On Controllability of Waves and Geometric Carleman Estimates Arick - - PowerPoint PPT Presentation

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On Controllability of Waves and Geometric Carleman Estimates

Arick Shao

Queen Mary University of London

LMS Hyperbolic Network Meeting Loughborough University 4 March, 2019

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Outline

• 1. Main problem: Controllability of wave equations.

Problem statement. Survey of methods.

• 2. New results: Control on time-dependent domains.

New observability and Carleman estimates. Lorentzian geometric ideas.

• 3. Discussion of Carleman estimates.

Ideas of proof. Current/future work.

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Controllability of Waves Background

Wave Equations

• Recall. Wave operators:

□ := −∂2

t + ∆x,

L := □ + ∇X + V. Types of wave equations: Free waves □φ = 0 Linear waves Lφ = F Nonlinear waves Lφ = G(φ, ∇φ) Geometric waves □ ⇒ □g := gαβ∇αβ Here, we focus on linear (and free) waves.

Waves present in many equations of physics:

Maxwell, Yang-Mills, Einstein, Euler equations.

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Controllability of Waves Background

Solving Wave Equations

Problem (Initial-boundary value problem; IBVP)

Given: Initial/fjnal times T− < T+ Spatial domain Ω ⊆ Rn open, bounded Unknown: φ : [T−, T+] × ¯ Ω → R Solve: Wave equation Lφ|(T−,T+)×Ω = F Initial data (φ, ∂tφ)|t=T− = (φ−

0 , φ− 1 )

Boundary data φ|(T−,T+)×∂Ω = φd

• r

∂vφ|(T−,T+)×∂Ω = φn

Theorem IBVP has unique solution φ.

(In appropriate function spaces.)

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Controllability of Waves Problem Statement

Let’s Be More Proactive

Question (Version 1) Can we steer solution to desired state?

Can we control what happens to φ? φ as model for physical phenomenon.

Question (Version 2)

From initial state (φ, ∂tφ)|t=T− = (φ−

0 , φ− 1 )...

...achieve fjnal state (φ, ∂tφ)|t=T+ = (φ+

0 , φ+ 1 ).

Unfortunately, we do not have superpowers.

In practice, can only directly afgect part of system.

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Controllability of Waves Problem Statement

Interior and Boundary Control

Interior control: Steer φ through forcing term.

Lφ = χωG, ω ⊆ Ω. Free to set G on (T−, T+) × ω.

Boundary control: Steer φ through boundary data.

Lφ = 0, Γ ⊆ (T−, T+) × ∂Ω. Dirichlet: φ|(T−,T+)×∂Ω = χΓφd. Neumann: ∂νφ|(T−,T+)×∂Ω = χΓφn.

Here, we focus on Dirichlet boundary control.

Other problems treated similarly.

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Controllability of Waves Problem Statement

Exact Controllability

Problem (Exact Dirichlet boundary controllability) Given:

Control region Γ ⊆ (T−, T+) × ∂Ω

Goal: ∀ initial/fjnal data (φ±

0 , φ± 1 ) ∈ L2(Ω) × H−1(Ω):

Find boundary data φd ∈ L2(Γ), such that... ...solution φ of IBVP (with initial and boundary data)... ...satisfjes (φ, ∂tφ)|t=T+ = (φ+

0 , φ+ 1 ).

• Q. Can solutions be controlled via Dirichlet boundary data?

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Controllability of Waves Problem Statement

Basic Observations

• Observation. Can take (φ+

0 , φ+ 1 ) = 0 or (φ− 0 , φ− 1 ) = 0.

Null controllability. (From time-reversibility of wave equation.)

From now on, assume (φ−

0 , φ− 1 ) = 0.

• Remark. Contrast with heat equations:

Null controllability ̸⇔ exact controllability.

• Observation. Finite speed of propagation.

Information from ∂Ω needs time to travel to all of Ω. Fundamental lower bound for T+ − T−.

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Controllability of Waves Survey of Methods

The Adjoint Problem

(P1) Consider linear operator

S : L2(Γ) → L2(Ω) × H−1(Ω), S(φd) = (φ, ∂tφ)|t=T+. L2 × H−1: transposition solutions. Goal: Show S has full range. Lφ|(T−,T+)×Ω = 0, (φ, ∂tφ)|t=T− = 0, φ|(T−,T+)×∂Ω = φd.

