In egalit es spectrales pour le contr ole des EDP lin eaires : - - PowerPoint PPT Presentation

In´ egalit´ es spectrales pour le contrˆ

• le des EDP lin´

eaires : groupe de Schr¨

• dinger contre semigroupe de la chaleur.

Luc Miller

Universit´ e Paris Ouest Nanterre La D´ efense, France

Pde’s, Dispersion, Scattering theory and Control theory, Monastir, June 13, 2013.

D’apr` es une collaboration avec Thomas Duyckaerts (Univ. Paris 13) :

Resolvent conditions for the control of parabolic equations, Journal of Functional Analysis 263 (2012), pp. 3641-3673. http://hal.archives-ouvertes.fr/hal-00620870

Luc Miller, Paris Ouest, France 1 / 20

Outline

1

Part 1: Background on the interior control of linear PDEs

2

Part 2: Resolvent conditions for parabolic equations

3

Part 3: The harmonic oscillator observed from a half-line

4

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements

Luc Miller, Paris Ouest, France 2 / 20

Control of the temperature f in a smooth domain M ⊂ Rd (Dirichlet), from a chosen source u acting in an open subset Ω ⊂ M during a time T.

M Ω Fast null-controllability

The heat O.D.E. in E = L2(M) with input u ∈ L2(R; E): ∂tf − ∆f = Ωu. ∀T > 0, ∀f (0) ∈ E, ∃u, f (T) = 0

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 3 / 20

Control of the temperature f in a smooth domain M ⊂ Rd (Dirichlet), from a chosen source u acting in an open subset Ω ⊂ M during a time T.

M Ω Fast null-controllability (at cost κT)

The heat O.D.E. in E = L2(M) with input u ∈ L2(R; E): ∂tf − ∆f = Ωu. ∀T > 0, ∀f (0) ∈ E, ∃u, f (T) = 0 and T u(t)2dt κTf (0)2.

• Fast final-observability (at cost κT)

(FinalObs) eT∆v2 κT T Ωet∆v2dt, v ∈ E, T > 0.

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 3 / 20

Links between heat/Schr¨

• dinger/waves controllability

∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂tf − ∆f = Ωu No No Schr¨

• dinger eq.

i∂tψ − ∆ψ = Ωu Yes No Wave eq. ∂2

t w − ∆w = Ωu

Yes Yes

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20

Links between heat/Schr¨

• dinger/waves controllability

∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂tf − ∆f = Ωu No No Schr¨

• dinger eq.

i∂tψ − ∆ψ = Ωu Yes No Wave eq. ∂2

t w − ∆w = Ωu

Yes Yes

1 ∃T, wave control ⇒ ∀T, heat control

(by the control transmutation method, cf. Russell, Phung, Miller).

2 ∃T, wave control ⇒ ∀T, Schr¨

• dinger control

(by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss).

3 ∃T, wave control ⇔ ∃T, wave group control: i ˙

ψ + √ −∆ψ = Ωu (by resolvent conditions, cf. Miller’12) This leads to the new question : Schr¨

• dinger control ⇒ heat control ?

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20

Links between heat/Schr¨

• dinger/waves controllability

∆ is the Laplacian on a bounded M with Dirichlet boundary conditions. Controllabilty of Restriction on Ω Restriction on T Heat eq. ∂tf − ∆f = Ωu No No Schr¨

• dinger eq.

i∂tψ − ∆ψ = Ωu Yes No Wave eq. ∂2

t w − ∆w = Ωu

Yes Yes

1 ∃T, wave control ⇒ ∀T, heat control

(by the control transmutation method, cf. Russell, Phung, Miller).

2 ∃T, wave control ⇒ ∀T, Schr¨

• dinger control

(by resolvent conditions, cf. Liu, Miller, Tucsnak-Weiss).

3 ∃T, wave control ⇔ ∃T, wave group control: i ˙

ψ + √ −∆ψ = Ωu (by resolvent conditions, cf. Miller’12) This leads to the new question : Schr¨

• dinger control ⇒ heat control ?

No but: Schr¨

• dinger ⇒ fractional diffusion ∂tf + (−∆)sf = Ωu, s > 1.

