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Theorem 3.7: abstract small-data perturbation result

PDF p. 14: Theorem 3.7; PDF p. 15: proof concludes

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary and connected exterior $Ω = R^{n} ∖ Θ$ , and its generalized reflected rays leave every compact set in finite time. Fix $T > 0$ , $s \in [1, 2]$ , $m \in {3, 5}$ and real $λ \neq 0$ . Let $H = H_{D}^{s} (Ω)$ be the Dirichlet spectral space, $W (t) = e^{i t Δ_{D}}$ its free group, and $φ$ the control cutoff of Control region and nested real cutoffs. Let $X_{T}$ satisfy the three abstract perturbation hypotheses of Abstract nonlinear control-space hypotheses, including its norm-one continuous inclusion into $C ([0, T]; H)$ .

Then there is $δ > 0$ such that, for every $v_{0}, v_{T} \in H$ with $∥ v_{0} ∥_{H}, ∥ v_{T} ∥_{H} \leq δ$ , there are a control $h \in C ([0, T]; H)$ and an integral solution $v \in X_{T}$ of $i \partial_{t} v + Δ_{D} v + λ ∣ v ∣^{m - 1} v = φ h$ with $v (0) = v_{0}$ and $v (T) = v_{T}$ . The integral solution is understood through the variation-of-constants formulation in Mild formulation of the controlled power NLS, with the homogeneous Dirichlet condition encoded by the Dirichlet realization.

Facts & Assumptions

[F1]

The geometry and cutoff have the meanings in Exterior and truncated obstacle domains and Control region and nested real cutoffs. The controlled equation and power nonlinearity are those of Linear and nonlinear internally controlled exterior systems.

[F2]

The perturbation hypotheses provide a Banach path space $X_{T}$ , its continuous inclusion into $C ([0, T]; H)$ , and the homogeneous, retarded-forcing and nonlinear estimates with a fixed constant $C_{0} \geq 1$ . Continuous paths and the $L^{1}$ forcing spaces use the conventions of Continuous and mixed time-space function spaces. Abstract nonlinear control-space hypotheses

[F3]

Under Choice and the stated geometry and regularity, Theorem 3.6: bounded continuous control operator supplies a bounded linear endpoint selector $G : H \times H \to C ([0, T]; H)$ for the linear controlled evolution. The Axiom of Choice

[F4]

With this selector, Endpoint-correcting nonlinear feedback map defines $N (v) = D (i g (v))$ , $h_{v} = G (v_{0}, v_{T} - N (v) (T))$ , and $T (v) = W (\cdot) v_{0} + N (v) + D (- i φ h_{v})$ , where $g (z) = λ ∣ z ∣^{m - 1} z$ and $D$ is the retarded forcing operator. These expressions belong to their stated spaces.

[F5]

The feedback estimates in Endpoint correction and small-ball contraction estimates give constants $C_{1}, C_{2}$ depending on the fixed data such that, with $M = 4 C_{1} δ$ and $4 C_{2} M^{m - 1} \leq 1$ , the feedback map has a fixed point in the closed radius- $M$ ball of $X_{T}$ whenever each endpoint norm is at most $δ$ . That fixed point has the prescribed endpoints.

[F6]

The variation-of-constants equivalence in Mild formulation of the controlled power NLS identifies a continuous path satisfying the displayed mild equation, with both nonlinear and control forcings Bochner integrable in $H$ , as an integral solution of the controlled abstract equation.

[F7]

The normed Dirichlet spectral space $H = H_{D}^{s} (Ω)$ and the free group $W (t) = e^{i t Δ_{D}}$ are those of Dirichlet spectral Sobolev scale and duality, under the same Choice and exterior-geometry hypotheses.

Proof

Given: The assumptions and notation of the Statement.

Goal: Construct a continuous control and an integral solution in $X_{T}$ with the prescribed endpoints for sufficiently small endpoint data.

1.1F2F3F4F7

The stated range $s \in [1, 2]$ is contained in the selector theorem's range. Choose its bounded endpoint selector $G$ and use it in the feedback construction. The three perturbation hypotheses apply to this same $X_{T}$ , free group and power nonlinearity, so the feedback map and its control $h_{v}$ are well defined as in F4.

1.2F4F5

The constants in F5 are fixed once the geometry, time, parameters, cutoff, selector and perturbation bounds are fixed. Since $m > 1$ , choose $δ > 0$ small enough that, for $M = 4 C_{1} δ$ , the inequality $4 C_{2} M^{m - 1} \leq 1$ holds. For any two endpoints with norms at most $δ$ , F5 then gives a fixed point $v \in X_{T}$ with $v (0) = v_{0}$ and $v (T) = v_{T}$ . Its selected control $h = h_{v}$ belongs to $C ([0, T]; H)$ .

2.1F2F4step 1.2

The fixed-point identity is $v = W (\cdot) v_{0} + D (i g (v)) + D (- i φ h)$ . The perturbation hypotheses give $g (v) \in L^{1} (0, T; H)$ ; the feedback construction gives $φ h \in L^{1} (0, T; H)$ . These are Bochner-integrable forcings, and the identity is exactly the mild equation in F6.

3.1F1F5F6step 1.2step 2.1∎

Applying F6 gives the asserted integral solution of the controlled equation with initial value $v_{0}$ . Its terminal value is $v_{T}$ by step 1.2. Thus the chosen $δ$ works for every endpoint pair in the stated balls. The argument asserts uniqueness only for the constructed feedback fixed point inside its chosen small ball, not for all controlled solutions.

Source notes

The frozen article, PDF pages 14–15, is cited for attribution of Theorem 3.7 and its feedback construction. The deduction above uses the stated selector, feedback estimates and mild-solution equivalence.

Depends on

Used by · 3

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF pages 14–15, Theorem 3.7