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Transversality Conditions for Boundary Constraints Defined by Differential Equations

Isaac M Ross

arXiv:2609.04084v1math.OCcs.NAcs.SYeess.SYmath.DGmath.NA

TL;DR

The paper asks how to state transversality conditions when boundary endpoints lie on dynamically defined orbits rather than specified algebraically. It develops conditions for differential boundary constraints using coordinated or uncoordinated clock times and weak adjoint covectors; the resulting conditions include parameter coupling and scalar natural boundary equations.

  • Problem

    Dynamically defined orbits may contain any point on the orbit rather than a specified endpoint, leaving a gap in algebraic boundary formulations and their transversality conditions.

    Source: §I · p. 2

  • Method

    The paper develops transversality conditions using coordinated or uncoordinated clock times and weak adjoint covectors.

    Source: C. Coordinated and Uncoordinated Clock… · p. 10, §V · p. 13

  • Results

    The new conditions couple transversality to optimality conditions for parameters and provide scalar equations for initial and final time-like variables.

    Source: §VI · p. 15

  • Takeaways & Limitations

    The framework defines differential boundary conditions and develops generic transversality conditions beyond algebraically specified endpoints.

    Source: §VIII · p. 24, §IX · p. 25

Abstract

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What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.

I. A Motivating Problem

The paper motivates transversality conditions for endpoints on dynamically generated orbits, which need not have algebraic descriptions. It frames the question within a general optimal-control problem and emphasizes that the goal extends beyond spacecraft trajectory optimization.

I. A Motivating Problem

  • Three-body orbits can be computed but cannot naturally be described by algebraic equations, unlike the conics used for two-body problems.The motivating task is to identify optimal departure and arrival points on these orbits rather than transfer between fixed points.Source: §I · p. 2
  • The motivating problem seeks necessary conditions for optimal departure and arrival points, which may also depend on the initial and final clock times.Source: §I · p. 2
  • The objective is to derive transversality conditions for boundary stipulations beyond the three-body problem and spacecraft trajectory optimization.Source: §I · p. 2

II. A Brief Review of Existing Transversality Conditions

  • The reviewed optimal-control formulation permits a general closed endpoint set S, including fixed points, curves, and nonconvex sets.The problem minimizes a Bolza cost subject to state dynamics and allowable controls.Source: §II · p. 3

A. Mathematical Preliminaries

The preliminaries develop tangent and normal cones for endpoint sets and identify the endpoint quantities used in transversality conditions.

A. Mathematical Preliminaries

  • The tangent cone collects feasible tangent directions, while the normal cone provides the corresponding endpoint normals.The paper uses the Bouligand (contingent) tangent cone and Mordukhovich (limiting) normal cone.Source: A. Mathematical Preliminaries · pp. 4–5
  • At a smooth boundary, the normal cone is perpendicular to the tangent hyperplane; nonsmooth boundaries can have convex or nonconvex normal cones.Source: A. Mathematical Preliminaries · pp. 4–5
  • Normal covectors belong to dual space, and their orthogonality to tangent vectors is expressed by a zero covector-vector pairing.Source: A. Mathematical Preliminaries · p. 5
  • The transversality conditions involve endpoint costates, endpoint values of the lower Hamiltonian, and the normal cone to the endpoint set.The Hamiltonian is minimized over allowable controls to define the lower Hamiltonian.Source: A. Mathematical Preliminaries · p. 5

( Minimize

The review presents standard Hamiltonian and normal-cone transversality conditions, then extends them to optimization parameters and algebraically constrained endpoints.

( Minimize

  • Minimizing the Hamiltonian over allowable controls defines an optimal control and a lower Hamiltonian that no longer depends on the control.Source: ( Minimize · pp. 5–6

B. Transversality Conditions for the Case When p is Not an Optimization Variable

C. Transversality Conditions for the General Case

D. Transversality Conditions When S Is Parameterized by Algebraic Inequalities

III. Foundations of Boundary Conditions Defined by Differential Equations

The section builds endpoint sets from differential-equation flows and side conditions, then distinguishes the local parameter s from the optimal-control clock t. That distinction permits endpoint parameterizations whose local time is either coordinated with or independent of transfer time.

