Covector mapping principle

A compatibility principle giving conditions under which discretizing an optimal-control problem and then dualizing yields covectors corresponding to a discretization of the continuous Pontryagin adjoint system.

Version

v1 · 2026-09-08 · History

Domain-specific #

3952

Origin domain

computational optimal control

Subdomain

computational optimal control

Core Idea

The principle addresses whether costates, multipliers, and transversality conditions recovered from a nonlinear program converge consistently to the continuous-time dual variables rather than being artifacts of discretization. A primal discretization maps continuous trajectories and controls to nodes; differentiation produces discrete multipliers, and a declared covector transformation and convergence theorem map them to approximations of continuous adjoints. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Covector mapping principle belongs to computational optimal control and is useful where the analyst can specify the typed computational optimal control carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the continuous control problem, discretization and quadrature, primal convergence, discrete nonlinear program, multiplier normalization, covector map, boundary and path constraints, regularity, and convergence mode are explicit. The scope is broad within that domain but bounded by the need for the continuous control problem, discretization and quadrature, primal convergence, discrete nonlinear program, multiplier normalization, covector map, boundary and path constraints, regularity, and convergence mode are explicit. Conceptual optimal-control theorem only; safety-critical trajectory and controller design require validated dynamics, numerical verification, constraint margins, and qualified engineering review.

Clarity

The abstraction clarifies a crowded vocabulary by making the continuous control problem, discretization and quadrature, primal convergence, discrete nonlinear program, multiplier normalization, covector map, boundary and path constraints, regularity, and convergence mode are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Covector mapping principle can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Covector mapping principle. Covector mapping principle compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed computational optimal control carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the continuous control problem, discretization and quadrature, primal convergence, discrete nonlinear program, multiplier normalization, covector map, boundary and path constraints, regularity, and convergence mode are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of computational optimal control because they reuse the typed computational optimal control carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, A primal discretization maps continuous trajectories and controls to nodes; differentiation produces discrete multipliers, and a declared covector transformation and convergence theorem map them to approximations of continuous adjoints., and type the carrier, state every parameter and convention in the definition, test that the continuous control problem, discretization and quadrature, primal convergence, discrete nonlinear program, multiplier normalization, covector map, boundary and path constraints, regularity, and convergence mode are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Covector mapping principle Domain-specific

Parents (1) — more general patterns this builds on

• Covector mapping principle is a kind of Equivalence-Preserving Rewriting Prime

  The proposed strict upward parent is prime:equivalence_preserving_rewriting.

Hierarchy paths (2) — routes to 2 parentless roots

• Covector mapping principle → Equivalence-Preserving Rewriting → Transformation → Function (Mapping)

Show alternative path (1)

Neighborhood in Abstraction Space

Covector mapping principle sits in a moderately populated region (44^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Feedback Control & Dynamical Systems (29 abstractions)

Nearest neighbors

• Optimal control — 0.90
• Trajectory optimization — 0.90
• Separation principle — 0.90
• Motion planning — 0.89
• Semi-infinite programming — 0.88

Computed from structural-signature embeddings · 2026-09-08