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Article items · 08 — Cubic and quintic time bounds from L4 Linfinity control
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 1–5.
1 · Prerequisites
- Complex polynomial weak derivatives and zero boundary preservation
- Continuous and mixed time-space function spaces
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- Integer-order Sobolev spaces are Banach
- A closed subspace of a Banach space is Banach
- Strongly measurable Banach-valued function
- Bochner integrability criterion
- Holder's inequality for integrals, including the endpoint cases
- Chain rule for globally Lipschitz scalar maps of Sobolev functions
- Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
- Dominated convergence
- Explicit compactly supported smooth cutoffs
- The Axiom of Choice
- Reachability and observation vocabulary for controlled evolutions
- Smooth manifolds and their smooth charts
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Riemannian metric and riemannian manifold
- Levi civita connection
- Fundamental theorem of riemannian geometry
- Geodesic of an affine connection
- Riemannian speed and length
- Independent generalized reflected-ray and GCC convention
- Periodic energy space on an interval
- Complex integer-order Sobolev notation
- Exterior and truncated obstacle domains
- The standard smooth step function
- Test function cutoffs and euclidean localization
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- Bounded C1 domains and their outward normals
- Classical normal derivative
- The Laplacian of a C² function and of a C² vector field
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
- Real and imaginary parts, complex conjugation, and modulus
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- The Axiom of Countable Choice (AC_ω)
- Diagonal gluing without a single-valued reflected flow
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Continuity of a map of topological spaces at a point and globally
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)
- A uniform limit of continuous functions is continuous, so C(X,Y) is closed in Y^X under the uniform metric
- Endpoint maps and exact steering from a null-control operator
- Banach space
- A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The exponential function is strictly increasing
- Realification preserves a complex norm and its metric
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- Compatible Banach pairs and their integrals
- Strongly continuous shifted orbits in finite Bochner norms
- Abstract retarded integration in compatible time spaces
- Scalar Banach-function retarded estimates by Fatou approximation
- Spatial Lq classes satisfy the scalar mixed-norm hypotheses
- Strongly continuous one-parameter unitary group
- Bochner integral norm inequality
- Minkowski's integral inequality
- Hilbert space
- Bochner-integrable function
- Uniform localized H2 algebra on a smooth exterior domain
- Weak product rule for bounded Sobolev functions
- Meyers–Serrin density on an arbitrary open set
- Weak Leibniz rule with a smooth factor
- Chain rule for a C¹ function with bounded derivative
- The Lᵖ trace operator on a bounded C¹ domain
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Higher-order Sobolev embedding
- Compact normalized compressed phase sets over bounded regions
- Unit-speed confinement before an artificial outer boundary
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- For an open subset of ℝⁿ, connectedness, path-connectedness and polygonal connectedness are equivalent
- Every path-connected space is connected, and every path component lies inside a component
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- For a nonempty set X and a metric space (Y,d) the uniform metric ρ̄(f,g) = supₓ min{d(f(x),g(x)), 1} is a metric on Y^X
- On a nonempty compact metric domain, the compact-open topology is the uniform topology
- The compact-open topology on C(X,Y) for a metric domain X, with subbasis S(K,V) = {f : f[K] ⊆ V}
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Herbert Koch and Daniel Tataru, On the Spectrum of Hyperbolic Semigroups, Comm. PDE 20 (1995), 901–937 · Author-uploaded full text: introduction, manifold/symbol setup and negative-inside defining function before (0.3), continuation paragraph after (P),(B),(H); §1 compressed-coordinate construction, (1.5), Definition 1.1, (1.6), Proposition 1.2(a),(b), and Proposition 1.4
- Compatible compact Banach triples
- Bochner spaces and bounded inclusion of value spaces
- Distributional time derivatives with values in a Banach space
- Finite-time bounds for Banach-valued derivatives
- The L2 pivot injects into the local negative Sobolev space
- Restriction Bessel-potential spaces on polyhedral domains
- Every real-order Bessel-potential completion is Hilbert
- A quotient of a Banach space by a closed subspace is Banach
- L² with the integral pairing is a Hilbert space
- Jacques Simon, Compact sets in the space Lᵖ(0,T;B), Annali di Matematica Pura ed Applicata 146 (1987), 65–96 · §8, embedding assumption (8.1), printed p.84, and Corollary 4 first clause, printed p.85; scanned PDF leaf 11 (zero-based page 10)
- The distance from a point to a nonempty compact set is attained at a point of that set, and two disjoint compact sets are at positive distance
- A convex subset of ℝᵐ contains every line segment between two of its points
- Every convex subset of ℝⁿ, in particular every ball and ℝⁿ itself, is path-connected and hence connected
- A constrained local extremum annihilates every velocity of a differentiable parametrization
- Weak shifted Dirichlet form and energy bound on an open set
- Collar and tail assembly of the exterior graph domain
- Symmetric, self-adjoint and essentially self-adjoint operators
- Distributional equation convention for complex Schrödinger solutions
- Real-order Bessel-potential completion Hˢ
- Real-order Hˢ as weighted Fourier distributions
- Complex Holder, Minkowski, and the quotient norm
- Integral over a measurable subset