A. A Basic Set Parameterized by a Differential Equation

  • An initial-value problem generates a curve of states indexed by s, but additional side conditions are needed to define a complete endpoint set.The solution passes through the specified initial point; the s-range can also be bounded.Source: A. A Basic Set Parameterized by a… · pp. 8–9

B. A Manifold Parameterized by Differential Equations

  • Starting points in P generate flow curves that form a manifold M, typically containing P, while the candidate endpoints need not lie in P.The points in P are called phantom initial conditions because they generate the flow but do not limit candidate endpoint choices.Source: B. A Manifold Parameterized by… · p. 9

C. Coordinated and Uncoordinated Clock Times

  • A differential-equation parameterization uses local time s, which may be distinct from the optimal-control clock t.For a unit circle, a parameterization using t need not make that parameter the transfer clock; an independent s parameter avoids that identification.Source: C. Coordinated and Uncoordinated Clock… · pp. 9–10
  • Clock times are coordinated when a relation φ(s,t)=0 uniquely links them, and uncoordinated when φ is identically zero.The synchronized case is φ(s,t)=s−t; intercept and rendezvous problems can require coordinated clocks.Source: C. Coordinated and Uncoordinated Clock… · p. 10

D. A Preliminary Mathematical Model for Endpoint Conditions Governed by Differential Equations

The preliminary endpoint model connects local parameter time and transfer clock time while adding side conditions to constrain the resulting sets. The formulated initial and final endpoint sets can use different vector fields, clock relations, and admissible time intervals.

D. A Preliminary Mathematical Model for Endpoint Conditions Governed by Differential Equations

  • The preliminary set model requires side conditions, including constraints on local time and, for uncoordinated clocks, on transfer time.These additions bridge the preliminary model and the endpoint sets defined in the next section.Source: D. A Preliminary Mathematical Model… · pp. 10–11

IV. A Formulation of Endpoint Conditions Given by Differential Equations

  • The initial and final endpoint sets are built from separate differential equations, allowing distinct vector fields and control strategies at each endpoint.A boundary vector field may be generated by an open- or closed-loop control law, not only by dynamics with no control action.Source: §IV · pp. 11–13
  • Each endpoint set can independently coordinate its local s-time with its clock time and impose separate admissible intervals for both.The initial and final clock relations may differ, for example when one endpoint is an uncoordinated manifold and the other is a synchronized rendezvous.Source: §IV · pp. 11–12
  • The formulation constrains the local parameter range and clock-time intervals, with discontinuous boundary vector fields requiring Carathéodory conditions.The clock-time intervals are chosen by the modeler or operator for practical reasons and are assumed not to conflict.Source: §IV · p. 12
  • The boundary parameters must be defined and constrained to construct Sode+, which provides the basis for the new transversality conditions.The parameter constraints are expressed through Cartesian products for the initial and final sets.Source: §IV · pp. 12–13

V. The New Transversality Conditions

The paper introduces weak adjoint covectors to formulate transversality conditions for endpoints defined by differential equations. The conditions are stated for the resulting optimal-control problem under differentiability assumptions on its cost functions.

V. The New Transversality Conditions

  • Unlike a strong adjoint covector’s pointwise constraint, the weak form tests the constraint functionally over trajectories satisfying the differential equation.The strong covector is defined by a pointwise constraint functional equation.Source: §V · p. 13
  • A weak adjoint covector is a constant covector ψ used to form window-indexed covectors that satisfy a weak constraint equation.Testing the constraint over every window enforces the same result as the strong form, using an alternative mathematical framework.Source: §V · p. 13
  • Theorem 4 gives transversality conditions for the differential-boundary problem, with continuously differentiable endpoint and running costs and weak covectors for the boundary equations.The theorem’s conditions also include clock-coordination multipliers.Source: §V · p. 14

VI. Discussion of the Main Result and Development of Some Corollaries

Theorem 4 couples transversality conditions to parameter optimality and supplies scalar boundary conditions for the local time-like variables. When the boundary differential equations vanish, its conditions reduce to the familiar static-boundary results.

