Ω Γ x ε1(x) ε2(x)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 22 / 46

boundary degeneracy higher space dimension

null controllability (n = 2)

σ(A(x)) = {λ1(x), λ2(x)} λ1(x) ∼ dΓ(x)α      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ Theorem (C, Martinez, Vancostenoble – CRAS 2009) 0 ≤ α < 2 null controllability holds α ≥ 2 null-controllability fails the proof uses topological lemma to construct adapted weight Carleman’s estimate to provide observability inequality Hardy’s inequality to control degenerate terms

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 23 / 46

boundary degeneracy higher space dimension

null controllability (n = 2)

σ(A(x)) = {λ1(x), λ2(x)} λ1(x) ∼ dΓ(x)α      ut − div(A(x)∇u) = χω(x)f(t, x) in QT u(x, 0) = u0(x) x ∈ Ω + b. c. on Γ Theorem (C, Martinez, Vancostenoble – CRAS 2009) 0 ≤ α < 2 null controllability holds α ≥ 2 null-controllability fails the proof uses topological lemma to construct adapted weight Carleman’s estimate to provide observability inequality Hardy’s inequality to control degenerate terms

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 23 / 46

boundary degeneracy inverse problems

Outline

1

Examples of degenerate parabolic equations

2

Existence, uniqueness, and regularity

3

Null controllability for boundary degeneracy Overview of null controllability Null controllability in one space dimension Null controllability in higher space dimension Application to inverse problems

4

Controllability for Grushin-type operators

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 24 / 46

boundary degeneracy inverse problems

inverse problems in 1 d

0 ≤ α < 2           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) want to ‘determine’ g by measurements of u stability estimates uniqueness results 1 T u(T ′, ·) ux(·, 1) g ← −

Figure: ‘boundary measurements’

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 25 / 46

boundary degeneracy inverse problems

inverse problems in 1 d

0 ≤ α < 2           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) want to ‘determine’ g by measurements of u stability estimates uniqueness results 1 T u(T ′, ·) ux(·, 1) g ← −

Figure: ‘boundary measurements’

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 25 / 46

boundary degeneracy inverse problems

references

uniformly parabolic case (α = 0) Bukhgeim & Klibanov (1981), Klibanov (1992), Isakov (1998), Klibanov & Timonov (2004) (H¨

• lder stability by local Carleman estimates)

Emanouilov & Yamamoto (1998, 2001) (Lipschitz stability by global Carleman estimates) degenerate case α ∈ [0, 2) C– Tort & Yamamoto (2010) (Lipschitz stability n = 1) C– Martinez & Vancostenoble (Lipschitz stability n = 2)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 26 / 46

boundary degeneracy inverse problems

references

uniformly parabolic case (α = 0) Bukhgeim & Klibanov (1981), Klibanov (1992), Isakov (1998), Klibanov & Timonov (2004) (H¨

• lder stability by local Carleman estimates)

Emanouilov & Yamamoto (1998, 2001) (Lipschitz stability by global Carleman estimates) degenerate case α ∈ [0, 2) C– Tort & Yamamoto (2010) (Lipschitz stability n = 1) C– Martinez & Vancostenoble (Lipschitz stability n = 2)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 26 / 46

boundary degeneracy inverse problems

references

uniformly parabolic case (α = 0) Bukhgeim & Klibanov (1981), Klibanov (1992), Isakov (1998), Klibanov & Timonov (2004) (H¨

• lder stability by local Carleman estimates)

Emanouilov & Yamamoto (1998, 2001) (Lipschitz stability by global Carleman estimates) degenerate case α ∈ [0, 2) C– Tort & Yamamoto (2010) (Lipschitz stability n = 1) C– Martinez & Vancostenoble (Lipschitz stability n = 2)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 26 / 46

boundary degeneracy inverse problems

Lipschitz stability: boundary measurements

           ut − (xαux)x = g(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) Theorem 0 ≤ α < 2 there exists t0 ∈ (0, T) and k ≥ 0 such that

• ∂g

∂t (t, x)

• ≤ k
• g(T ′, x)
• where

T ′ = T + t0 2 = ⇒ g2

L2(QT ) ≤ C

T

t0

|utx(t, 1)|2dt + C 1 |(xαux(T ′, x))x|2dx where C = C(α, k, t0, T)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 27 / 46

boundary degeneracy inverse problems

Lipschitz stability: boundary measurements

           ut − (xαux)x = g(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) Theorem 0 ≤ α < 2 there exists t0 ∈ (0, T) and k ≥ 0 such that

• ∂g

∂t (t, x)

• ≤ k
• g(T ′, x)
• where

T ′ = T + t0 2 = ⇒ g2

L2(QT ) ≤ C

T

t0

|utx(t, 1)|2dt + C 1 |(xαux(T ′, x))x|2dx where C = C(α, k, t0, T)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 27 / 46

boundary degeneracy inverse problems

uniqueness: boundary measurements

           ut − (xαux)x = f(x)r(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) (IP) r ∈ C1([0, T] × [0, 1]) given such that at T ′ = T + t0 2 we have r(T

′, x) ≥ d > 0

∀x ∈ [0, 1] (∗) f1, f2 ∈ L2(0, 1) u1, u2 ∈ L2(0, 1) solutions of (IP) = ⇒ f1 − f22

