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Theorem 1.4: linearized control around a trajectory

PDF p. 5: Theorem 1.4; PDF p. 19: Proof of Theorem 1.4

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , $Ω = R^{n} ∖ Θ$ is the connected exterior of a nonempty compact obstacle with smooth boundary, and $Θ$ is non-trapping, meaning that every generalized reflected ray leaves every compact set in finite time. Let $T > 0$ and $λ \in R ∖ {0}$ . Assume (W): $w \in C ([0, T]; H_{D}^{2} (Ω) \cap W^{2, \infty} (Ω))$ solves $i w_{t} + Δ w + λ ∣ w ∣^{2} w = 0$ distributionally, with the homogeneous Dirichlet condition encoded by $H_{D}^{2} (Ω) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ .

Choose $R > 0$ with $Θ \subset B_{R} (0)$ and the real smooth cutoff $φ$ of Control region and nested real cutoffs, zero for $∣ x ∣ \leq R$ , positive for $∣ x ∣ > R$ , and equal to one for $∣ x ∣ \geq R + 1$ . Assume (UCPw): an adjoint solution $v \in C ([0, T]; H_{0}^{1} (Ω))$ of $i v_{t} + Δ v + 2 λ ∣ w ∣^{2} v - λ w^{2} \overline{v} = 0$ that vanishes on $(0, T) \times {x \in Ω : ∣ x ∣ > R}$ is identically zero.

For $s \in {1, 2}$ and arbitrary $z_{0}, z_{T} \in H_{D}^{s} (Ω)$ , there is a control $h \in C ([0, T]; H_{D}^{s} (Ω))$ and a corresponding solution $z \in C ([0, T]; H_{D}^{s} (Ω))$ of

$i z_{t} + Δ z + 2 λ ∣ w ∣^{2} z + λ w^{2} \overline{z} = φ h, z ∣_{\partial Ω} = 0, z (0) = z_{0},$

such that $z (T) = z_{T}$ . Here $H_{D}^{1} (Ω) = H_{0}^{1} (Ω)$ , and continuity is in the relevant spectral norm. The solution is in the mild class of Real HUM operator for the trajectory linearization; no restriction to dimensions two and three is imposed on this linear statement.

Facts & Assumptions

[F1]

Under Choice and (H), (W), and (UCPw), Coercivity and bounded trajectory control operator gives, for $E = H_{D}^{s} (Ω)$ and $s \in {1, 2}$ , a bounded real-linear map $G_{w} : E \times E \to C ([0, T]; E)$ selecting a control that steers the linearized system between the prescribed endpoints.

[F2]

In Real HUM operator for the trajectory linearization, the selected control is $G_{w} (x_{0}, x_{T}) = h_{a}$ , where $a = S_{w}^{- 1} (x_{0} - P (0, T) x_{T})$ ; the backward controlled solution starts at $S_{w} a$ and ends at zero, and the homogeneous evolution $P (t, T) x_{T}$ ends at $x_{T}$ . The equation and cutoff are those of Linearized equation and its real-duality adjoint and Control region and nested real cutoffs.

[F3]

The scale identifications and path-space meaning are supplied by Dirichlet spectral Sobolev scale and duality and Continuous and mixed time-space function spaces. The hypotheses and their conventions are those of Exterior and truncated obstacle domains, Reference trajectory hypothesis W, Conditional energy-level unique continuation hypothesis, and The Axiom of Choice.

Proof

Given: The assumptions and endpoints in the statement.

Goal: Construct a continuous control and a mild solution with the two prescribed endpoints.

1.1givenF1F2F3

Put $E = H_{D}^{s} (Ω)$ and set $h = G_{w} (z_{0}, z_{T})$ . The hypotheses of F1 are exactly those assumed, so $h \in C ([0, T]; E)$ . In F2 set $a = S_{w}^{- 1} (z_{0} - P (0, T) z_{T})$ ; the inverse exists by F1. The corresponding backward controlled solution $z_{a}$ belongs to $C ([0, T]; E)$ , has $z_{a} (T) = 0$ , and has $z_{a} (0) = z_{0} - P (0, T) z_{T}$ .

2.1F2F3step 1.1∎

Define $z (t) = z_{a} (t) + P (t, T) z_{T}$ . By the real linearity of the equation and the mild evolution in F2, this is a mild solution with control $h$ , and it is continuous in $E$ . At time zero the two terms sum to $z_{0}$ ; at time $T$ they sum to $z_{T}$ . The scale and boundary conventions in F3 give the stated formulation.

Source notes

The original article is cited for attribution of Theorem 1.4 and its HUM method. The argument above applies the independently stated local control-operator proposition; it does not take the article's theorem as a premise.

Depends on

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 5, Theorem 1.4; page 19, proof of Theorem 1.4