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Article items · 12 — Linearized propagator bounds in the nonlinear control spaces
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 15–26.
1 · Prerequisites
- Exterior and truncated obstacle domains
- Non-trapping obstacle
- Reference trajectory hypothesis W
- Dirichlet spectral Sobolev scale and duality
- Energy control space with a time-integrable supremum norm
- Complex integer-order Sobolev notation
- Continuous and mixed time-space function spaces
- Time-dependent potential and conjugation bounds on the scale
- Forward and backward real-linear propagators
- Free and retarded bounds in the energy control space
- Scale isometries and free Schrödinger evolution
- Energy dispersive bounds under bounded trajectory perturbations
- The Axiom of Choice
- Control region and nested real cutoffs
- Resolvent-regularized adjoint and its forcing
- Cutoff multipliers through order two and by duality
- Positive spectral Hilbert scale: duality and interpolation
- Lemma 4.2: smoothing of the regularized adjoint
- Integer resolvent commutator and localized first-order bounds
- Spectral interpolation and lower-order absorption
- Local negative Sobolev space in the compactness argument
- Weak convergence notation in normed state spaces
- Exterior spectral domains and the critical trace index
- Fractional compactness on bounded collar domains
- General Banach-triple Aubin–Lions–Simon compactness
- Complex completeness, density, and inner product: the consumer interface
- Dominated convergence
- Weakly convergent sequences are norm bounded
- Dirichlet Laplacian and positive shift
- Distributional time derivatives with values in a Banach space
- Hahn-Banach dominated extension theorem for real vector spaces
- AC supplies the countable and dependent choices used in Banach integration
- Weak Leibniz rule with a smooth factor
- Bochner integrability criterion
- Bounded linear maps commute with Bochner integration
- Invisible initial-data spaces
- Regularized lower-order observation remainder
- Endpoint multiplier bounds extended to negative spectral orders
- Lemma 3.2: compactness of the remainder
- Lemma 3.3: observability modulo a compact remainder
- Lemma 3.4: free unique continuation on the scale
- Lifting zero-order observability to the order-two endpoint
- Hilbert spaces are reflexive
- Reflexivity is equivalent to weak subsequential compactness of bounded sequences
- AC implies DC implies countable choice
- The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter
- States compatible with the uncontrolled obstacle boundary
- Low-order identifications of the Dirichlet scale
- Linear and nonlinear internally controlled exterior systems
- Cubic NLS and boundary control formulations
- Distributional equation convention for complex Schrödinger solutions
- Lemma 3.11: extension preserving the obstacle Dirichlet condition
- Collar traces and extension across an artificial boundary
- The sharp trace theorem: boundedness and range in the fractional space
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Locally constant invisible spaces are Laplacian invariant
- Linearized equation and its real-duality adjoint
- Conditional energy-level unique continuation hypothesis
- Lemma 4.5: invisible solutions acquire energy regularity
- Scale Riesz map and free HUM operator
- Linear forced well-posedness by Duhamel formula
- Observation lower bounds give Hilbert control right inverses
- Realification preserves a complex norm and its metric
- Bochner Lp is complete for a Banach value space
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Holder's inequality for integrals, including the endpoint cases
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- Integrable real and complex functions, and their integrals
- The Lebesgue integral is linear on L¹(μ)
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Hilbert space
- Fundamental theorem of calculus for Banach-valued continuous curves
- Homogeneous half-derivative local smoothing
- Real HUM operator for the trajectory linearization
- Real pairing cancellation for the linearized-adjoint pair
- Bounded real-linear perturbations preserve solution estimates
- Realification of Hilbert pairings and conjugate-linear coefficients
- Weak shifted Dirichlet form and energy bound on an open set
- Evolution triples and the parabolic energy space
- The evolution-space energy identity and continuous H trace
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Exterior Dirichlet realization, H² domain and form domain
- Classical derivatives agree with weak derivatives
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Test function space d of an open set
- Distribution
- Local finite order characterization of distributions
- Multiplication of a distribution by a smooth function
- Support of a distribution
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Abstract nonlinear control-space hypotheses
- Average convergence for a continuous Banach-valued function
- Linearity of the Bochner integral
- Quadratic-cubic remainder after subtracting a trajectory
- Duhamel formula for a real-linear propagator
- A bounded endpoint selector yields small nonlinear exact steering
- A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
- Quadratic-cubic estimates near a reference trajectory
- Infinitesimal generator of a C0-semigroup
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- Variation of constants for the inhomogeneous abstract Cauchy problem
- Complex Lp completeness and almost-everywhere subsequences
- Mild formulation of the controlled power NLS
- Fractional Hilbert Sobolev algebra on a smooth exterior domain
- Complex polynomial weak derivatives and zero boundary preservation
- Bochner integral norm inequality
- If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y)
- Exterior H2 algebra and polynomial nonlinear estimates
- Absolute continuity of the integral
- Cubic and quintic energy-space nonlinear time estimates
- Fubini's theorem for L¹ functions on a sigma-finite product
- Independent generalized reflected-ray and GCC convention
- Closed compactness of arbitrary exterior generalized-ray segments
- Uniform hitting for a compact closed family of ray segments
- An outward Euclidean exit cannot return
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Reachability and observation vocabulary for controlled evolutions
- Nontrapping cutoff resolvent estimates
- Imported homogeneous and retarded exterior smoothing
- Unit-speed confinement before an artificial outer boundary
- Compact normalized compressed phase sets over bounded regions
- A sequence in a metric space has at most one limit
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
PropositionDependency level 15
Linearized propagator bounds in the nonlinear control spaces
LemmaDependency level 16
Lemma 4.3: perturbed observability modulo a remainder
LemmaDependency level 16
Lemma 4.4: compactness for regularized perturbed trajectories
PropositionDependency level 16
Proposition 3.5: full-scale free observability
PropositionDependency level 16
Transfer from exterior internal control to outer boundary control
RemarkDependency level 16
Scope of the spectral unique continuation argument
PropositionDependency level 17
Proposition 4.6: conditional adjoint observability
PropositionDependency level 17
Free HUM identity, coercivity and invertibility
RemarkDependency level 17
Proposed transfer of the smoothing-control mechanism
PropositionDependency level 18
Coercivity and bounded trajectory control operator
RemarkDependency level 18
Remark 6: exact use and zero-trajectory case of unique continuation
TheoremDependency level 18
Theorem 3.6: bounded continuous control operator
DefinitionDependency level 19
Endpoint-correcting nonlinear feedback map
DefinitionDependency level 19
Endpoint feedback around a trajectory
RemarkDependency level 19
Localization and analytic scope of the free control construction
TheoremDependency level 19
Theorem 1.2: linear control on the full stated scale
TheoremDependency level 19
Theorem 1.4: linearized control around a trajectory
PropositionDependency level 20
Endpoint correction and small-ball contraction estimates
PropositionDependency level 20
Small-data contraction around a nonzero trajectory
TheoremDependency level 21
Theorem 1.5: cubic internal control near a trajectory
TheoremDependency level 21
Theorem 3.7: abstract small-data perturbation result
CorollaryDependency level 22
Remark 5: controllability above the Sobolev algebra threshold
PropositionDependency level 22
Proposition 3.10: order-two local control
PropositionDependency level 22
Proposition 3.9: energy-space local control
CorollaryDependency level 23
Corollary 1.6: quintic control near zero
TheoremDependency level 23
Theorem 1.3: cubic internal control near zero
TheoremDependency level 24
Theorem 1.1: exact outer boundary controllability
RemarkDependency level 25
Non-trapping and geometric control from infinity
RemarkDependency level 26