L2(0,1) ≤ C

T

t0

|(u1 − u2)tx(t, 1)|2dt + C 1 |(xα(u1(T ′, x) − u2(T ′, x))x)x|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 28 / 46

boundary degeneracy inverse problems

uniqueness: boundary measurements

           ut − (xαux)x = f(x)r(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) (IP) r ∈ C1([0, T] × [0, 1]) given such that at T ′ = T + t0 2 we have r(T

′, x) ≥ d > 0

∀x ∈ [0, 1] (∗) f1, f2 ∈ L2(0, 1) u1, u2 ∈ L2(0, 1) solutions of (IP) = ⇒ f1 − f22

L2(0,1) ≤ C

T

t0

|(u1 − u2)tx(t, 1)|2dt + C 1 |(xα(u1(T ′, x) − u2(T ′, x))x)x|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 28 / 46

boundary degeneracy inverse problems

uniqueness: boundary measurements

           ut − (xαux)x = f(x)r(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) (IP) r ∈ C1([0, T] × [0, 1]) given such that at T ′ = T + t0 2 we have r(T

′, x) ≥ d > 0

∀x ∈ [0, 1] (∗) f1, f2 ∈ L2(0, 1) u1, u2 ∈ L2(0, 1) solutions of (IP) = ⇒ f1 − f22

L2(0,1) ≤ C

T

t0

|(u1 − u2)tx(t, 1)|2dt + C 1 |(xα(u1(T ′, x) − u2(T ′, x))x)x|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 28 / 46

boundary degeneracy inverse problems

uniqueness: boundary measurements

           ut − (xαux)x = f(x)r(t, x) QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) (IP) r ∈ C1([0, T] × [0, 1]) given such that at T ′ = T + t0 2 we have r(T

′, x) ≥ d > 0

∀x ∈ [0, 1] (∗) f1, f2 ∈ L2(0, 1) u1, u2 ∈ L2(0, 1) solutions of (IP) = ⇒ f1 − f22

L2(0,1) ≤ C

T

t0

|(u1 − u2)tx(t, 1)|2dt + C 1 |(xα(u1(T ′, x) − u2(T ′, x))x)x|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 28 / 46

boundary degeneracy inverse problems

locally distributed measurements

           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) g ← − u(T ′, ·) ω 1 T ut

r r r r r r r

stability estimates uniqueness results

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 29 / 46

boundary degeneracy inverse problems

locally distributed measurements

           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) g ← − u(T ′, ·) ω 1 T ut

r r r r r r r

stability estimates uniqueness results

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 29 / 46

boundary degeneracy inverse problems

locally distributed measurements

           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) g ← − u(T ′, ·) ω 1 T ut

r r r r r r r

stability estimates uniqueness results

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 29 / 46

boundary degeneracy inverse problems

locally distributed measurements

           ut − (xαux)x = g QT = (0, T) × (0, 1) u (t, 1) = 0 t ∈ (0, T) u (t, 0) = 0 for 0 ≤ α < 1 (xαux)|x=0 = 0 for 1 ≤ α < 2 t ∈ (0, T) u (0, x) = u0(x) x ∈ (0, 1) g ← − u(T ′, ·) ω 1 T ut

r r r r r r r

stability estimates uniqueness results

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 29 / 46

control with interior degeneracy

Grushin-type operators

Ω := (−1, 1) × (0, 1), ω ⊂ (a, b) × (0, 1) with 0 < a < b < 1 −1 1 x Ω ω

♥

1 y a b γ > 0        ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 30 / 46

control with interior degeneracy

Grushin-type operators

Ω := (−1, 1) × (0, 1), ω ⊂ (a, b) × (0, 1) with 0 < a < b < 1 −1 1 x Ω ω

♥

1 y a b γ > 0        ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 30 / 46

control with interior degeneracy

existence and uniqueness of solutions

In H = L2(Ω) with scalar product (f, g) =

• Ω
• fxgx + |x|2γfygy
• dxdy

V = C∞

0 (Ω)

a(f, g) = −(f, g) ∀f, g ∈ V D(A) = {f ∈ V : ∃ c > 0 such that |a(f, h)| ≤ chH ∀h ∈ V} and Af, h = a(f, h) ∀h ∈ V A : D(A) ⊂ H → H generator of a semigroup etA of contractions in H Theorem T > 0 , u0 ∈ L2(Ω) , f ∈ L2((0, T) × Ω) = ⇒ ∃! u ∈ C([0, T]; L2(Ω)) : ∀t ∈ (0, T) , φ ∈ C2([0, T] × Ω)

• Ω

[u(t)φ(t) − u(0)φ(0)] = t

• Ω

u(∂tφ + ∂2

xφ + |x|2γ∂2 yφ) + χωfφ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 31 / 46

control with interior degeneracy

existence and uniqueness of solutions

In H = L2(Ω) with scalar product (f, g) =

• Ω
• fxgx + |x|2γfygy
• dxdy

V = C∞

0 (Ω)

a(f, g) = −(f, g) ∀f, g ∈ V D(A) = {f ∈ V : ∃ c > 0 such that |a(f, h)| ≤ chH ∀h ∈ V} and Af, h = a(f, h) ∀h ∈ V A : D(A) ⊂ H → H generator of a semigroup etA of contractions in H Theorem T > 0 , u0 ∈ L2(Ω) , f ∈ L2((0, T) × Ω) = ⇒ ∃! u ∈ C([0, T]; L2(Ω)) : ∀t ∈ (0, T) , φ ∈ C2([0, T] × Ω)

