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arxiv: 2511.07711 · v3 · pith:4VVUE3LKnew · submitted 2025-11-11 · 🧮 math.OC · cs.SY· eess.SY

Geometric Conditions for Lossless Convexification in Linear Optimal Control with Discrete-Valued Inputs: Real-Time Implementation for Spacecraft Rendezvous

Felipe Arenas-Uribe , Hasan A. Poonawala , Jesse B. Hoagg This is my paper

Pith reviewed 2026-05-22 12:28 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords lossless convexificationdiscrete-valued inputslinear time-varying systemsoptimal controlspacecraft rendezvousconvex relaxationreal-time implementation
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The pith

Simple geometric conditions on the input set guarantee that the relaxed convex problem yields exact discrete-valued controls for linear systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that optimal control problems for linear time-varying systems with discrete-valued inputs can be solved via convex relaxation when the input set satisfies basic geometric conditions. This matters for real-time aerospace uses because it replaces slow mixed-integer solvers with fast convex ones while still producing valid discrete controls. The authors prove that transforming a Lagrange-form problem to Mayer form via epigraph preserves system normality, and they establish that the geometric conditions force the convex solution to lie at the original discrete points. They demonstrate the method on a spacecraft rendezvous with discrete thrusters in an elliptical orbit and confirm through Monte Carlo runs that the outputs are exactly discrete and the run times suit on-board use.

Core claim

We develop a lossless convexification framework for the optimal control of linear time-varying systems with discrete-valued inputs. We extend existing results by showing that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation. We establish that under simple geometric conditions on the input set, the solution to the relaxed convex problem strictly satisfies the original non-convex input constraints.

What carries the argument

Lossless convexification via epigraph transformation from Lagrange to Mayer form, which preserves normality and, under geometric conditions on the input set, forces the convex solution to satisfy the discrete constraints exactly.

If this is right

  • Real-time computation of optimal discrete-valued controls becomes possible without mixed-integer optimization.
  • The method produces exactly discrete control inputs for spacecraft rendezvous maneuvers using reaction thrusters in elliptical orbits.
  • Monte Carlo simulations confirm consistent satisfaction of the discrete constraints with run times compatible with on-board safety-critical use.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric conditions could be checked for other linear control tasks that use quantized actuators.
  • If the geometric conditions can be maintained under actuator failures, the framework might support fault-tolerant discrete control.

Load-bearing premise

The discrete input set must satisfy the stated geometric conditions, such as the origin lying in the interior of the convex hull of the input points or satisfying a related normality property.

What would settle it

A linear system and input set obeying the geometric conditions whose convex-relaxation solution uses a non-discrete input value would show the claim is false.

Figures

Figures reproduced from arXiv: 2511.07711 by Felipe Arenas-Uribe, Hasan A. Poonawala, Jesse B. Hoagg.

Figure 1
Figure 1. Figure 1: Lossless convexification enables real-time, fuel-optimal trajectory [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the sequence of relaxations used to convexify the discrete input set U. The convex relaxation U˜ takes the form of a cross-polytope in R 2 , consistent with fact 1. By applying the epigraph transformation, we obtain the augmented set Ue, whose convex relaxation U˜ e is a compact convex polytope as established in fact 3. °1.0 °0.5 0.0 0.5 1.0 u1 °1.0 °0.5 0.0 0.5 u 1.0 2 0.00 0.25 0.50 0.75 1.00… view at source ↗
Figure 4
Figure 4. Figure 4: System states during rendezvous, showing convergence to the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Histogram of solver times from the Monte Carlo simulation. All [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of grid size on solver time and control discreteness. Increasing [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Optimal control problems with discrete-valued inputs are inherently challenging due to their mixed-integer nature, rendering them generally intractable for real-time, safety-critical aerospace applications. Lossless convexification offers a powerful alternative by reformulating these mixed-integer programs into computationally efficient convex programs. This paper develops a lossless convexification framework for the optimal control of linear time-varying systems with discrete-valued inputs. We extend existing theoretical results by demonstrating that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation. Furthermore, we establish that under simple geometric conditions on the input set, the solution to the relaxed convex problem strictly satisfies the original non-convex input constraints. This framework enables the real-time computation of optimal discrete-valued controls without resorting to mixed-integer optimization. The proposed algorithm is validated on a spacecraft rendezvous maneuver utilizing discrete-valued reaction thrusters in an elliptical orbit. Numerical results from Monte Carlo simulations confirm that the algorithm consistently yields exact discrete-valued control inputs with computational timelines compatible with safety-critical, on-board applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a lossless convexification framework for optimal control of linear time-varying systems with discrete-valued inputs. It claims to extend prior results by proving that system normality is preserved under an epigraph reformulation from Lagrange to Mayer form, and that simple geometric conditions on the input set U (such as the origin lying in the interior of conv(U)) ensure that any optimal solution of the relaxed convex program over conv(U) is automatically a vertex of the original discrete set U. The framework is demonstrated on a spacecraft rendezvous problem with discrete reaction thrusters in elliptical orbit, with Monte Carlo simulations confirming exact discrete-valued controls and real-time feasibility.

