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Control duality and nonlinear estimates
Develop the real Hilbert observation-to-control chain and endpoint feedback construction alongside polynomial, time-integrability and exterior algebra bounds needed for nonlinear endpoint and trajectory control.
1 · Prerequisites
- Integer-order Sobolev spaces and their norms
- The space Lᵖ(μ) as the quotient by null functions
- Sobolev extension domains and extension operators
- Weak derivative of a locally integrable function
- Meyers–Serrin density on an arbitrary open set
- Integer-order Sobolev spaces are Banach
- Weak partial derivatives lower the Sobolev order
- Classical derivatives agree with weak derivatives
- Choice function
- Complex norm and complex normed space
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complete metric space: every Cauchy sequence converges in the space
- Linear map between vector spaces over the same field
- Finite, countably infinite, countable, uncountable
- The well-ordering principle
- A nonempty set is at most countable iff it is a surjective image of ℕ
- The transpose of a bounded operator
- The spaces B(X,Y) and B(X) of bounded linear operators
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Real and complex inner-product spaces and their induced length
- Vector space over a field
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
- The Sobolev norm descends to equivalence classes
- Bochner integrability criterion
- The Banach-valued simple integral is well defined
- Additivity of the nonnegative Lebesgue integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Complex modulus restricts to real absolute value
- Fourier Bessel-potential and restriction-quotient Sobolev spaces
- The weighted Fourier model is an injective Hilbert representation
- The Matérn spectral integral, normalization, and radial formula
- Smooth compactly supported functions of an open set are dense in L²
- Schwartz space and its seminorms
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ
- Monotone convergence for the integral
- Dominated convergence
- Complex integration by parts on intervals and decaying lines
- The complex exponential is entire and its complex derivative is itself
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Explicit compactly supported smooth cutoffs
- Locally integrable functions as regular distributions
- Leibniz rule for distributions
- Multiplication of a distribution by a smooth function
- Uniqueness of a weak derivative as an almost-everywhere class
- Complex Lp classes and Euclidean test-function conventions
- Chain rule for globally Lipschitz scalar maps of Sobolev functions
- Linearity, locality, and commutation of weak derivatives
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Cauchy sequence in a metric space
- The recursion theorem
- For |r| < 1 the sequence rᵏ is null, and for |r| > 1 the sequence |r|ᵏ diverges to +∞
- Integer powers aᵐ
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- A sequence in a metric space has at most one limit
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
- Metric continuity characterisations, with countable choice for the sequential converse
- Factorisation of bⁿ - aⁿ, and the resulting Lipschitz estimate
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- The principle of mathematical induction
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Inverses of positives are positive, and reciprocation reverses order
- Algebra of limits: sums, scalar multiples, products and quotients
- Laws of integer exponents
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Limits and Cauchy sequences of reals
- The rationals embed densely in the reals
- Bounded Cᵏ domains and boundary charts
- Integer-order Sobolev extension from a half-space
- Cᵏ boundary flattening preserves local W^k,p
- Bounded restriction and cutoff localisation in Sobolev spaces
- Compactly supported Sobolev functions extend by zero in every integer order
- Weighted tempered-distribution characterization of Hˢ
- Real-order Bessel-potential completion Hˢ
- The Bessel completion embeds canonically in tempered distributions
- Real powers of the Japanese bracket act on Schwartz space
- Smooth polynomially bounded multipliers on schwartz space
- Regular distribution from a locally integrable function
- Fourier transform acts continuously on Schwartz space
- Schwartz derivatives are integrable
- L1 Fourier inversion with an integrable transform
- The L1 transform is bounded and uniformly continuous
- Fubini's theorem for L¹ functions on a sigma-finite product
- The notation Hᵏ and the reserved zero-boundary symbol
- Local Hölder and scaled C-two-alpha norms on balls
- The Sobolev conjugate exponent and the scaling identity
- Morrey's inequality for p>n
- The p=1 Gagliardo-Nirenberg-Sobolev inequality
- The Gagliardo-Nirenberg-Sobolev inequality for 1<p<n
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- A Euclidean bump for a compact set inside an open set
- Interior mollification commutes with weak derivatives
- Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
- If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz
- Complex Lp completeness and almost-everywhere subsequences
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- Conjugate exponents, including the endpoint conventions
- The function space Lᵖ(μ) for 0 < p < ∞
- The space L^∞(μ) of essentially bounded measurable functions
- The essential supremum is attained as the least essential bound
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- Young's inequality for conjugate real exponents
- A nonnegative integral over a null set vanishes
- The Lᵖ trace operator on a bounded C¹ domain
- The trace agrees with classical restriction for continuous Sobolev functions
- The trace commutes with smooth cutoffs and is chart local
- The Gauss-Green integration-by-parts formula with Sobolev traces
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Finite ambient partitions near compact sets
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞
- Fundamental theorem of calculus for absolutely continuous functions
- Complex translation, convolution, approximate identities, and mollification
- Bounded, coercive and symmetric sesquilinear forms
- Real and imaginary parts, complex conjugation, and modulus
- Hilbert space
- A coercive form operator is bounded below
- Coercivity makes a small form step a strict contraction
- A bounded form is represented by a unique bounded operator
- Orthogonal decomposition by a closed subspace
- The dual space X^* of a normed space and its dual norm
- Linear subspace of a vector space
- Orthogonality and the orthogonal complement
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
Corollary
The Sobolev space W^k,p is an algebra above the critical index
Definition
The Axiom of Choice
Definition
Banach space
Definition
A bounded linear operator between normed spaces
Definition
The Axiom of Countable Choice (AC_ω)
Definition
The Hilbert-space adjoint of a bounded operator
Definition
Realification of a complex vector space by restriction of scalars
Definition
Zero-boundary Sobolev space as a norm closure
Lemma
Bochner integral norm inequality
Lemma
Realification preserves a complex norm and its metric
Lemma
Schwartz functions are dense in the Fourier Bessel-potential norm
Lemma
Test function cutoffs and euclidean localization
Lemma
Weak Leibniz rule with a smooth factor
Lemma
Weak product rule for bounded Sobolev functions
Theorem
A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
Theorem
Bounded Cᵏ domains admit integer-order Sobolev extension
Theorem
Real-order Hˢ as weighted Fourier distributions
Theorem
Fourier inversion on Schwartz space
Theorem
Fourier transform of a product with one integrable transform
Theorem
Higher-order Sobolev embedding
Theorem
Hilbert-adjoint identities
Theorem
Holder's inequality for integrals, including the endpoint cases
Theorem
The kernel of the trace is the closure of the test functions
Theorem
The Lax--Milgram theorem
Theorem