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 4 / 20

Abstract semigroup framework: t → e−tA observed by C.

Hilbert spaces E (states), F (observations). Semigroup e−tA on E. Bounded (in this talk) operator C ∈ L(E, F) (defines what is observed). Its adjoint C ∗ defines how the input u : t → F acts in order to control.

Example (Heat on the domain M observed on Ω ⊂ M)

A = −∆ 0, E = F = L2(M), D(A) = H2(M) ∩ H1

0(M), C = Ω.

Fast null-controllability of ∂tf + A∗f = C ∗u, with u ∈ L2(R; F)

∀T > 0, ∀f (0) ∈ E, ∃u, f (T) = 0 and T u(t)2dt κTf (0)2.

• Fast final-observability (at cost κT)

(FinalObs) e−TAv2 κT T Ce−tAv2dt, v ∈ E, T > 0.

Part 1: Background on the interior control of linear PDEs Luc Miller, Paris Ouest, France 5 / 20

Resolvent conditions for control (= Hautus tests)

From spectral to dynamic inequalities by (unitary) Fourier transform on E.

Recall: Huang-Pr¨ uss’84 test for exponential stability of t → e−tA

(A − λ)−1 m, Re λ < 0.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20

Resolvent conditions for control (= Hautus tests)

From spectral to dynamic inequalities by (unitary) Fourier transform on E.

Recall: Huang-Pr¨ uss’84 test for exponential stability of t → e−tA

(A − λ)−1 m, Re λ < 0.

Hautus test for observability of t → eitA, A = A∗, by C, for some T

v2 m(A − λ)v2 + ˜ mCv2, v ∈ D(A), λ ∈ R. Zhou-Yamamoto’97 (Huang-Pr¨ uss). Burq-Zworski’04 (⇒). Miller’05 (⇔): T > π√m and κT = 2 ˜ mT/(T 2 − mπ2).

Recall: Observability of t → eitA, A = A∗, by C for some T means

(ExactObs) v2 κT T CeitAv2dt, v ∈ E.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20

Resolvent conditions for control (= Hautus tests)

From spectral to dynamic inequalities by (unitary) Fourier transform on E.

Recall: Huang-Pr¨ uss’84 test for exponential stability of t → e−tA

(A − λ)−1 m, Re λ < 0.

Hautus test for observability of t → eitA, A = A∗, by C, for some T

v2 m(A − λ)v2 + ˜ mCv2, v ∈ D(A), λ ∈ R. Zhou-Yamamoto’97 (Huang-Pr¨ uss). Burq-Zworski’04 (⇒). Miller’05 (⇔): T > π√m and κT = 2 ˜ mT/(T 2 − mπ2).

Similar Hautus test for wave ¨ w + Aw = C ∗f , A > 0, for some T

v2 m λ (A − λ)v2 + ˜ mCv2, v ∈ D(A), λ ∈ R∗. Liu’97 (Huang-Pr¨ uss), Miller’05 (⇐), R.T.T.Tucsnak’05, Miller’12.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 6 / 20

Sufficient resolvent conditions for t → e−tA, A > 0

Recall (ExactObs) for Schr¨

• dinger ˙

ψ − iAψ = 0 ⇒ (Res) with δ = 1 ⇒ (Res) with δ = 0 ⇔ (ExactObs) for wave ¨ w + Aw = 0.

Theorem (Duyckaerts-Miller’11: Main Result)

If the resolvent condition with power-law factor : ∃m > 0, (Res) v2 mλδ 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0, holds for some δ ∈ [0, 1), then observability (FinalObs) holds for all T > 0 with the control cost estimate κT cec/T β for β = 1+δ

1−δ and some c > 0.

Here C is bounded, or admissible to some degree (cf. our paper).

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 7 / 20

Sufficient resolvent conditions for t → e−tA, A > 0

Recall (ExactObs) for Schr¨

• dinger ˙

ψ − iAψ = 0 ⇒ (Res) with δ = 1 ⇒ (Res) with δ = 0 ⇔ (ExactObs) for wave ¨ w + Aw = 0.