• Ω

[u(t)φ(t) − u(0)φ(0)] = t

• Ω

u(∂tφ + ∂2

xφ + |x|2γ∂2 yφ) + χωfφ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 31 / 46

control with interior degeneracy

existence and uniqueness of solutions

In H = L2(Ω) with scalar product (f, g) =

• Ω
• fxgx + |x|2γfygy
• dxdy

V = C∞

0 (Ω)

a(f, g) = −(f, g) ∀f, g ∈ V D(A) = {f ∈ V : ∃ c > 0 such that |a(f, h)| ≤ chH ∀h ∈ V} and Af, h = a(f, h) ∀h ∈ V A : D(A) ⊂ H → H generator of a semigroup etA of contractions in H Theorem T > 0 , u0 ∈ L2(Ω) , f ∈ L2((0, T) × Ω) = ⇒ ∃! u ∈ C([0, T]; L2(Ω)) : ∀t ∈ (0, T) , φ ∈ C2([0, T] × Ω)

• Ω

[u(t)φ(t) − u(0)φ(0)] = t

• Ω

u(∂tφ + ∂2

xφ + |x|2γ∂2 yφ) + χωfφ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 31 / 46

control with interior degeneracy

approximate controllability

approximate controllability ⇐ ⇒ unique continuation Garofalo (1993): unique continuation for elliptic operator A = ∂2 + |x|2γ∂2

y

for parabolic operators: Proposition (BCG 2012) Let T > 0, γ > 0, let ω ⊂ (0, 1) × (0, 1), and let g ∈ C([0, T]; H) ∩ L2(0, T; V) be a weak solution of

• ∂tg − ∂2

x g − |x|2γ∂2 yg = 0

(t, x, y) ∈ (0, ∞) × Ω g(t, x, y) = 0 (t, x, y) ∈ (0, ∞) × ∂Ω If g ≡ 0 on (0, T) × ω, then g ≡ 0 on (0, T) × Ω.

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 32 / 46

control with interior degeneracy

approximate controllability

approximate controllability ⇐ ⇒ unique continuation Garofalo (1993): unique continuation for elliptic operator A = ∂2 + |x|2γ∂2

y

for parabolic operators: Proposition (BCG 2012) Let T > 0, γ > 0, let ω ⊂ (0, 1) × (0, 1), and let g ∈ C([0, T]; H) ∩ L2(0, T; V) be a weak solution of

• ∂tg − ∂2

x g − |x|2γ∂2 yg = 0

(t, x, y) ∈ (0, ∞) × Ω g(t, x, y) = 0 (t, x, y) ∈ (0, ∞) × ∂Ω If g ≡ 0 on (0, T) × ω, then g ≡ 0 on (0, T) × Ω.

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 32 / 46

control with interior degeneracy

approximate controllability

approximate controllability ⇐ ⇒ unique continuation Garofalo (1993): unique continuation for elliptic operator A = ∂2 + |x|2γ∂2

y

for parabolic operators: Proposition (BCG 2012) Let T > 0, γ > 0, let ω ⊂ (0, 1) × (0, 1), and let g ∈ C([0, T]; H) ∩ L2(0, T; V) be a weak solution of

• ∂tg − ∂2

x g − |x|2γ∂2 yg = 0

(t, x, y) ∈ (0, ∞) × Ω g(t, x, y) = 0 (t, x, y) ∈ (0, ∞) × ∂Ω If g ≡ 0 on (0, T) × ω, then g ≡ 0 on (0, T) × Ω.

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 32 / 46

control with interior degeneracy

approximate controllability

approximate controllability ⇐ ⇒ unique continuation Garofalo (1993): unique continuation for elliptic operator A = ∂2 + |x|2γ∂2

y

for parabolic operators: Proposition (BCG 2012) Let T > 0, γ > 0, let ω ⊂ (0, 1) × (0, 1), and let g ∈ C([0, T]; H) ∩ L2(0, T; V) be a weak solution of

• ∂tg − ∂2

x g − |x|2γ∂2 yg = 0

(t, x, y) ∈ (0, ∞) × Ω g(t, x, y) = 0 (t, x, y) ∈ (0, ∞) × ∂Ω If g ≡ 0 on (0, T) × ω, then g ≡ 0 on (0, T) × Ω.

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 32 / 46

control with interior degeneracy

null controllability

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) −1 1 x Ω ω

♥

1 y a b Theorem (BCG 2012) 0 < γ < 1 ⇒ (G) null controllable ∀T > 0 γ > 1 ⇒ (G) not null controllable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G) is

null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy

null controllability

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) −1 1 x Ω ω

♥

1 y a b Theorem (BCG 2012) 0 < γ < 1 ⇒ (G) null controllable ∀T > 0 γ > 1 ⇒ (G) not null controllable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G) is

null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy

null controllability

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) −1 1 x Ω ω

♥

1 y a b Theorem (BCG 2012) 0 < γ < 1 ⇒ (G) null controllable ∀T > 0 γ > 1 ⇒ (G) not null controllable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G) is

null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy

null controllability

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) −1 1 x Ω ω

♥

1 y a b Theorem (BCG 2012) 0 < γ < 1 ⇒ (G) null controllable ∀T > 0 γ > 1 ⇒ (G) not null controllable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G) is

null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy

null controllability

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) −1 1 x Ω ω

♥

1 y a b Theorem (BCG 2012) 0 < γ < 1 ⇒ (G) null controllable ∀T > 0 γ > 1 ⇒ (G) not null controllable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G) is

null controllable ∀ T > T ∗ not null controllable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 33 / 46

control with interior degeneracy

• bservability

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗)