(P2) Adjoint of S:

S∗ : L2(Ω) × H1

0(Ω) → L2(Γ),

S∗(ψ+

1 , ψ+ 0 ) = ∂νψ|Γ.

H1

0 × L2: weak solutions.

L∗ψ|(T−,T+)×Ω = 0, (ψ, ∂tψ)|t=T+ = (ψ+

0 , ψ+ 1 ),

ψ|(T−,T+)×∂Ω = 0.

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Controllability of Waves Survey of Methods

Observability and Controllability

(Dolecki–Russell) By duality and the closed range theorem:

Null/exact controllability ⇔ S is surjective ⇔ ∥ξ∥ ≲ ∥S∗ξ∥. Last statement known as observability inequality (for (P2) and L∗): ∥(ψ, ∂tψ)(T+)∥H1×L2 ≲ ∥∂νψ∥L2(Γ).

Main goal: prove observability inequality. (J.-L. Lions) Hilbert uniqueness method (HUM)

Modern machinery for (observability ⇔ controllability). Methods for generating control φd (e.g. as minimiser of functional). Can fjnd φd minimising L2(Γ)-norm.

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Controllability of Waves Survey of Methods

Methods for Observability I

• I. Fourier series methods

Applies to one spatial dimension: (−∂2

t + ∂2 x + α)ψ = 0.

Apply variants of Ingham’s inequality... ...to Fourier representation of solution.

Theorem (Ingham) Consider sequence (λn)n∈Z of real numbers, with λn+1 − λn ≥ γ > 0, n ∈ Z. If T > π

γ, then for any sequence (cn)n∈Z, we have

∑

n

|cn|2 ≲T,γ ∫ T

−T

• ∑

n

cneiλnt

• 2

dt.

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Controllability of Waves Survey of Methods

Methods of Observability II

• II. Multiplier methods

Applies to all dimensions: □ψ = 0 (and small perturbations)

• Idea. Integrate by parts the RHS of

0 = ∫

[T−,T+]×Ω

□ψS∗ψ.

Theorem (Ho, Lions, ...) Fix x0 ∈ Rn, and assume

T+ − T− > 2 sup

x∈∂Ω

|x − x0|.

Then, for L := □, observability holds with:

Γ := (T−, T+) × {x ∈ ∂Ω | (x − x0) · ν > 0}.

• Note. (t, x) ∈ Γ ⇔ −

→ x0x is leaving Ω at x.

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Controllability of Waves Survey of Methods

Methods for Observability III

• III. Carleman estimates

Weighted (spacetime) integral estimates, with free real parameter. Robust technique—extends multiplier methods to general L. Extends to some geometric wave equations on R × M. ∥eλF∇t,xψ∥2

L2 + ∥eλFψ∥2 L2 ≲ λ−2∥eλF□ψ∥2 L2 + . . . ,

λ ≫ 1. Take λ large ⇒ absorb lower-order terms.

Theorem (Lasiecka–Triggiani–Zhang, Zhang, ...) Previous theorem holds for general L.

But, for technical reasons, also requires x0 ̸∈ ¯ Ω.

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Controllability of Waves Survey of Methods

Methods for Observability IV

• IV. Microlocal methods

Most precise, optimal (w.r.t. control region) results. For time-independent (or time-analytic) equations. Extends to geometric wave equations on R × M.

Theorem (Bardos–Lebeau–Rauch, Burq, ...) Observability on Γ ⇔ geometric control condition.

Each bicharacteristic (null geodesic) in [T−, T+] × Ω hits Γ. Bicharacteristics refmect ofg boundary via geometric optics. (Le Rousseau–Lebeau–Terpolilli–Trélat) Time-dependent Γ.

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Controllability of Waves Survey of Methods

Carleman vs. Microlocal Methods

Microlocal methods ⇒ stronger results

Optimal control region. Must assume PDE is time-analytic.

Carleman methods ⇒ more robust results

No analyticity required for PDE. Fails to achieve geometric control condition in general.

Current interests: robustness, non-analytic equations

Applications to time-dependent domains, nonlinear waves.

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Moving Boundaries The Problem Statement

Time-Dependent Domains

• Q. What about time-dependent domains?