Theorem (Duyckaerts-Miller’11: Main Result)

If the resolvent condition with power-law factor : ∃m > 0, (Res) v2 mλδ 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0, holds for some δ ∈ [0, 1), then observability (FinalObs) holds for all T > 0 with the control cost estimate κT cec/T β for β = 1+δ

1−δ and some c > 0.

Here C is bounded, or admissible to some degree (cf. our paper).

Theorem (Duyckaerts-Miller’11: Schr¨

• dinger to heat)

If (ExactObs) for Schr¨

• dinger t → eitA holds for some T,

then (FinalObs) for “higher-order” heat t → e−tAγ, γ > 1 holds for all T.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 7 / 20

Sufficient resolvent conditions for t → e−tA, A > 0

log-improvement of λδ, δ < 1, into λ/(ϕ(λ))2, ϕ(λ) = (log λ)α, α > 1.

Theorem (Duyckaerts-Miller’11: Main Result, log-improved)

If the resolvent condition with logarithmic factor : ∃m > 0, v2 mλ (ϕ(λ))2 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0, holds for some α > 1, then observability (FinalObs) holds for all T > 0. Here C is bounded, or admissible for the wave equation ¨ w + Aw = 0.

Theorem (Duyckaerts-Miller’11: Schr¨

• dinger to heat, log-improved)

If (ExactObs) for Schr¨

• dinger t → eitA holds for some T,

then (FinalObs) for “higher-order” heat t → e−tAϕ(1+A), α > 1, T > 0.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 8 / 20

Application to the control of diffusions in a potential well

A = −∆ + V on E = L2(R), D(A) =

• u ∈ H2(R) | Vu ∈ L2(R)
• .

V (x) = x2k, k ∈ N, k > 0. C = Ω = (−∞, x0), x0 ∈ R.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 9 / 20

Application to the control of diffusions in a potential well

A = −∆ + V on E = L2(R), D(A) =

• u ∈ H2(R) | Vu ∈ L2(R)
• .

V (x) = x2k, k ∈ N, k > 0. C = Ω = (−∞, x0), x0 ∈ R.

Theorem (Miller at CPDEA, IHP’10)

v2 mλ1/k 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0, and the decay of the first coefficient cannot be improved.

Theorem (Duyckaerts-Miller’11)

The diffusion in the potential well V (x) = x2k, k ∈ N, k > 1, ∂tφ − ∂2

xφ − V φ = Ωu,

φ(0) = φ0 ∈ L2(R), u ∈ L2([0, T] × R), is null-controllable in any time, i.e. ∀T > 0, ∀φ0, ∃u such that φ(T) = 0.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 9 / 20

Necessary resolvent conditions for any semigroup t → e−tA

Example (worst resolvent condition for the Laplacian on a manifold)

−A is the Laplacian on the unit sphere S2 =

• x2 + y2 + z2 = 1
• ,

C = Ω is the complement a neighborhood of the great circle {z = 0}. en(x, y, z) = (x + iy)n: (A − λn)en = 0 and ∃a > 0, en aea√λnCen. This leads to the resolvent condition with exponential factor : ∃m > 0, (Res) v2 mem(Re λ)α (A − λ)v2 + Cv2 , v ∈ D(A), Re λ > 0.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 10 / 20

Necessary resolvent conditions for any semigroup t → e−tA

Example (worst resolvent condition for the Laplacian on a manifold)

−A is the Laplacian on the unit sphere S2 =

• x2 + y2 + z2 = 1
• ,

C = Ω is the complement a neighborhood of the great circle {z = 0}. en(x, y, z) = (x + iy)n: (A − λn)en = 0 and ∃a > 0, en aea√λnCen. This leads to the resolvent condition with exponential factor : ∃m > 0, (Res) v2 mem(Re λ)α (A − λ)v2 + Cv2 , v ∈ D(A), Re λ > 0.

Theorem (Duyckaerts-Miller’11)

If (FinalObs) holds for some T > 0 then (Res) holds with α = 1. If (FinalObs) holds for all T ∈ (0, T0] with the control cost κT = cec/T β for some β > 0, c > 0, T0 > 0, then (Res) holds with α =

β β+1 < 1.

Still valid for C ∈ L(D(A), F) admissible, i.e. T Ce−tAv2dt kTv2.

