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
- Realification of a complex vector space by restriction of scalars
- Realification preserves a complex norm and its metric
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Hilbert space
- The Axiom of Countable Choice (AC_ω)
- Riesz representation for Hilbert spaces
- L² with the integral pairing is a Hilbert space
- The Hilbert-space adjoint of a bounded operator
- The essential supremum is attained as the least essential bound
- The dual space X^* of a normed space and its dual norm
- A bounded linear operator between normed spaces
- Measure spaces
- The essential supremum of a measurable function with respect to a measure
- Complex Lp classes and Euclidean test-function conventions
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Additivity of the nonnegative Lebesgue integral
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- Observation lower bounds give Hilbert control right inverses
- Reachability and observation vocabulary for controlled evolutions
- The Axiom of Choice
- Bochner L2 on a finite interval is a separable Hilbert space
- Bochner integrability criterion
- Bochner integral norm inequality
- Linearity of the Bochner integral
- Bounded linear maps commute with Bochner integration
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Strongly measurable Banach-valued function
- Banach-valued simple function and integral
- Continuous and mixed time-space function spaces
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- Finite and countable subadditivity of measures
- Hilbert-adjoint identities
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Holder's inequality for integrals, including the endpoint cases
- Complex polynomial weak derivatives and zero boundary preservation
- 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
- 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
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
Lemma
Realification of Hilbert pairings and conjugate-linear coefficients
Lemma
Endpoint maps and exact steering from a null-control operator
Lemma