U = ∪

T−<τ<T+

({τ} × Ωτ). U has moving boundary: Ub = ∪

T−<τ<T+

({τ} × ∂Ωτ).

Assume boundary Ub is timelike:

Ub “moves at less than wave/characteristic speed”. Ub appropriate for boundary data.

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Moving Boundaries The Problem Statement

Well-Posedness Revisited

Theorem (IBVP)

Domain: Spacetime domain U Timelike boundary Ub Can solve: Wave equation Lφ|U = 0 Initial data (φ, ∂tφ)|t=T− = (φ−

0 , φ− 1 )

Boundary data φ|Ub = φd Uniquely for: φ : U → R

Idea: Apply change of variables:

U ⇒ R × Ω. L ⇒ linear (time-dependent) hyperbolic PDE.

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Moving Boundaries The Problem Statement

Controllability Revisited

Problem (Exact Dirichlet boundary controllability) Given:

Control region Γ ⊆ Ub

Goal: ∀ initial/fjnal data (φ±

0 , φ± 1 ) ∈ L2 × H−1:

Find φd ∈ L2(Γ), such that... ... solution φ of IBVP (with initial and boundary data)... ... satisfjes (φ, ∂tφ)|t=T+ = (φ+

0 , φ+ 1 ).

HUM theory still applies:

Exact controllability ⇔... Observability inequality for adjoint system: ∥(ψ, ∂tψ)(T+)∥H1×L2 ≲ ∥∂νψ∥L2(Γ).

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Moving Boundaries The Problem Statement

Some Existing Results

Literature for time-dependent domains is sparse. General n: (Only for L = □).

(Bardos–Chen) U expanding. (Miranda) U self-similar, asymptotically cylindrical.

n = 1: Recent works (only for L = □).

Ub = two lines (optimal). Ub = line + curve. (Cui–Jiang–Wang, Sun–Li–Lu, Wang–He–Li, ...)

Missing: General results in any dimension.

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Moving Boundaries The Problem Statement

The Main Estimate (Rough Statement)

Theorem (S., 2018)

Consider general L on moving domain U. Fix x0 ∈ Rn, and assume T+ − T− > R+ + R−, R± := sup

x∈Ub∩{t=T±}

|x − x0|. Then, we have observability (for adjoint problem): ∥(ψ, ∂tψ)(T±)∥2

H1×L2 ≲

∫

Y

|∂νψ|2. The observation region Y is “much improved” and satisfjes: Y is a “much smaller”, time-dependent set.

(More precise statement later.)

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Moving Boundaries The Problem Statement

Some Remarks

• 1. Comparison to previous results (n = 1):

Recovers all existing results in n = 1. Applies to general Ub = curve + curve. Recovers optimal time (GCC) for all Ub.

• 2. Comparison to previous results (general n):

First result for general timelike Ub and general L. Improves existing Carleman/multiplier results for static U. Y proper subset of {(x − x0) · ν > 0}. In general, does not achieve GCC. x0 can be outside or inside of domain.

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Moving Boundaries A Preliminary Multiplier Result

Minkowski Geometry

First, consider free waves (L := □):

Can handle using multiplier methods.

• Idea. □ naturally associated with Minkowski geometry:

M := R1+n, g := −dt2 + d(x1)2 + · · · + d(xn)2.

Geometric setting of special relativity. □ = gαβ∇αβ—analogue of ∆ in Euclidean geometry.

Many aspects of Riemannian geometry have Lorentzian analogues:

Connection, curvature, divergence theorem, integration by parts.

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Moving Boundaries A Preliminary Multiplier Result

Classical Multiplier Result

Main step: Integrate by parts, starting with

0 = ∫

[T−,T+]×Ω

□ψS0

∗ψ,

S0

∗ψ := (x − x0) · ∇xψ + n − 1

2 ψ.

Combine with energy conservation ⇒

(T+ − T−)E(T±) ≤ 2R · E(T±) + 1 2 ∫

(T−,T+)×∂Ω

[(x − x0) · ν]|∂νψ|2. E(t) = 1

2

∫

{t}×Ω |∇t,xψ|2.

R = supx∈∂Ω |x − x0|. Green ⇒ T+ − T− > 2R. Orange ⇒ Γ := {(x − x0) · ν > 0}.