Part 2: Resolvent conditions for parabolic equations Luc Miller, Paris Ouest, France 10 / 20

The harmonic oscillator observed from a half-line Ω ⊂ R

Disproves : controllability of Schr¨

• dinger eq. ⇒ controllability of heat eq.

∂tφ − ∂2

xφ + x2φ = Ωu

versus i∂tψ − ∂2

xψ + x2ψ = Ωu

Here Ω = (−∞, x0), x0 ∈ R, and A = −∂2

x + x2 on E = L2(R) = F.

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 11 / 20

The harmonic oscillator observed from a half-line Ω ⊂ R

Disproves : controllability of Schr¨

• dinger eq. ⇒ controllability of heat eq.

∂tφ − ∂2

xφ + x2φ = Ωu

versus i∂tψ − ∂2

xψ + x2ψ = Ωu

Here Ω = (−∞, x0), x0 ∈ R, and A = −∂2

x + x2 on E = L2(R) = F.

Theorem (Miller at CPDEA, IHP’10)

Observability (FinalObs) for heat t → e−tA does not hold for any time. Observability (ExactObs) for Schr¨

• dinger t → eitA holds for some time.

Eigenvalues are λn = 2n + 1. N.b. 1

λn = +∞ but dim F = 1.

Eigenfunctions en are en(x) = cn (∂x − x)n e−x2/2 = cnHn(x)e−x2/2, where cn = (√π2n(n!))−1/2, Hn = (−1)nex2∂n

x e−x2 are the Hermite polynomials.

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 11 / 20

Sketch of proof 1: non-observability for the heat semigroup

Harmonic oscillator A = −∂2

x + x2 observed from a half line Ω = (−∞, x0).

Disprove (FinalObs) +∞

−∞

|(e−TAv)(x)|2dx κ2

T

T x0

−∞

|(e−tAv)(x)|2dxdt, by taking the Dirac mass at y / ∈ Ω as initial data v and letting y → ∞.

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 12 / 20

Sketch of proof 1: non-observability for the heat semigroup

Harmonic oscillator A = −∂2

x + x2 observed from a half line Ω = (−∞, x0).

Disprove (FinalObs) +∞

−∞

|(e−TAv)(x)|2dx κ2

T

T x0

−∞

|(e−tAv)(x)|2dxdt, by taking the Dirac mass at y / ∈ Ω as initial data v and letting y → ∞. More precisely, v(x) = e−εA(x, y), where ε is a small time, e−tA(x, y) is the kernel of the operator etA. Hence (e−tAv)(x) = e−(t+ε)A(x, y).

Bound from below the fundamental state e0 hence the final state

e−TAv e−(T+ε)λ0|e0(y)| cT exp

• −y2

2

• .

Bound from above the kernel hence the observation: Mehler formula

e−tA(x, y) = e−t

• π(1 − e−4t))

exp

• −1 + e−4t

1 − e−4t x2 + y2 2 + 2e−2t 1 − e−4t xy

• .

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 12 / 20

Sketch of proof 2: observability for the Schr¨

• dinger group

Harmonic oscillator A = −∂2

x + x2 observed from a half line Ω = (−∞, x0).

Prove (Res) using a semiclassical reduction and microlocal propagation.

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 13 / 20

Sketch of proof 2: observability for the Schr¨

• dinger group

Harmonic oscillator A = −∂2

x + x2 observed from a half line Ω = (−∞, x0).

Prove (Res) using a semiclassical reduction and microlocal propagation. By the change of variable u(y) = v(x), y = √ hx, h = 1/λ, (Res) v2 m(A − λ)v2 + mΩv2, v ∈ D(A), λ > 0, reduces to the semiclassical resolvent condition +∞

−∞

|u(y)|2dy m h2 +∞

−∞

|−h2u′′(y) + (y2 − 1)u(y)|2dy + m √

hx0 −∞

|u(y)|2dy, u ∈ C ∞

0 (R),

h ∈ (0, 1].

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 13 / 20

Sketch of proof 2: observability for the Schr¨

• dinger group

Harmonic oscillator A = −∂2

x + x2 observed from a half line Ω = (−∞, x0).

