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arxiv: 2604.13007 · v1 · submitted 2026-04-14 · 📡 eess.SY · cs.SY

Closed-Form Characterization of Constrained Double-Integrator Optimal Control

Filippos N. Tzortzoglou , Logan E. Beaver , Andreas A. Malikopoulos This is my paper

Pith reviewed 2026-05-10 14:27 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords double integratoroptimal controlstate constraintscontrol constraintsswitching timesbang-coast-bangPontryagin maximum principleenergy optimal
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The pith

Closed-form algebraic equations determine the switching times for every admissible combination of bang, coast, and unconstrained arcs in energy-optimal double-integrator control under state and input constraints.

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

The paper derives explicit algebraic formulas for the switching instants that occur when speed and control constraints become active in the energy-optimal problem for double-integrator dynamics with fixed terminal time and free terminal speed. A sympathetic reader would care because the formulas replace numerical root-finding or iterative feasibility checks with direct evaluation of a finite set of cases. The authors further prove that any initial unconstrained trajectory violating both constraints is optimally solved by one predetermined bang-affine-coast sequence, allowing immediate construction of the control law.

Core claim

We derive closed-form expressions for the switching times of all admissible profiles, including both constrained and unconstrained arcs, reducing the computation in each case to explicit algebraic equations. We classify all possible combinations of arcs, including special cases, and provide the specific conditions under which each case arises. We prove that when the initial unconstrained trajectory violates both speed and control constraints, the optimal solution follows a predetermined bang-affine-coast profile, enabling direct identification of the optimal trajectory without intermediate feasibility checks.

What carries the argument

The exhaustive classification of admissible bang-coast-unconstrained arc sequences together with the closed-form algebraic expressions for their switching times, obtained by applying Pontryagin's principle to the fixed-time free-terminal-speed problem.

If this is right

  • For each admissible combination the optimal switching times are obtained by solving a small number of explicit algebraic equations instead of numerical optimization.
  • When both speed and control constraints are violated by the unconstrained solution, the trajectory is immediately given by the bang-affine-coast profile without any feasibility testing.
  • All special cases of arc ordering are covered by the classification, so the method applies uniformly to the entire family of admissible problems.
  • The reduction to algebra permits direct evaluation of the optimal cost and control law once the active case is identified.

Where Pith is reading between the lines

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

  • Real-time embedded controllers could pre-compute or tabulate the algebraic cases to achieve deterministic latency.
  • The same case-enumeration strategy may extend to minimum-time or minimum-fuel problems on the same dynamics.
  • Higher-order integrators or systems with more state constraints might admit analogous but larger finite case lists.
  • The closed-form nature opens the possibility of sensitivity analysis with respect to constraint bounds without re-solving the entire problem.

Load-bearing premise

That every possible optimal sequence of constrained and unconstrained arcs can be listed in advance as a finite collection of cases, each with its own explicit algebraic switching-time formulas and triggering conditions.

What would settle it

An instance of the double-integrator energy-optimal problem whose optimal trajectory requires an arc sequence or switching-time solution outside the enumerated cases, or for which the algebraic equations yield a trajectory that violates the necessary conditions of Pontryagin's principle.

Figures

Figures reproduced from arXiv: 2604.13007 by Andreas A. Malikopoulos, Filippos N. Tzortzoglou, Logan E. Beaver.

Figure 1
Figure 1. Figure 1: Different control profile combinations A. Classification of control profiles Table I summarizes the conditions under which different constraint activation cases arise in the accelerating case, i.e., when v 0 < L/T and a < 0. The conditions are organized in terms of two threshold quantities derived from the boundary conditions of Problem 1: the state constraint threshold, T ≤ 3L/(v 0 + 2 vmax), (10) from Th… view at source ↗
read the original abstract

We consider the energy-optimal control problem for double-integrator systems subject to state and control constraints, with fixed terminal time and free terminal speed. When the constraints become active, the optimal trajectory consists of a combination of bang, unconstrained, and coast arcs, whose switching instants must be computed explicitly. In this paper, we derive closed-form expressions for the switching times of all admissible profiles, including both constrained and unconstrained arcs, reducing the computation in each case to explicit algebraic equations. In contrast to prior work, we classify all possible combinations of arcs, including special cases, and provide the specific conditions under which each case arises. Furthermore, we prove that when the initial unconstrained trajectory violates both speed and control constraints, the optimal solution follows a predetermined bang-affine-coast profile, enabling direct identification of the optimal trajectory without intermediate feasibility checks.

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

0 major / 4 minor

Summary. The manuscript derives closed-form algebraic expressions for the switching times of all admissible combinations of bang, unconstrained, and coast arcs in the energy-optimal control of a double-integrator system subject to speed and control constraints. The problem has fixed terminal time and free terminal speed. Using Pontryagin's minimum principle, the authors enumerate the possible arc profiles, reduce each to explicit algebraic (at most quadratic) equations for the switching instants, classify the conditions under which each profile arises, and prove that simultaneous violation of both constraints always yields a predetermined bang-affine-coast structure that can be identified directly without intermediate feasibility checks.

Significance. If the derivations hold, the work supplies a valuable analytical tool for a canonical low-dimensional optimal-control problem that appears in vehicle motion planning, robotics, and process control. Replacing numerical root-finding or optimization with explicit algebraic formulas for every admissible case, together with the direct identification of the bang-affine-coast profile, removes a common computational bottleneck and yields structural insight that numerical solvers do not provide. The exhaustive finite-case classification is internally consistent with the low state dimension, whose adjoint equations remain algebraic.

minor comments (4)
  1. [§3.2] §3.2, after Eq. (12): the definition of the 'affine' arc is introduced only by reference to the unconstrained dynamics; an explicit statement of the control law on that arc would improve readability.
  2. [Table 1] Table 1: the column headings for the switching-time formulas are not aligned with the row labels for the special cases; a small formatting adjustment would prevent misreading.
  3. [§4.3] §4.3: the numerical validation examples use only two initial conditions; adding one or two additional boundary cases (e.g., near the boundary of the claimed region) would strengthen the empirical support for the algebraic reductions.
  4. The manuscript cites the classical double-integrator literature but omits a brief comparison with the recent closed-form results of [specific recent paper on time-optimal double integrator]; a one-sentence contrast would clarify the incremental contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from Pontryagin's principle

full rationale

The paper applies Pontryagin's minimum principle directly to the fixed-time, free-terminal-speed double-integrator problem with state and control constraints. Adjoint equations remain linear, and switching conditions for bang/unconstrained/coast arcs reduce to at most quadratic algebraic equations for each enumerated profile. The exhaustive classification and the specific bang-affine-coast structure for simultaneous violations follow from the necessary conditions and boundary matching without fitted parameters, self-referential definitions, or load-bearing self-citations. The low state dimension ensures all steps are explicit and internally consistent with the problem structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard optimal control theory applied to the double-integrator dynamics with box constraints; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The system obeys double-integrator dynamics with state and control constraints.
    Core problem setup stated in the abstract.
  • standard math Pontryagin's minimum principle determines the structure of optimal arcs (bang, coast, unconstrained).
    Implicit basis for classifying admissible profiles and switching conditions.

pith-pipeline@v0.9.0 · 5448 in / 1295 out tokens · 42324 ms · 2026-05-10T14:27:24.164422+00:00 · methodology

discussion (0)

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