• bservable in [0, T] × ω

∃CT > 0 such that ∀v0 ∈ L2(Ω)

• Ω

|v(T, x, y)|2dxdy ≤ CT T

• ω

|v(t, x, y)|2dxdy (O) Theorem (BCG 2012) 0 < γ < 1 ⇒ (G∗)

• bservable

∀T > 0 γ > 1 ⇒ (G∗) not observable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G∗) is

• bservable

∀ T > T ∗ not observable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 34 / 46

control with interior degeneracy

• bservability

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗)

• bservable in [0, T] × ω

∃CT > 0 such that ∀v0 ∈ L2(Ω)

• Ω

|v(T, x, y)|2dxdy ≤ CT T

• ω

|v(t, x, y)|2dxdy (O) Theorem (BCG 2012) 0 < γ < 1 ⇒ (G∗)

• bservable

∀T > 0 γ > 1 ⇒ (G∗) not observable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G∗) is

• bservable

∀ T > T ∗ not observable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 34 / 46

control with interior degeneracy

• bservability

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗)

• bservable in [0, T] × ω

∃CT > 0 such that ∀v0 ∈ L2(Ω)

• Ω

|v(T, x, y)|2dxdy ≤ CT T

• ω

|v(t, x, y)|2dxdy (O) Theorem (BCG 2012) 0 < γ < 1 ⇒ (G∗)

• bservable

∀T > 0 γ > 1 ⇒ (G∗) not observable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G∗) is

• bservable

∀ T > T ∗ not observable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 34 / 46

control with interior degeneracy

• bservability

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗)

• bservable in [0, T] × ω

∃CT > 0 such that ∀v0 ∈ L2(Ω)

• Ω

|v(T, x, y)|2dxdy ≤ CT T

• ω

|v(t, x, y)|2dxdy (O) Theorem (BCG 2012) 0 < γ < 1 ⇒ (G∗)

• bservable

∀T > 0 γ > 1 ⇒ (G∗) not observable γ = 1 & ω = (a, b) × (0, 1) = ⇒ ∃T ∗ a2/2 such that (G∗) is

• bservable

∀ T > T ∗ not observable ∀ T < T ∗

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 34 / 46

control with interior degeneracy

method: Fourier decomposition

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗) v(t, x, y) =

∞

• n=1

vn(t, x)ϕn(y) with ϕn(y) := √ 2 sin(nπy) where vn(t, x) := 1

0 v(t, x, y)ϕn(y)dy

satisfies    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• Ω

|v(T, x, y)|2dxdy =

∞

• n=1

1

−1

|vn(T, x)|2dx

• ω=(a,b)×(0,1)

|v(t, x, y)|2dxdy =

∞

• n=1

b

a

|vn(t, x)|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 35 / 46

control with interior degeneracy

method: Fourier decomposition

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗) v(t, x, y) =

∞

• n=1

vn(t, x)ϕn(y) with ϕn(y) := √ 2 sin(nπy) where vn(t, x) := 1

0 v(t, x, y)ϕn(y)dy

satisfies    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• Ω

|v(T, x, y)|2dxdy =

∞

• n=1

1

−1

|vn(T, x)|2dx

• ω=(a,b)×(0,1)

|v(t, x, y)|2dxdy =

∞

• n=1

b

a

|vn(t, x)|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 35 / 46

control with interior degeneracy

method: Fourier decomposition

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗) v(t, x, y) =

∞

• n=1

vn(t, x)ϕn(y) with ϕn(y) := √ 2 sin(nπy) where vn(t, x) := 1

0 v(t, x, y)ϕn(y)dy

satisfies    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• Ω

|v(T, x, y)|2dxdy =

∞

• n=1

1

−1

|vn(T, x)|2dx

• ω=(a,b)×(0,1)

|v(t, x, y)|2dxdy =

∞

• n=1

b

a

|vn(t, x)|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 35 / 46

control with interior degeneracy

method: Fourier decomposition

   ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) (G∗) v(t, x, y) =

∞

• n=1

vn(t, x)ϕn(y) with ϕn(y) := √ 2 sin(nπy) where vn(t, x) := 1

0 v(t, x, y)ϕn(y)dy

satisfies    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• Ω

|v(T, x, y)|2dxdy =

∞

• n=1

1

−1

|vn(T, x)|2dx

• ω=(a,b)×(0,1)

|v(t, x, y)|2dxdy =

∞

• n=1

b

a

|vn(t, x)|2dx

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 35 / 46

control with interior degeneracy

uniform observability

   ∂tv − ∂2

x v − |x|2γ∂2 y v = 0

(0, T) × Ω v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) t ∈ (0, T) v(0, x, y) = v0(x, y) (x, y) ∈ Ω (G∗) −1 1 x Ω ω 1 y a b    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• bservability for (G∗) in ω

⇐ ⇒ uniform observability for (G∗

n) in (a, b)