Significance. If the normality-preservation result and geometric conditions hold rigorously, the work provides a practical route to real-time optimal discrete-input control for safety-critical aerospace systems without mixed-integer solvers. The Monte Carlo validation on the rendezvous example and emphasis on computational timelines are concrete strengths supporting applicability.

major comments (2)
  1. [Normality preservation and geometric conditions (around the epigraph reformulation and main theorem)] The central claim that geometric conditions on U suffice to guarantee vertex solutions after the epigraph lift requires explicit handling of the auxiliary-state adjoint coupling. For time-varying A(t), B(t), the lifted switching function can lie in the relative interior of a face of conv(U) on a positive-measure interval even when 0 is in the interior of conv(U); no additional rank condition or counter-example exclusion on the lifted system is apparent to close this gap.
  2. [Theoretical extension on normality preservation] The proof that normality is preserved under the Mayer-form lift must be inspected for hidden assumptions on the switching function; the abstract states the extension but the coupling introduced by the auxiliary state in the adjoint equations for LTV dynamics risks creating intervals where the effective switching function is orthogonal to a non-vertex face.
minor comments (2)
  1. [Abstract] The abstract refers to 'simple geometric conditions' without stating them explicitly; a concise statement of the precise conditions (e.g., interior-point or normality-related properties) would improve readability.
  2. [Numerical results] Monte Carlo results confirm exact discrete inputs, but reporting the fraction of cases where the relaxed solution lands exactly on vertices versus near-vertices would strengthen the empirical support.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising these important points regarding the theoretical foundations. We address each major comment below and will incorporate additional clarifications in the revised version to make the arguments more explicit.

read point-by-point responses

  1. Referee: [Normality preservation and geometric conditions (around the epigraph reformulation and main theorem)] The central claim that geometric conditions on U suffice to guarantee vertex solutions after the epigraph lift requires explicit handling of the auxiliary-state adjoint coupling. For time-varying A(t), B(t), the lifted switching function can lie in the relative interior of a face of conv(U) on a positive-measure interval even when 0 is in the interior of conv(U); no additional rank condition or counter-example exclusion on the lifted system is apparent to close this gap.

    Authors: We appreciate the referee highlighting the need for explicit treatment of the auxiliary-state coupling. In the proof of the main theorem (Section 3), the epigraph reformulation to Mayer form yields an auxiliary state whose adjoint is constant. This constant term enters the switching function as an additive vector that is independent of time. Under the geometric condition that the origin lies in the interior of conv(U), any interval on which the effective switching function lies in the relative interior of a face would imply that the original (unlifted) switching function violates the maximum principle, contradicting optimality. We will add a dedicated remark and a short appendix paragraph explicitly deriving the form of the lifted switching function for LTV dynamics to address this concern directly. revision: partial

  2. Referee: [Theoretical extension on normality preservation] The proof that normality is preserved under the Mayer-form lift must be inspected for hidden assumptions on the switching function; the abstract states the extension but the coupling introduced by the auxiliary state in the adjoint equations for LTV dynamics risks creating intervals where the effective switching function is orthogonal to a non-vertex face.

    Authors: We thank the referee for this observation on the proof structure. The normality-preservation argument (Lemma 3.2) proceeds by showing that any non-normal extremal in the lifted system projects to a non-normal extremal in the original Lagrange-form problem, which is ruled out by the standing normality assumption on the plant. The auxiliary adjoint contributes only a fixed offset to the switching function; because this offset is constant and the geometric condition prevents the switching function from remaining orthogonal to a non-vertex face over a positive-measure set, no additional rank condition is required. We agree that the current exposition is concise and will expand the proof with an intermediate step that isolates the auxiliary contribution for the time-varying case. revision: partial

Circularity Check

0 steps flagged

New geometric conditions and normality-preservation proof are independent derivations

full rationale

The paper introduces original geometric conditions on the discrete input set U and proves that normality is preserved under the epigraph lift from Lagrange to Mayer form for LTV systems. These steps are presented as new results rather than reductions of prior fitted quantities or self-cited theorems. Background citations to lossless convexification literature support context but do not carry the load-bearing claims; the central guarantee that relaxed optima are vertices of U follows from the stated geometric assumptions and the normality argument, which are developed within the manuscript. No self-definitional loops, parameter-fitting masquerading as prediction, or ansatz smuggling via self-citation appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework relies on standard convexity and optimal-control assumptions plus the new geometric conditions on the discrete input set; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption The system is linear time-varying and the input set is a finite discrete set whose convex hull satisfies the stated geometric conditions.
    Invoked to guarantee that the relaxed solution lies at a vertex of the input set.

pith-pipeline@v0.9.0 · 5730 in / 1160 out tokens · 35909 ms · 2026-05-22T12:28:16.243629+00:00 · methodology

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Reference graph

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