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Moving Boundaries A Preliminary Multiplier Result

Adaptation to Time-Dependent Setting

• 1. Replace reference point x0 by reference event (t0, x0).

View problem in terms of spacetime geometry.

• 2. Replace classical multiplier S0

∗ψ by

S∗ψ := [(x − x0) · ∇xψ + (t − t0) ∂tψ] + n − 1 2 ψ. Principal part ≃ (Minkowski) gradient of f0 := 1

4[|x − x0|2 − (t − t0)2].

∂t-part: corresponds to energy conservation arguments.

• 3. Apply Lorentzian version of integration by parts:

N: outward-pointing Minkowksi unit normal to Ub. (Euclidean normal, with t-component reversed.)

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Moving Boundaries A Preliminary Multiplier Result

The Multiplier Result

Theorem (S., 2018)

Consider L := □ on moving domain U. Assume T+ − T− > R+ + R−, R± := sup

x∈Ub∩{t=T±}

|x − x0|. Then, we have observability (for adjoint problem): ∥(ψ, ∂tψ)(T±)∥2

H1×L2 ≲

∫

Γ

|Nψ|2. Γ = Ub ∩ {Nf0 > 0}. N: outward-pointing Minkowski unit normal on Ub. f0 := 1

4[|x − x0|2 − (t − t0)2], where...

...t0 satisfjes t0 − T− > R− and T+ − t0 > R+.

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Moving Boundaries A Preliminary Multiplier Result

Some Remarks

n = 1: Recovers all previous results, and in full generality.

Handles arbitrary Ub = 2 timelike curves. Recovers optimal T+ − T−.

General n: Handles general domains.

No assumptions on growth or asymptotics of U. Requires much smaller T+ − T− than before (optimal in many cases).

{Nf0 > 0} generalises {(x − x0) · ν > 0}:

U time-independent Nf0 > 0 ⇔ (x − x0) · ν > 0 U “expanding” from t0 Need smaller Γ U “contracting” from t0 Need larger Γ

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Moving Boundaries Improved Carleman Estimates

Return to Carleman Estimates

• Q. What about general L?

In particular, interested in non-analytic coeffjcients. (This is also forced by moving boundary.)

• Goal. Establish global Carleman estimate:

∥eλF(∇t,xψ, ψ)∥2

L2 ≲ λ−2∥eλF□ψ∥2 L2 + ∥eλFNψ∥2 L2(Γ).

Integrate by parts the expression

∫ (eλF□ψ)S∗(eλFψ) Multiplier estimate for eλF□e−λF and ψ⋆ := eλFψ.

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Moving Boundaries Improved Carleman Estimates

The Carleman Weight

The weight eλF is constructed from f0:

(† Not quite true)

f0 = 1

4[|x − x0|2 − (t − t0)2].

eλF := f λ

0 e2λbf 1/2

.

Level sets of f0 are hyperboloids:

f0 = 0: null cone about (t0, x0). f0 = c > 0: one-sheeted hyperboloids. f0 = c < 0: two-sheeted hyperboloids.

D0 := {f0 > 0}: null cone exterior.

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Moving Boundaries Improved Carleman Estimates

A Novel Feature

• New. eλF vanishes at f0 = 0.

No boundary terms at f0 = 0. Estimate supported on null cone exterior D0 = {f0 > 0}.

This (roughly) yields the Carleman estimate:

∥eλF(∇t,xψ, ψ)∥2

L2(U∩D0) ≲ λ−2∥eλF□ψ∥2 L2(U∩D0) +

∫

Ub∩D0

e2λF(Nf0)|Nψ|2. Energy inequalities ⇒ observability, with Γ := Γ0 = Ub ∩ {Nf0 > 0} ∩ D0. (Classical Carleman methods: Γ ≈ Ub ∩ {Nf0 > 0}.) Extra restriction of Γ to D0.

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Moving Boundaries Improved Carleman Estimates

Some Pictures

Left: Ub. Red dot: (t0, x0). Purple: Ub ∩ D0. Green: Γ0 := Ub ∩ {Nf0 > 0} ∩ D0.

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Moving Boundaries Improved Carleman Estimates

A Relativistic Interpretation

Nf0 > 0 can be interpreted relativistically:

Fix (tb, xb) ∈ Ub ∩ D0.