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Exact boundary controllability of the nonlinear Schrödinger equation · Abstract
- Vortex Lines in an Imperfect Bose Gas · §2, printed pages 451–452, equations (5)–(9) and validity condition (16)
- The negative Sobolev space H⁻¹(Ω)
- Exact boundary controllability of the nonlinear Schrödinger equation · Theorem 1.1, journal page 4131
- Exact Controllability for the Schrödinger Equation · Machtyngier (1994), printed pp.24–25, equations (1.1), (1.3), Theorem 1.1 and real-inner-product conventions at the end of §1
- Geometric control in the presence of a black box · arXiv math/0304184v3, Theorem 1 and (1.1), PDF p.2; §6.2, PDF p.16
- Nicolas Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes · §0.1, printed p.6; Theorem 0.1 and equation (0.1), printed p.7; also Numdam archival scan https://www.numdam.org/article/MSMF₁993₂₅5_₃₀.pdf, PDF pp.5–6
- The published Riemann change-of-variables theorem already gives the Lebesgue formula for continuous compactly supported integrands
- A totally differentiable map with zero derivative on a convex open set is constant
- Euclidean balls have positive finite Lebesgue measure
- Stabilization of Schrödinger equation in exterior domains · Introduction, ambient-dimension convention, equation (0.5), Definition 1 and Theorem 2, printed pages 570–572
- Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains · Publisher abstract
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- A self-adjoint operator generates a strongly continuous unitary group
- Spectral form domain and core of a semibounded operator
- Spectral cutoff cores and unitary identifications of completed scales
- Exact scale interpolation and endpoint operator transport
- Bochner Lp is complete for a Banach value space
- Projection valued measure
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Monotone convergence for the integral
- Every nonnegative measurable function admits an explicit increasing sequence of simple approximations
- Admissible spectral strip profiles and the exact interpolation upper bound
- Well-defined strip norm and the action of a common endpoint operator
- A single collar-reflection and free-tail exterior extension
- Restriction quotient and Bochner interpolation for the exterior extension
- J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Compatible couples and sums: PDF pp.4–6; §2.2, PDF pp.9–10 (strip definition, Banach property and exact operator bound); Example 2.7(i), PDF p.11 (Bochner interpolation); Definitions 3.1, 3.3, 3.6 and Theorem 3.7, PDF pp.15–17 (whole-space Bessel convention and identity).
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- The Gauss-Green integration-by-parts formula with Sobolev traces
- The sharp trace theorem: boundedness and range in the fractional space
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Finite ambient partitions near compact sets
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
- The completed product measure
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Complex Lp completeness and almost-everywhere subsequences
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- Every section of a product-measurable function is measurable
- Riesz representation for Hilbert spaces
- Closed semibounded symmetric forms and their associated operators
- Semibounded closed-form representation and perturbation
- The Sobolev space H¹ is a Hilbert space
- Smooth compactly supported functions of an open set are dense in L²
- Continuous free convolution on a finite time interval
- Short-interval real Volterra contraction
- Global real evolution, cocycle and inverse
- Bochner variation of constants for real evolution
- Transfer of a compatible spacetime norm through the free forcing map
- Gronwall's integral inequality with variable and constant coefficients
- Linearity of the Bochner integral
- Star-shaped multiplier signs
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
LemmaDependency level 1
Cubic and quintic time bounds from L4 Linfinity control
RemarkDependency level 1
Prior compact-manifold control under GCC and unique continuation
RemarkDependency level 1
Prior interval and rectangle NLS control results
DefinitionDependency level 2
Control region and nested real cutoffs
DefinitionDependency level 2
Cubic NLS and boundary control formulations
DefinitionDependency level 2
Non-trapping obstacle
LemmaDependency level 2
Uniform hitting for a compact closed family of ray segments
LemmaDependency level 2
A bounded endpoint selector yields small nonlinear exact steering
LemmaDependency level 2
Homogeneous bounds transfer to L1 retarded forcing
LemmaDependency level 2
Tame bounds for the quadratic-cubic trajectory remainder
PropositionDependency level 2
Closed compactness of arbitrary exterior generalized-ray segments
RemarkDependency level 2
Authors' qualified novelty claims
TheoremDependency level 2
General Banach-triple Aubin–Lions–Simon compactness
ExampleDependency level 3
A ray trapped between two convex obstacles
LemmaDependency level 3
Exterior Dirichlet realization, H² domain and form domain
Open problemDependency level 3
Outer Neumann boundary controllability
Open problemDependency level 3
Question A: general boundary exact control
Open problemDependency level 3
Question B prime: outer boundary control
PostulateDependency level 3
Cubic NLS as a nonlinear dispersive wave model
RemarkDependency level 3
Prior full and partial boundary control results reported in the introduction
RemarkDependency level 3
Prior exterior-domain dispersive and Cauchy results
RemarkDependency level 3
Prior damped exterior Schrödinger results
DefinitionDependency level 4
Dirichlet Laplacian and positive shift
LemmaDependency level 4
Lifted spectral measure and isometric Bochner scale interpolation
LemmaDependency level 4
An explicit common interpolation model for the fractional Strichartz targets
LemmaDependency level 4
Localized multiplier estimate for the exterior Dirichlet normal trace
LemmaDependency level 5
Positive spectral Hilbert scale: duality and interpolation
LemmaDependency level 5