1

−1

|vn(T, x)|2dx ≤ C T b

a

|vn(t, x)|2dxdt ∀n ≥ 1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 36 / 46

control with interior degeneracy

uniform observability

   ∂tv − ∂2

x v − |x|2γ∂2 y v = 0

(0, T) × Ω v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) t ∈ (0, T) v(0, x, y) = v0(x, y) (x, y) ∈ Ω (G∗) −1 1 x Ω ω 1 y a b    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

• bservability for (G∗) in ω

⇐ ⇒ uniform observability for (G∗

n) in (a, b)

1

−1

|vn(T, x)|2dx ≤ C T b

a

|vn(t, x)|2dxdt ∀n ≥ 1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 36 / 46

control with interior degeneracy

step 1: dissipation speed

define An : D(An) ⊂ L2(−1, 1) → L2(−1, 1) by D(An) := H2 ∩ H1

0(−1, 1) ,

Anϕ := −ϕ′′ + (nπ)2|x|2γϕ λn := the first eigenvalue of An so that

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) satisfies 1

−1

|vn(T, x)|2dx ≤ e−λn(T−t) 1

−1

|vn(t, x)|2dx ∀t ∈ [0, T] (Dn) Lemma (dissipation speed) (ub) ∀γ > 0 ∃c∗ > 0 such that λn ≤ c∗n

2 1+γ

(lb) ∀γ ∈ (0, 1] ∃c∗ > 0 such that λn ≥ c∗n

2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 37 / 46

control with interior degeneracy

step 1: dissipation speed

define An : D(An) ⊂ L2(−1, 1) → L2(−1, 1) by D(An) := H2 ∩ H1

0(−1, 1) ,

Anϕ := −ϕ′′ + (nπ)2|x|2γϕ λn := the first eigenvalue of An so that

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) satisfies 1

−1

|vn(T, x)|2dx ≤ e−λn(T−t) 1

−1

|vn(t, x)|2dx ∀t ∈ [0, T] (Dn) Lemma (dissipation speed) (ub) ∀γ > 0 ∃c∗ > 0 such that λn ≤ c∗n

2 1+γ

(lb) ∀γ ∈ (0, 1] ∃c∗ > 0 such that λn ≥ c∗n

2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 37 / 46

control with interior degeneracy

step 1: dissipation speed

define An : D(An) ⊂ L2(−1, 1) → L2(−1, 1) by D(An) := H2 ∩ H1

0(−1, 1) ,

Anϕ := −ϕ′′ + (nπ)2|x|2γϕ λn := the first eigenvalue of An so that

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) satisfies 1

−1

|vn(T, x)|2dx ≤ e−λn(T−t) 1

−1

|vn(t, x)|2dx ∀t ∈ [0, T] (Dn) Lemma (dissipation speed) (ub) ∀γ > 0 ∃c∗ > 0 such that λn ≤ c∗n

2 1+γ

(lb) ∀γ ∈ (0, 1] ∃c∗ > 0 such that λn ≥ c∗n

2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 37 / 46

control with interior degeneracy

step 1: dissipation speed

define An : D(An) ⊂ L2(−1, 1) → L2(−1, 1) by D(An) := H2 ∩ H1

0(−1, 1) ,

Anϕ := −ϕ′′ + (nπ)2|x|2γϕ λn := the first eigenvalue of An so that

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) satisfies 1

−1

|vn(T, x)|2dx ≤ e−λn(T−t) 1

−1

|vn(t, x)|2dx ∀t ∈ [0, T] (Dn) Lemma (dissipation speed) (ub) ∀γ > 0 ∃c∗ > 0 such that λn ≤ c∗n

2 1+γ

(lb) ∀γ ∈ (0, 1] ∃c∗ > 0 such that λn ≥ c∗n

2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 37 / 46

control with interior degeneracy

failure of (UO): γ > 1 and γ = 1

take eigenfunctions wn of An associated with λn −w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 vn(t, x) := e−λntwn(x) solution to

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) (UO) fails if can provide upper bound such that T b

a |vn(t, x)|2dxdt

1

−1 |vn(T, x)|2dx

= e2λnT − 1 2λn b

a

|wn(x)|2dx → 0 (n → ∞) technical because (nπ)2|x|2γ − λn changes sign in [−1, 1]

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 38 / 46

control with interior degeneracy

failure of (UO): γ > 1 and γ = 1

take eigenfunctions wn of An associated with λn −w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 vn(t, x) := e−λntwn(x) solution to

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) (UO) fails if can provide upper bound such that T b

a |vn(t, x)|2dxdt

1

−1 |vn(T, x)|2dx

= e2λnT − 1 2λn b

a

|wn(x)|2dx → 0 (n → ∞) technical because (nπ)2|x|2γ − λn changes sign in [−1, 1]

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 38 / 46

control with interior degeneracy

failure of (UO): γ > 1 and γ = 1

take eigenfunctions wn of An associated with λn −w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 vn(t, x) := e−λntwn(x) solution to

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) (UO) fails if can provide upper bound such that T b

a |vn(t, x)|2dxdt

1

−1 |vn(T, x)|2dx

= e2λnT − 1 2λn b

a

|wn(x)|2dx → 0 (n → ∞) technical because (nπ)2|x|2γ − λn changes sign in [−1, 1]

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 38 / 46

control with interior degeneracy

failure of (UO): γ > 1 and γ = 1

take eigenfunctions wn of An associated with λn −w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 vn(t, x) := e−λntwn(x) solution to

