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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
- Continuous and mixed time-space function spaces
- Reachability and observation vocabulary for controlled evolutions
- Endpoint maps and exact steering from a null-control operator
- Banach space
- The Axiom of Choice
- 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‖
- Cubic and quintic time bounds from L4 Linfinity control
- Uniform localized H2 algebra on a smooth exterior domain
- Complex polynomial weak derivatives and zero boundary preservation
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- Weak product rule for bounded Sobolev functions
- Meyers–Serrin density on an arbitrary open set
- Test function cutoffs and euclidean localization
- Weak Leibniz rule with a smooth factor
- Chain rule for a C¹ function with bounded derivative
- Explicit compactly supported smooth cutoffs
- Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
- Dominated convergence
- Holder's inequality for integrals, including the endpoint cases
- Strongly measurable Banach-valued function
- Bochner integrability criterion
- Integer-order Sobolev spaces are Banach
- A closed subspace of a Banach space is Banach
- The Lᵖ trace operator on a bounded C¹ domain
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Higher-order Sobolev embedding
- A compactly supported common L²/H² extension from a bounded smooth domain
- Whole-space interpolation of one extension–restriction operator
- Restriction-quotient interpolation from a common endpoint retraction
- A common exterior extension and explicit collar–tail retraction
- Well-defined strip norm and the action of a common endpoint operator
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
A bounded endpoint selector yields small nonlinear exact steering
Lemma
Tame bounds for the quadratic-cubic trajectory remainder
Lemma