Prove (Res) using a semiclassical reduction and microlocal propagation. By the change of variable u(y) = v(x), y = √ hx, h = 1/λ, (Res) v2 m(A − λ)v2 + mΩv2, v ∈ D(A), λ > 0, reduces to the semiclassical resolvent condition +∞

−∞

|u(y)|2dy m h2 +∞

−∞

|−h2u′′(y) + (y2 − 1)u(y)|2dy + m √

hx0 −∞

|u(y)|2dy, u ∈ C ∞

0 (R),

h ∈ (0, 1]. Arguing by contradiction, introduce a semiclassical measure (= Wigner measure) in phase space (x, ξ) ∈ R2: it is supported on

• x2 + ξ2 = 1
• ,

invariant by rotation and supported in {x 0}.

Part 3: The harmonic oscillator observed from a half-line Luc Miller, Paris Ouest, France 13 / 20

An observability estimate for sums of eigenfunctions

M Ω

• M

|v(x)|2dx ˜ cec

√ λ

• Ω

|v(x)|2dx, for all λ > 0 and v =

• µλ

eµ, where

• −∆eµ = µeµ
• n M

eµ = 0

• n ∂M .

Lebeau-Robbiano’95 (Carleman estimates), Lebeau-Jerison’96, Lebeau-Zuazua’98.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 14 / 20

The direct Lebeau-Robbiano strategy

We may write the previous spectral observability estimate concisely with spectral subspaces of the Dirichlet Laplacian Eλ = Spanµλ eµ: v ˜ aea

√ λΩv,

v ∈ Eλ, λ > 0. More generally Eλ may be defined by some functional calculus. For example, when A is self-adjoint: Eλ = 1A<λ E.

Observability on spectral subspaces (with power α ∈ (0, 1))

(SpecObs) v ˜ aeaλαCv, v ∈ Eλ, λ λ0 > 0. ⇓

Fast final-observability (at cost κT ˜ cec/T β, β =

α 1−α)

(FinalObs) e−TAv2 κT T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 15 / 20

“Dynamic spectral inequality” for the direct L.-R. strategy

Observability on spectral subspaces (with power α ∈ (0, 1))

(SpecObs) v ˜ aeaλαCv, v ∈ Eλ, λ λ0 > 0. ⇓ β > 0

Dynamic observability on spectral subspaces (α ∈ (0, 1))

e−TAv2 ˜ aeaλα+b/T β T Ce−tAv2dt, v ∈ Eλ, T > 0, λ λ0. ⇓ β =

α 1−α

Fast final-observability (at cost κT ˜ cec/T β, β =

α 1−α)

(FinalObs) e−TAv2 κT T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 16 / 20

Sufficient resolvent conditions for t → e−tA, A > 0

Now we are ready to sketch the proof of the main result, which we recall:

Theorem (Duyckaerts-Miller’11: Main Result)

If the resolvent condition with power-law factor : ∃m > 0, (Res) v2 mλδ 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0, holds for some δ ∈ [0, 1), then observability (FinalObs) holds for all T > 0 with the control cost estimate κT cec/T β for β = 1+δ

1−δ and some c > 0.

In this talk, we consider only δ = 1/3 to simplify the computations.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 17 / 20

Sketch of proof of the Main Result (with δ = 1/3)

Recall A > 0, Eλ = 1A<λ E hence Eλ2 = 1√

A<λ E.

(Res) v2 mλ1/3 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0. ⇒ (FinalObs) e−TAv2 cec/T 2 T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3)

Recall A > 0, Eλ = 1A<λ E hence Eλ2 = 1√

A<λ E.

(Res) v2 mλ1/3 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0. ⇒ v2 mλ2/3 ( √ A − λ)v2 + Cv2 , v ∈ D( √ A), λ > 0. ⇒ (FinalObs) e−TAv2 cec/T 2 T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3)

Recall A > 0, Eλ = 1A<λ E hence Eλ2 = 1√

A<λ E.

(Res) v2 mλ1/3 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0. ⇒ v2 mλ2/3 ( √ A − λ)v2 + Cv2 , v ∈ D( √ A), λ > 0. ⇒ Controllability of waves on Eλ2 × Eλ2 for times ∼ λ1/3 at cost ∼ λ1/3. ⇒ (FinalObs) e−TAv2 cec/T 2 T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3)

Recall A > 0, Eλ = 1A<λ E hence Eλ2 = 1√

A<λ E.

