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Lemma 4.3: perturbed observability modulo a remainder

PDF p. 18: Lemma 4.3; (4.4)

Statement

Assume the Axiom of Choice and hypotheses (H) and (W): $n \geq 2$ , $Ω = R^{n} ∖ Θ$ is the connected exterior of a nonempty compact smooth nontrapping obstacle, $T > 0$ , and $w$ is the reference trajectory specified in Reference trajectory hypothesis W. Let $H_{D}^{s} (Ω)$ be the Dirichlet spectral scale, $R = (I - Δ_{D})^{- 1}$ , and $U (t)$ the adjoint evolution in Resolvent-regularized adjoint and its forcing. Let $φ, χ, ψ$ be the cutoffs of Control region and nested real cutoffs, so $χ = 1 - φ$ and $0 \leq χ \leq ψ$ . For each $σ \in {- 1, - 2}$ there is $C > 0$ , depending on the fixed data but not on $v_{0}$ , such that every $v_{0} \in H_{D}^{σ} (Ω)$ , with $v (t) = U (t) v_{0}$ and $y (t) = R v (t)$ , satisfies

$∥ v_{0} ∥_{H_{D}^{σ}}^{2} \leq C (\int_{0}^{T} ∥ φ v (t) ∥_{H_{D}^{σ}}^{2} d t + \int_{0}^{T} ∥ ψ y (t) ∥_{L^{2}}^{2} d t) .$

Facts & Assumptions

[F1]

The exterior geometry and nontrapping condition are those of Exterior and truncated obstacle domains and Non-trapping obstacle, and (W) is Reference trajectory hypothesis W. The scale and resolvent satisfy $R : H_{D}^{s} \to H_{D}^{s + 2}$ isometrically Dirichlet spectral Sobolev scale and duality. Choice is assumed as in The Axiom of Choice.

[F2]

The cutoffs satisfy $χ = 1 - φ$ , $0 \leq χ \leq ψ$ , and $χ \in C_{c}^{\infty} (R^{n})$ Control region and nested real cutoffs. Multiplication by these cutoffs is bounded on the relevant scale levels Cutoff multipliers through order two and by duality. Time integrals below are squared Bochner norms as specified in Continuous and mixed time-space function spaces.

[F3]

The regularized adjoint is $y = R U (t) v_{0}$ with $y_{0} = R v_{0}$ , and compatible constructions agree across scale orders Resolvent-regularized adjoint and its forcing. The adjoint evolution and its inverse are uniformly bounded on each level between $- 2$ and $2$ Forward and backward real-linear propagators. Higher scale levels embed continuously into lower ones Positive spectral Hilbert scale: duality and interpolation.

[F4]

For $0 \leq θ \leq 1$ , localized smoothing gives $\int_{0}^{T} ∥ χ y (t) ∥_{H_{D}^{θ + 1 / 2}}^{2} d t \leq C ∥ y_{0} ∥_{H_{D}^{θ}}^{2}$ whenever $v_{0} \in H_{D}^{θ - 2}$ Lemma 4.2: smoothing of the regularized adjoint. Spectral interpolation gives $∥ y_{0} ∥_{H_{D}^{1 / 2}}^{2} \leq ε ∥ y_{0} ∥_{H_{D}^{1}}^{2} + C_{ε} ∥ y_{0} ∥_{2}^{2}$ Spectral interpolation and lower-order absorption.

[F5]

At either $σ = - 1$ or $σ = - 2$ , the localized resolvent commutator gives $∥ φ R v ∥_{H_{D}^{σ + 2}} \leq ∥ φ v ∥_{H_{D}^{σ}} + C ∥ ψ R v ∥_{2}$ Integer resolvent commutator and localized first-order bounds.

Proof

[F1] The exterior geometry and nontrapping condition are those of Exterior and truncated obstacle domains and Non-trapping obstacle, and (W) is Reference trajectory hypothesis W. The scale and resolvent satisfy $R : H_{D}^{s} \to H_{D}^{s + 2}$ isometrically Dirichlet spectral Sobolev scale and duality. Choice is assumed as in The Axiom of Choice.

[F2] The cutoffs satisfy $χ = 1 - φ$ , $0 \leq χ \leq ψ$ , and $χ \in C_{c}^{\infty} (R^{n})$ Control region and nested real cutoffs. Multiplication by these cutoffs is bounded on the relevant scale levels Cutoff multipliers through order two and by duality. Time integrals below are squared Bochner norms as specified in Continuous and mixed time-space function spaces.