• ∂tvn − ∂2

xvn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) (UO) fails if can provide upper bound such that T b

a |vn(t, x)|2dxdt

1

−1 |vn(T, x)|2dx

= e2λnT − 1 2λn b

a

|wn(x)|2dx → 0 (n → ∞) technical because (nπ)2|x|2γ − λn changes sign in [−1, 1]

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 38 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

comparison argument

−w′′

n (x) +

• (nπ)2|x|2γ − λn
• wn(x) = 0

x ∈ (−1, 1) wn(±1) = 0 , wn ≥ 0 , wnL2(−1,1) = 1 restrict to [xn, 1] with xn :=

• λn

(nπ)2

1

2γ → 0 as n → ∞

equation yields upper bound |w′

n(xn)| √xnλn

by comparison argument    −W ′′

n (x) + [(nπ)2x2γ − λn]Wn(x) 0

Wn(1) 0 , W ′

n(xn) < −√xnλn

= ⇒ b

a

w2

n dx

b

a

W 2

n dx

construct Cn > 0 such that Wn(x) := Cne−Cγnxγ+1 satisfies e2λnT − 1 2λn b

a

|wn|2dx ≤ e2λnT 2λn b

a

|Wn|2dx ≤ e2n( λn

n T−Cγ) R(n)

rational

conclude e2n( λn

n T−Cγ) → 0 because of dissipation speed λn ≤ c∗n 2 1+γ

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 39 / 46

control with interior degeneracy

step 2: Carleman estimate

For n 1 introduce operator Png := ∂g ∂t − ∂2g ∂x2 + (nπ)2|x|2γg (t, x) ∈ (0, T) × (−1, 1) Theorem Let γ ∈ (0, 1] and let 0 < a < b 1 Then there exist β ∈ C1([−1, 1]; R∗

+) and C1, C2 > 0 such that

C1 T 1

−1

• M

t(T − t)

• ∂g

∂x (t, x)

• 2 +

M3 (t(T − t))3

• g(t, x)
• 2
• e

− Mβ(x)

t(T−t) dxdt

• T

1

−1

|Png|2e

− Mβ(x)

t(T−t) dxdt +

T b

a

M3 (t(T − t))3 |g(t, x)|2e

− Mβ(x)

t(T−t) dxdt

for every n 1, T > 0 and g ∈ C0([0, T]; L2(−1, 1)) ∩ L2(0, T; H1

0(−1, 1)) where

M := C2 max{T + T 2; nT 2}

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 40 / 46

control with interior degeneracy

step 2: Carleman estimate

For n 1 introduce operator Png := ∂g ∂t − ∂2g ∂x2 + (nπ)2|x|2γg (t, x) ∈ (0, T) × (−1, 1) Theorem Let γ ∈ (0, 1] and let 0 < a < b 1 Then there exist β ∈ C1([−1, 1]; R∗

+) and C1, C2 > 0 such that

C1 T 1

−1

• M

t(T − t)

• ∂g

∂x (t, x)

• 2 +

M3 (t(T − t))3

• g(t, x)
• 2
• e

− Mβ(x)

t(T−t) dxdt

• T

1

−1

|Png|2e

− Mβ(x)

t(T−t) dxdt +

T b

a

M3 (t(T − t))3 |g(t, x)|2e

− Mβ(x)

t(T−t) dxdt

for every n 1, T > 0 and g ∈ C0([0, T]; L2(−1, 1)) ∩ L2(0, T; H1

0(−1, 1)) where

M := C2 max{T + T 2; nT 2}

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 40 / 46

control with interior degeneracy

step 2: Carleman estimate

For n 1 introduce operator Png := ∂g ∂t − ∂2g ∂x2 + (nπ)2|x|2γg (t, x) ∈ (0, T) × (−1, 1) Theorem Let γ ∈ (0, 1] and let 0 < a < b 1 Then there exist β ∈ C1([−1, 1]; R∗

+) and C1, C2 > 0 such that

C1 T 1

−1

• M

t(T − t)

• ∂g

∂x (t, x)

• 2 +

M3 (t(T − t))3

• g(t, x)
• 2
• e

− Mβ(x)

t(T−t) dxdt

• T

1

−1

|Png|2e

− Mβ(x)

t(T−t) dxdt +

T b

a

M3 (t(T − t))3 |g(t, x)|2e

− Mβ(x)

t(T−t) dxdt

for every n 1, T > 0 and g ∈ C0([0, T]; L2(−1, 1)) ∩ L2(0, T; H1

0(−1, 1)) where

M := C2 max{T + T 2; nT 2}

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 40 / 46

control with interior degeneracy

uniform observability

Carleman estimate yields uniform observability for    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

Theorem Let γ ∈ (0, 1) and let 0 < a < b 1 Then there exists C > 0 such that for every T > 0, n 1, and v0,n ∈ L2(−1, 1) 1

−1

vn(T, x)2dx CT 2e

C

• 1+T

− 1+γ 1−γ

T

b

a

vn(t, x)2dxdt (UO) Theorem γ = 1 and 0 < a < b 1 = ⇒ ∃ T1 a2/2 such that ∀ T > T1 (G∗

n) is uniformly observable with respect to n on (a, b) in time T

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 41 / 46

control with interior degeneracy

uniform observability

Carleman estimate yields uniform observability for    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