• Idea. f0 is Lorentz-invariant.

Apply Lorentz boost about (t0, x0):

Inertial coordinates (t′, x′). (t0, x0) and (tb, xb) simultaneous.

• Observation. Nf0|(tb,xb) > 0 is the same as

(x′ − x0) · ν′ > 0. ν′: spatial part of N w.r.t. (t′, x′).

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Moving Boundaries Improved Carleman Estimates

A Pseudoconvexity Issue

Main requirement for Carleman estimates is pseudoconvexity.

Level sets of f0 (barely) fail to be pseudoconvex.

Thus, can only obtain a degenerate Carleman estimate:

∥eλFψ∥2

L2(U∩D0) ≲ λ−2∥eλF□ψ∥2 L2(U∩D0) +

∫

Ub∩D0

e2λF(Nf0)|Nψ|2. No H1-control for ψ ⇒ no observability. Thus, cannot build weight eλF from f0.

• Idea. Perturb f0 → fε such that:

Level sets of fε are pseudoconvex. D0 still corresponds to fε > 0.

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Moving Boundaries Improved Carleman Estimates

The Classical Approach

Classical Carleman observability results:

fε := 1 4[|x − x0|2 − (1 − ε)(t − t0)2]. Hyperboloids for waves with slower speed.

• Drawback. Not well-adapted to characteristics of wave equation.

fε = 0 no longer the null cone about (t0, x0). Cannot obtain Carleman estimates supported on D0.

• Drawback. Only works for (t0, x0) ̸∈ U.

(Arises from dealing with the region {f0 < 0}.)

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Moving Boundaries Improved Carleman Estimates

A Conformal Viewpoint

New idea. Perturb geometry rather than f0:

Consider “warped” Minkowski metric: gε := g + O(ε2), ε > 0. Level sets of f0 are pseudoconvex w.r.t. gε-geometry.

• Observation. (D0, g), (D0, gε) conformally related.

Pseudoconvexity is conformally invariant. fε: pullback of f0 through conformal isometry.

• Strategy. 2 steps for proof of Carleman estimate:

1

Prove “warped” Carleman estimate on (D0, gε).

2

Pull “warped” Carleman estimate back to (D0, g).

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Moving Boundaries Improved Observability

The Main Observability Estimate

Theorem (S., 2018)

Consider general L on moving domain U. Assume T+ − T− > R+ + R−, R± := sup

x∈Ub∩{t=T±}

|x − x0|. Then, we have observability (for adjoint problem): ∥(ψ, ∂tψ)(T±)∥2

H1×L2 ≲

∫

Y

|Nψ|2. Y: any open subset of boundary with Y ⊇ Ub ∩ {Nf0 > 0} ∩ D0. f0 and t0 as before (t0 − T− > R− and T+ − t0 > R+).

• Remark. Any such Y suffjces:

Perturbation f0 → fε can be arbitrarily small.

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Moving Boundaries Improved Observability

Some Remarks

• 1. Improves upon previous multiplier results:

Extends moving-boundary observability from □ to general L. (ε-discrepancy between Γ and region from multiplier methods.)

• 2. Improves classical Carleman observability, even for static domains:

Further restricts observation region Γ to D0. Γ is time-dependent. (t0, x0) allowed to lie in U.

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Moving Boundaries Improved Observability

Current/Future Work

• 0. Analogous results for interior control

In progress (Vaibhav Jena)

• 1. Control for geometric wave equations.

Wave equations on Lorentzian settings unexplored. In progress (with Vaibhav Jena)

• 2. Nonlinear wave equations.

Results for non-analytic equations ⇒ possibilities for nonlinear applications.

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Moving Boundaries Improved Observability

Singular Wave Equations

• 3. Control for wave equations with critically singular potentials.

Cylindrical domain [T−, T+] × BRn(0, 1). Singular wave equation [ □ − c (1 − r)2 ] φ + ∇Xφ + Vφ = 0. Potential is critically singular at boundary. No existing observability results when n > 1.

Theorem (Enciso, Vergara, S.; 2019)

n ̸= 2 and T+ − T− suffjciently large ⇒ Dirichlet boundary observability.

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The End

Thank you for your attention!

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