[F3] The regularized adjoint is $y = R U (t) v_{0}$ with $y_{0} = R v_{0}$ , and compatible constructions agree across scale orders Resolvent-regularized adjoint and its forcing. The adjoint evolution and its inverse are uniformly bounded on each level between $- 2$ and $2$ Forward and backward real-linear propagators. Higher scale levels embed continuously into lower ones Positive spectral Hilbert scale: duality and interpolation.

[F4] For $0 \leq θ \leq 1$ , localized smoothing gives $\int_{0}^{T} ∥ χ y (t) ∥_{H_{D}^{θ + 1 / 2}}^{2} d t \leq C ∥ y_{0} ∥_{H_{D}^{θ}}^{2}$ whenever $v_{0} \in H_{D}^{θ - 2}$ Lemma 4.2: smoothing of the regularized adjoint. Spectral interpolation gives $∥ y_{0} ∥_{H_{D}^{1 / 2}}^{2} \leq ε ∥ y_{0} ∥_{H_{D}^{1}}^{2} + C_{ε} ∥ y_{0} ∥_{2}^{2}$ Spectral interpolation and lower-order absorption.

[F5] At either $σ = - 1$ or $σ = - 2$ , the localized resolvent commutator gives $∥ φ R v ∥_{H_{D}^{σ + 2}} \leq ∥ φ v ∥_{H_{D}^{σ}} + C ∥ ψ R v ∥_{2}$ Integer resolvent commutator and localized first-order bounds.

Given: The hypotheses, cutoffs, index and datum in the statement. Put $τ = σ + 2 \in {0, 1}$ and $y_{0} = R v_{0}$ .

Goal: Establish the displayed estimate, uniformly in the datum.

1.1F1F2F3given

By the inverse evolution bound at order $τ - 2$ and the resolvent isometry, $∥ y_{0} ∥_{H_{D}^{τ}} \leq C ∥ y (t) ∥_{H_{D}^{τ}}$ for every $t \in [0, T]$ . The same argument at order $- 2$ gives $∥ y_{0} ∥_{2} \leq C ∥ y (t) ∥_{2}$ . These statements apply simultaneously by scale embedding and compatibility. Since $y = φ y + χ y$ , integration and the triangle inequality yield $∥ y_{0} ∥_{H_{D}^{τ}}^{2} \leq C \int_{0}^{T} (∥ φ y ∥_{H_{D}^{τ}}^{2} + ∥ χ y ∥_{H_{D}^{τ}}^{2}) d t$ and $∥ y_{0} ∥_{2}^{2} \leq C \int_{0}^{T} (∥ φ y ∥_{2}^{2} + ∥ ψ y ∥_{2}^{2}) d t$ . In the second inequality use $0 \leq χ \leq ψ$ .

2.1F2F3F4step 1.1

If $τ = 0$ , the pointwise cutoff inequality in step 1.1 directly controls the $χ y$ term by $ψ y$ . If $τ = 1$ , the datum belongs to $H_{D}^{- 3 / 2}$ by the scale embedding. Apply smoothing with $θ = 1 / 2$ , then interpolation, to get $\int_{0}^{T} ∥ χ y ∥_{H_{D}^{1}}^{2} d t \leq C ∥ y_{0} ∥_{H_{D}^{1 / 2}}^{2} \leq C ε ∥ y_{0} ∥_{H_{D}^{1}}^{2} + C_{ε} ∥ y_{0} ∥_{2}^{2}$ . Insert the second estimate of step 1.1, use $∥ φ y ∥_{2} \leq ∥ φ y ∥_{H_{D}^{1}}$ , and choose $ε$ small enough to absorb the first term into the left side. Thus in both cases $∥ y_{0} ∥_{H_{D}^{τ}}^{2} \leq C \int_{0}^{T} (∥ φ y ∥_{H_{D}^{τ}}^{2} + ∥ ψ y ∥_{2}^{2}) d t$ .

3.1F1F5step 2.1∎

Apply the commutator estimate to $v (t)$ at the chosen endpoint, square it and integrate. The resulting bound for the $φ y$ term in step 2.1 is a constant times the sum of the two integrals in the goal. Finally $∥ y_{0} ∥_{H_{D}^{τ}} = ∥ v_{0} ∥_{H_{D}^{σ}}$ by the resolvent isometry. This proves the estimate.

Source notes

The article is cited for attribution of Lemma 4.3 and its proof strategy. The deduction above uses the stated suppliers and does not use the article's lemma as a premise.

Depends on

Used by · 1

Proposition 4.6: conditional adjoint observability

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF page 18, Lemma 4.3 and equation (4.4)