Alphabeta MathPersonal research
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
- The Lᵖ trace operator on a bounded C¹ domain
- The half-space trace lies in the fractional Slobodeckij space
- A bounded right inverse of the half-space trace by normal mollification
- The fractional Sobolev space on a compact C¹ boundary
- Chart independence of the fractional boundary norm
- The trace commutes with smooth cutoffs and is chart local
- The coordinate-direction form of the Slobodeckij seminorm
- Cᵏ boundary flattening preserves local W^k,p
- Finite ambient partitions near compact sets
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Holder's inequality for integrals, including the endpoint cases
- Integer-order Sobolev spaces and their norms
- The Axiom of Choice
- Weak Leibniz rule with a smooth factor
- The space L^∞(μ) of essentially bounded measurable functions
- The function space Lᵖ(μ) for 0 < p < ∞
- The space Lᵖ(μ) as the quotient by null functions
- The ACL characterisation of W^1,p
- Absolute continuity on almost every coordinate line
- Absolute continuity on a compact interval
- Linearity, locality, and commutation of weak derivatives
- A Lipschitz function after an absolutely continuous function is absolutely continuous
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
- Fundamental theorem of calculus for absolutely continuous functions
- Chain rule for an indefinite integral after an absolutely continuous composition
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Generalized Holder inequality puts products into Lʳ
- Finite-measure Lʳ includes into Lᵖ for p < r
- The essential supremum is attained as the least essential bound
- Lᵖ and L^∞ are vector spaces for p ≥ 1
- Continuous functions on Euclidean spaces are Borel measurable
- Composition with a Borel measurable outer map preserves measurability
- Test function cutoffs and euclidean localization
- The Lebesgue integral is linear on L¹(μ)
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Finite, sigma-finite, and semifinite measures
- AC supplies the countable and dependent choices used in Banach integration
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- Limit superior and limit inferior of a real sequence as infₙ sup_k ≥ n xₖ and supₙ inf_k ≥ n xₖ in ℝ̄
- A real sequence converges to L ∈ ℝ iff liminf xₖ = limsup xₖ = L, and diverges to ±∞ iff both equal ±∞
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- Lebesgue outer measure on ℝⁿ
- Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume
- Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of ℝⁿ is the infimum of the measures of the open sets containing it
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Every complete ordered field is Archimedean
- The Axiom of Countable Choice (AC_ω)
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra
- Borel representatives make the convolution integrand Borel measurable
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Conjugate exponents, including the endpoint conventions
- Banach space
- A bounded linear operator between normed spaces
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complex norm and complex normed space
- Hilbert space
- The Hilbert-space adjoint of a bounded operator
- Hilbert-adjoint identities
- The Lax--Milgram theorem
- 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
- Normed vector space over an absolutely valued field
- The reverse triangle inequality in a normed space
- Continuous functions on a closed interval attain their extrema
- Vector addition and scalar multiplication are continuous in a normed space
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Zero-boundary Sobolev space as a norm closure
- Weak product rule for bounded Sobolev functions
- Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
- Dominated convergence
- Explicit compactly supported smooth cutoffs
- Bounded Cᵏ domains and boundary charts
- Bounded Cᵏ domains admit integer-order Sobolev extension
- The Sobolev space W^k,p is an algebra above the critical index
- Higher-order Sobolev embedding
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
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
Theorem
The sharp trace theorem: boundedness and range in the fractional space
Theorem
Chain rule for globally Lipschitz scalar maps of Sobolev functions
Theorem
Young's convolution inequality under Countable Choice
Definition
Reachability and observation vocabulary for controlled evolutions
Lemma
Observation lower bounds give Hilbert control right inverses
Lemma
Complex polynomial weak derivatives and zero boundary preservation
Lemma