VI. Discussion of the Main Result and Development of Some Corollaries

  • Theorem 4 couples transversality conditions to optimality conditions for parameters pa and pb, which Theorem 1 alone cannot support.The weak adjoint covectors can also independently help verify the extremality of a computed solution.Source: §VI · p. 15
  • Equations (42c) and (42d) provide scalar natural boundary conditions for the local time-like variables sa and sb.Each boundary state already satisfies an Nx-dimensional differential equation, leaving only one free local time-like variable at each endpoint.Source: §VI · p. 15

A. Case 1: Vanishing Boundary Differential Equations

  • Vanishing boundary differential equations do not necessarily fix the initial endpoint; the resulting set corresponds to panel (c), unless Pa is a singleton.The local time-like variable s♯a is effectively a dummy variable that does not directly constrain x0 or t0.Source: A. Case 1: Vanishing Boundary… · pp. 15–16
  • With vanishing boundary differential equations, Theorem 4 simplifies to conditions equivalent to those obtained independently using Theorem 2.The paper presents this equivalence as an independent validation of Theorem 4.Source: A. Case 1: Vanishing Boundary… · p. 17
  • In this case, the boundary constraints become static specifications on pa and pb, and the resulting conditions agree with Theorem 2.The discussion derives an initial condition equivalent to the standard Hamiltonian value condition and an initial transversality condition matching Theorem 4.Source: A. Case 1: Vanishing Boundary… · pp. 16–17

B. Case 2: Uncoordinated Boundary Clock Times

For uncoordinated boundary clock times, the conditions can require a weak adjoint covector to be orthogonal to the boundary differential equation’s vector field. This differs from manifold-based orthogonality, and the result extends beyond integrable boundary equations.

B. Case 2: Uncoordinated Boundary Clock Times

  • When boundary clock times are uncoordinated and unconstrained, and costs do not depend on the local times, the weak adjoint covector is orthogonal to the boundary vector field.The conditions apply to the vector fields defining endpoint conditions, not algebraic equations defining the boundary, and weak adjoint covectors are not the endpoint costates.Source: B. Case 2: Uncoordinated Boundary… · p. 17
  • Figure 10’s example states that the initial costate is orthogonal to the vector field at the point (x, y).The figure depicts the vector field with single-headed arrows and the initial costate with double-headed arrows.Source: B. Case 2: Uncoordinated Boundary… · p. 18, Figure 10 · p. 18
  • The vector-field orthogonality condition is equivalent to manifold-based transversality in the integrable circle example, although the multipliers differ dimensionally.Eliminating the endpoint multiplier for the algebraic circle constraint yields the same condition.Source: B. Case 2: Uncoordinated Boundary… · pp. 18–19
  • The result is more general than equivalence to algebraic-manifold conditions: it holds even when boundary differential equations are nonintegrable.The paper identifies boundary conditions in the restricted three-body problem as practical examples.Source: B. Case 2: Uncoordinated Boundary… · p. 19

C. Case 3: Synchronized Boundary Clock Times

For synchronized boundary clock times, the paper derives simplified transversality and Hamiltonian value conditions under specified assumptions. These conditions show that free clock times do not always yield the familiar zero final Hamiltonian.

C. Case 3: Synchronized Boundary Clock Times

  • For synchronized, unconstrained local times absent from endpoint and running costs, Corollary 3 reduces Theorem 4’s transversality and Hamiltonian value conditions.The passage supplies the reduction’s assumptions but not the resulting equations.Source: C. Case 3: Synchronized Boundary Clock… · p. 20
  • When Pa and Pb are singletons, the boundary optimization points are no longer optimization variables, and the conditions simplify.The passage indicates that Corollary 4 considers free clock times, minimum transfer-time normal problems, and problems without endpoint cost.Source: C. Case 3: Synchronized Boundary Clock… · pp. 20–21
  • With free clock times, Corollary 4’s Hamiltonian value conditions do not necessarily give H[tf] = 0.The paper distinguishes these synchronized-time conditions from the familiar zero-Hamiltonian condition for uncoordinated times.Source: C. Case 3: Synchronized Boundary Clock… · p. 21
  • For minimum transfer-time “normal” problems, defined by E(x0, t0, xf, tf) = tf − t0 and ν0 = 1, Corollary 4 does not exhibit the familiar condition that the final lower-Hamiltonian value is −1.Source: C. Case 3: Synchronized Boundary Clock… · p. 21

VII. A Proof of Theorem 4 with Insights on ψ0 and ψf

The proof of Theorem 4 is organized into three lemmas, which also offer insight into how the weak adjoint covectors arise.