Theorem Let γ ∈ (0, 1) and let 0 < a < b 1 Then there exists C > 0 such that for every T > 0, n 1, and v0,n ∈ L2(−1, 1) 1

−1

vn(T, x)2dx CT 2e

C

• 1+T

− 1+γ 1−γ

T

b

a

vn(t, x)2dxdt (UO) Theorem γ = 1 and 0 < a < b 1 = ⇒ ∃ T1 a2/2 such that ∀ T > T1 (G∗

n) is uniformly observable with respect to n on (a, b) in time T

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 41 / 46

control with interior degeneracy

uniform observability

Carleman estimate yields uniform observability for    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

Theorem Let γ ∈ (0, 1) and let 0 < a < b 1 Then there exists C > 0 such that for every T > 0, n 1, and v0,n ∈ L2(−1, 1) 1

−1

vn(T, x)2dx CT 2e

C

• 1+T

− 1+γ 1−γ

T

b

a

vn(t, x)2dxdt (UO) Theorem γ = 1 and 0 < a < b 1 = ⇒ ∃ T1 a2/2 such that ∀ T > T1 (G∗

n) is uniformly observable with respect to n on (a, b) in time T

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 41 / 46

control with interior degeneracy

uniform observability

Carleman estimate yields uniform observability for    ∂tvn − ∂2

x vn + (nπ)2|x|2γvn = 0

(t, x) ∈ (0, T) × (−1, 1) vn(t, ±1) = 0 t ∈ (0, T) vn(0, x) = v0,n(x) x ∈ (−1, 1) (G∗

n)

Theorem Let γ ∈ (0, 1) and let 0 < a < b 1 Then there exists C > 0 such that for every T > 0, n 1, and v0,n ∈ L2(−1, 1) 1

−1

vn(T, x)2dx CT 2e

C

• 1+T

− 1+γ 1−γ

T

b

a

vn(t, x)2dxdt (UO) Theorem γ = 1 and 0 < a < b 1 = ⇒ ∃ T1 a2/2 such that ∀ T > T1 (G∗

n) is uniformly observable with respect to n on (a, b) in time T

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 41 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1)

apply technique by Benabdallah-Dermenjian-Le Rousseau (2007) ϕn(y) := √ 2 sin(nπy) y ∈ [0, 1] , n 1 recall Proposition (Lebeau-Robbiano) Let c, d ∈ R be such that c < d There exists C > 0 such that, for every n 1 and (bk)1kn ∈ Rn,

n

• k=1

|bk|2 CeCn d

c

• n
• k=1

bkϕk(y)

• 2

dy

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 42 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1)

apply technique by Benabdallah-Dermenjian-Le Rousseau (2007) ϕn(y) := √ 2 sin(nπy) y ∈ [0, 1] , n 1 recall Proposition (Lebeau-Robbiano) Let c, d ∈ R be such that c < d There exists C > 0 such that, for every n 1 and (bk)1kn ∈ Rn,

n

• k=1

|bk|2 CeCn d

c

• n
• k=1

bkϕk(y)

• 2

dy

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 42 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1)

apply technique by Benabdallah-Dermenjian-Le Rousseau (2007) ϕn(y) := √ 2 sin(nπy) y ∈ [0, 1] , n 1 recall Proposition (Lebeau-Robbiano) Let c, d ∈ R be such that c < d There exists C > 0 such that, for every n 1 and (bk)1kn ∈ Rn,

n

• k=1

|bk|2 CeCn d

c

• n
• k=1

bkϕk(y)

• 2

dy

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 42 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1)

apply technique by Benabdallah-Dermenjian-Le Rousseau (2007) ϕn(y) := √ 2 sin(nπy) y ∈ [0, 1] , n 1 recall Proposition (Lebeau-Robbiano) Let c, d ∈ R be such that c < d There exists C > 0 such that, for every n 1 and (bk)1kn ∈ Rn,

n

• k=1

|bk|2 CeCn d

c

• n
• k=1

bkϕk(y)

• 2

dy

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 42 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (continued)

study observability for adjoint problem on finite dimensional subspaces    ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) ∈ Ej (G∗) by Carleman estimate and Lebeau-Robbiano’s lemma Hn := L2(−1, 1) ⊗ ϕn (n 1) Ej := ⊕n2j Hn (j 0) Proposition Let γ ∈ (0, 1), and let a, b, c, d ∈ R be such that 0 < a < b < 1 and 0 < c < d < 1 Then there exists C > 0 such that for every T > 0 and v0 ∈ Ej (j 1)

• Ω

v(T, x, y)2dxdy e

C

• 2j +T

− 1+γ 1−γ

T

• ω

v(t, x, y)2dxdydt where ω := (a, b) × (c, d)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 43 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (continued)

study observability for adjoint problem on finite dimensional subspaces    ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) ∈ Ej (G∗) by Carleman estimate and Lebeau-Robbiano’s lemma Hn := L2(−1, 1) ⊗ ϕn (n 1) Ej := ⊕n2j Hn (j 0) Proposition Let γ ∈ (0, 1), and let a, b, c, d ∈ R be such that 0 < a < b < 1 and 0 < c < d < 1 Then there exists C > 0 such that for every T > 0 and v0 ∈ Ej (j 1)