(Res) v2 mλ1/3 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0. ⇒ v2 mλ2/3 ( √ A − λ)v2 + Cv2 , v ∈ D( √ A), λ > 0. ⇒ Controllability of waves on Eλ2 × Eλ2 for times ∼ λ1/3 at cost ∼ λ1/3. ⇒ Controllability of heat on Eλ2 for all T > 0 at cost ∼ eλ2/3/T. ⇒ (FinalObs) e−TAv2 cec/T 2 T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

Sketch of proof of the Main Result (with δ = 1/3)

Recall A > 0, Eλ = 1A<λ E hence Eλ2 = 1√

A<λ E.

(Res) v2 mλ1/3 1 λ(A − λ)v2 + Cv2

• ,

v ∈ D(A), λ > 0. ⇒ v2 mλ2/3 ( √ A − λ)v2 + Cv2 , v ∈ D( √ A), λ > 0. ⇒ Controllability of waves on Eλ2 × Eλ2 for times ∼ λ1/3 at cost ∼ λ1/3. ⇒ Controllability of heat on Eλ2 for all T > 0 at cost ∼ eλ2/3/T. ⇒ Controllability of heat on Eλ for all T > 0 at cost ∼ eλ1/3/T, but λ1/3/T λα + 1/T β where α = 2/3 and β = 2 satisfy β =

α 1−α,

hence the direct Lebeau-Robbiano strategy in the previous slide applies. ⇒ (FinalObs) e−TAv2 cec/T 2 T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 18 / 20

The direct Lebeau-Robbiano Strategy: log-improvement

Here A is self-adjoint, Eλ = 1A<λ E, C is bounded or admissible.

Theorem (Duyckaerts-Miller’11: logarithmic L.-R. strategy)

Logarithmic observability on spectral subspaces with α > 2 v2 aeaλ/((log(log λ))α log λ)Cv2, v ∈ Eλ, λ λ0 > e. ⇓ (FinalObs) e−TAv2 κT T Ce−tAv2dt, v ∈ E, T > 0.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 19 / 20

The direct Lebeau-Robbiano Strategy: log-improvement

Here A is self-adjoint, Eλ = 1A<λ E, C is bounded or admissible.

Theorem (Duyckaerts-Miller’11: logarithmic L.-R. strategy)

Logarithmic observability on spectral subspaces with α > 2 v2 aeaλ/((log(log λ))α log λ)Cv2, v ∈ Eλ, λ λ0 > e. ⇓ (FinalObs) e−TAv2 κT T Ce−tAv2dt, v ∈ E, T > 0.

Theorem (Duyckaerts-Miller’11: logarithmic anomalous diffusion)

Let ϕ(λ) = (log λ)α, α > 1 or ϕ(λ) = (log(log λ))α log λ, α > 2. The following anomalous diffusion is null-controllable in any time T > 0: ∂tφ + √ −∆ϕ( √ −∆)φ = Ωu, φ(0) = φ0 ∈ L2(M), u ∈ L2([0, T] × M).

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 19 / 20

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Downloads on http://hal.archives-ouvertes.fr/aut/luc+miller/. About the direct Lebeau-Robbiano strategy: On the cost of fast control for heat-like semigroups: spectral inequalities and transmutation, PICOF’10, hal-00459601. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, DCDS’10, hal-00411846. Seidman’08. Tenenbaum-Tucsnak’10. About the Hautus test for conservative PDE’s: Tucsnak-Weiss, Observation and Control for Operator Semigroups, Birkh¨ auser Advanced Texts: Basel Textbooks, ’09. Resolvent conditions for the control of unitary groups and their approximations, JST’12, hal-00620772. Ervedoza’08 (approximation). Jacob-Zwart’09 (other semigroups). About the Hautus test for parabolic PDE’s: Resolvent conditions for the control of parabolic equations, Joint work with Thomas Duyckaerts, JFA’12, hal-00620870.

Part 4: The Lebeau-Robbiano strategy and logarithmic improvements Luc Miller, Paris Ouest, France 20 / 20