VII. A Proof of Theorem 4 with Insights on ψ0 and ψf

  • Three lemmas organize the proof of Theorem 4 and provide insight into how the weak adjoint covectors ψ0 and ψf arise.Source: §VII · p. 21

A. Development of the Three Lemmas Required for a Proof of Theorem 4

The lemmas recast differential boundary conditions in a form usable by the transversality theorems and extend the multiplier treatment to algebraic inequalities combined with abstract sets.

A. Development of the Three Lemmas Required for a Proof of Theorem 4

  • The first lemma establishes an equivalent integral form for the differential boundary constraints, making them usable in the proof of Theorem 4.It rewrites the endpoint sets using the differential equation's integral form.Source: A. Development of the Three Lemmas… · p. 22
  • Lemma 2 gives transversality conditions for boundary sets combining algebraic inequalities with an abstract set, using a multiplier rule.Its proof follows from Theorems 2 and 3; the associated multiplier rule is derived from first principles in cited work.Source: A. Development of the Three Lemmas… · p. 22
  • The generalized derivative of the distance function to the abstract set yields its normal cone under technical conditions.The distance function is nonsmooth, and the technical conditions are outside this paper's scope.Source: A. Development of the Three Lemmas… · p. 22

B. Proof of Theorem 4

Theorem 4's proof applies the multiplier rule to endpoint integral constraints, yielding transversality relations that include their multipliers. These multipliers are weak, rather than strong, adjoint covectors for the boundary differential equations.

B. Proof of Theorem 4

  • The final costate equals the endpoint-Lagrangian derivative with respect to xf plus the multiplier for the final integral constraint.The displayed relation is λ(tf) = ν0 ∂xf E(x0, t0, pa; xf, tf, pb) + ψ̃f.Source: B. Proof of Theorem 4 · p. 23

C. Insights on ψ0 and ψf as Weak Adjoint Covectors for the Boundary Differential Equations

  • The multipliers ψ0 and ψf enforce the initial and final integral constraints, respectively, but are not the usual strong adjoint covectors for differential constraints.The weak covector arises from multiplying the constraint by the indicator function over its relevant interval.Source: C. Insights on ψ0 and ψf as Weak… · pp. 23–24

VIII. Theorem 5: A Practical Extension of the Main Result

Theorem 5 extends Theorem 4 to boundary conditions that combine differential equations with algebraic constraints. The conclusion emphasizes that the resulting generic conditions use weak adjoint covectors and can be used to validate computed extremals.

VIII. Theorem 5: A Practical Extension of the Main Result

  • Theorem 5 extends Theorem 4 to boundary conditions that add algebraic constraints, with the transversality conditions holding after replacing E by the extended endpoint function.The added set must not over-constrain the boundary problem.Source: §VIII · p. 24
  • The extended endpoint function combines the differential-boundary endpoint Lagrangian with the algebraic endpoint constraints.It is formed as ν0 E plus νT e, with ν satisfying the stated complementarity condition.Source: §VIII · p. 24
  • The algebraic endpoint set allows t0 and tf to be constrained jointly, including a flight-time constraint that cannot be imposed by separately constraining those times in the stated three-body setting.The paper notes that a flight-time constraint appears in one problem definition in.Source: §VIII · p. 25

IX. Conclusions

  • The paper's conclusions identify weak adjoint covectors as indispensable and state that the generic transversality conditions can validate computed extremals.The conditions are stated in terms of the vector field defining the boundary differential equations; practical problems may combine algebraic and differential boundary conditions.Source: §IX · p. 25
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