• Ω

v(T, x, y)2dxdy e

C

• 2j +T

− 1+γ 1−γ

T

• ω

v(t, x, y)2dxdydt where ω := (a, b) × (c, d)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 43 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (continued)

study observability for adjoint problem on finite dimensional subspaces    ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) ∈ Ej (G∗) by Carleman estimate and Lebeau-Robbiano’s lemma Hn := L2(−1, 1) ⊗ ϕn (n 1) Ej := ⊕n2j Hn (j 0) Proposition Let γ ∈ (0, 1), and let a, b, c, d ∈ R be such that 0 < a < b < 1 and 0 < c < d < 1 Then there exists C > 0 such that for every T > 0 and v0 ∈ Ej (j 1)

• Ω

v(T, x, y)2dxdy e

C

• 2j +T

− 1+γ 1−γ

T

• ω

v(t, x, y)2dxdydt where ω := (a, b) × (c, d)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 43 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (continued)

study observability for adjoint problem on finite dimensional subspaces    ∂tv − ∂2

x v − |x|2γ∂2 yv = 0

v(t, ±1, y) = 0 , v(t, x, 0) = 0 = v(t, x, 1) v(0, x, y) = v0(x, y) ∈ Ej (G∗) by Carleman estimate and Lebeau-Robbiano’s lemma Hn := L2(−1, 1) ⊗ ϕn (n 1) Ej := ⊕n2j Hn (j 0) Proposition Let γ ∈ (0, 1), and let a, b, c, d ∈ R be such that 0 < a < b < 1 and 0 < c < d < 1 Then there exists C > 0 such that for every T > 0 and v0 ∈ Ej (j 1)

• Ω

v(T, x, y)2dxdy e

C

• 2j +T

− 1+γ 1−γ

T

• ω

v(t, x, y)2dxdydt where ω := (a, b) × (c, d)

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 43 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

null controllability for γ ∈ (0, 1) (completed)

   ∂tu − ∂2

x u − |x|2γ∂2 y u = χω(x, y)f(t, x, y)

u(t, ±1, y) = 0 , u(t, x, 0) = 0 = u(t, x, 1) u(0, x, y) = u0(x, y) (G) fix 0 < ρ < 1−γ

1+γ and let K = K(ρ) > 0 be such that K ∞ j=1 2−jρ = T

let Tj := K2−jρ and let (aj)j∈N be defined by a0 = 0 , aj+1 = aj + 2Tj

• n [aj, aj + Tj] apply control f such that ΠEj u(aj + Tj, ·) = 0 and

fL2(aj ,aj +Tj ;L2(Ω)) Cju(aj, ·)L2(Ω) with Cj := eC

• 2j +T

− 1+γ 1−γ j

• and u(aj + Tj, ·)L2(Ω) (1 +
• TjCj)u(aj, ·)L2(Ω)

no control on [aj + Tj, aj+1] ⇒ u(aj+1, ·)L2(Ω) e−λ2j Tj u(aj + Tj, ·)L2(Ω) combining above inequalities to conclude u(aj+1, ·)L2(Ω) exp

• 2j
• k=1
• ln(1 +
• TkCk) − C(2k)

2 1+γ Tk

• →−∞ as j→∞
• u0L2(Ω)
• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 44 / 46

control with interior degeneracy

concluding remarks . . .

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

−1 1 x Ω ω ♥ 1 y a b we have proved that null controllability holds in any positive time when γ ∈ (0, 1) and ω ⊂ (0, 1) × (0, 1) holds only in large time when γ = 1 and ω = (a, b) × (0, 1) does not hold when degeneracy is too strong, i.e. γ > 1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 45 / 46

control with interior degeneracy

concluding remarks . . .

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

−1 1 x Ω ω ♥ 1 y a b we have proved that null controllability holds in any positive time when γ ∈ (0, 1) and ω ⊂ (0, 1) × (0, 1) holds only in large time when γ = 1 and ω = (a, b) × (0, 1) does not hold when degeneracy is too strong, i.e. γ > 1

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 45 / 46

control with interior degeneracy

help needed!

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

to study null controllability for γ = 1 and more general ω sharp estimate of T ∗ for γ = 1 (T ∗ = a2/2?) muldimensional configurations and/or boundary control

thank you for your attention

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 46 / 46

control with interior degeneracy

help needed!

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

to study null controllability for γ = 1 and more general ω sharp estimate of T ∗ for γ = 1 (T ∗ = a2/2?) muldimensional configurations and/or boundary control

thank you for your attention

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 46 / 46

control with interior degeneracy

help needed!

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

to study null controllability for γ = 1 and more general ω sharp estimate of T ∗ for γ = 1 (T ∗ = a2/2?) muldimensional configurations and/or boundary control

thank you for your attention

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 46 / 46

control with interior degeneracy

help needed!

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

to study null controllability for γ = 1 and more general ω sharp estimate of T ∗ for γ = 1 (T ∗ = a2/2?) muldimensional configurations and/or boundary control

thank you for your attention

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 46 / 46

control with interior degeneracy

help needed!

∂tu − ∂2

xu − |x|2γ∂2 yu = χω(x, y)f(x, y, t)

to study null controllability for γ = 1 and more general ω sharp estimate of T ∗ for γ = 1 (T ∗ = a2/2?) muldimensional configurations and/or boundary control

thank you for your attention

• P. Cannarsa (Rome Tor Vergata)

degenerate parabolic operators April 4, 2012 46 / 46