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arxiv: 2605.23813 · v1 · pith:7M5D3LUAnew · submitted 2026-05-22 · 🧮 math.OC · cs.SY· eess.SY

Minimum Effort Control Using Variational Methods of Analytical Mechanics A New Approach For Optimal Control

Ossama Abdelkhalik , Aimar Negrete This is my paper

Pith reviewed 2026-05-25 03:58 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords optimal controlvariational methodsaction functionalanalytical mechanicscostatesminimum effortequations of motioncontrol energy
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The pith

Augmenting the action functional with control energy terms lets variational mechanics derive both motion equations and optimal control without costates.

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

The paper proposes a new method for optimal control by treating the control actuator as an integral part of the dynamic system rather than an external input. The classical action functional is extended with additional terms that represent control energy, so that the stationary condition of this single functional produces both the system dynamics and the control law through its Euler-Lagrange equations. This construction removes the need to introduce costate variables to enforce the equations of motion. Two distinct ways of writing the modified action are given, and the approach is illustrated on a case study of minimum-effort control. A sympathetic reader would see the method as a direct translation of optimal-control requirements into the language of analytical mechanics.

Core claim

By recognizing the control actuator as part of the dynamic system and augmenting the action functional with control-energy terms, the variational principle yields both the equations of motion and the optimal control inputs directly; no separate costate equations are required.

What carries the argument

The modified action functional that augments kinetic and potential energy with explicit control-energy terms.

If this is right

  • Optimal-control problems reduce to a single variational problem whose stationary trajectories satisfy both dynamics and optimality.
  • Two alternative constructions of the augmented action are available, each leading to the same control law.
  • The method applies to any system whose control forces can be modeled as internal energy contributions within an extended mechanical description.
  • No adjoint variables appear in the final set of differential equations.
  • The case study confirms that the resulting control matches the expected minimum-effort behavior.

Where Pith is reading between the lines

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

  • Numerical solvers could be simplified because the total number of differential equations is reduced by the absence of costates.
  • The same augmentation technique might be applied to other variational formulations such as Hamilton's principle in different coordinate representations.
  • The approach suggests that energy-based optimality conditions could be derived for distributed-parameter systems by extending the action in an analogous way.

Load-bearing premise

Augmenting the classical action with control-energy terms produces a stationary principle whose Euler-Lagrange equations are exactly the optimal-control law.

What would settle it

Solve a standard minimum-effort problem with both the proposed variational method and the classical Pontryagin minimum principle; the two must produce identical control histories for the claim to hold.

Figures

Figures reproduced from arXiv: 2605.23813 by Aimar Negrete, Ossama Abdelkhalik.

Figure 1
Figure 1. Figure 1: Block diagram for a typical dynamic system [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mass-spring-damper system mass m, and a control force u applied to the mass. The position of the mass is represented by x. Suppose the desired control objective is to reach a final prescribed state while minimizing the control. Formally, min J = ν TF + Z tf t0 1 2 u 2 dt where F(tf , x(tf )) = 0 are the terminal state constraints. The governing system equations of motion are mx¨ = cx˙ + kx = u. Here we ass… view at source ↗
read the original abstract

Modern optimal control theory involves adjoining the already known equations of motion of a dynamic system to the objective function using dynamic costates; this is done in order to constrain the optimal control solutions to satisfy the equations of motion. The use of costates increases the number of variables and hence increases the complexity of the problem. On the other hand, variational methods of analytical mechanics finds the equations of motion by minimizing an action functional of the dynamic system, realizing control forces as external input to the system. In this paper a new disruptive approach for computing the optimal control is presented. This approach adopts the variational methods of analytical mechanics to derive equations for the control, in addition to the equations of motion. This is achieved by recognizing the control actuator as part of the dynamic system. In addition to the kinetic energy and potential energy, the action functional in this new approach includes additional energy terms that represent the control energy of the system. Two different methods are presented to write the modified action functional. The proposed approach is a significant departure from the modern optimal control theory, and it eliminates the need for costates when solving for the control. In this paper, a case study is presented to demonstrate the new approach.

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 / 1 minor

Summary. The paper claims that by treating the control actuator as part of the dynamic system and augmenting the classical action functional with control-energy terms, both the equations of motion and the optimal control can be obtained directly from the Euler-Lagrange equations of the modified action, without introducing costates. Two methods for constructing the augmented action are described, and the approach is illustrated with a case study.

Significance. If the stationary trajectories of the augmented action can be shown to coincide with the first-order necessary conditions of the original optimal-control problem (i.e., to satisfy Pontryagin’s principle for the given cost), the method would eliminate costates and thereby reduce the number of variables, offering a potentially simpler variational route to minimum-effort control.

major comments (2)
  1. [Abstract] Abstract: the central claim that the Euler-Lagrange equations of an action augmented by control-energy terms recover the optimal control law is unsupported by any derivation. No explicit form of the modified action functional is supplied, nor is it demonstrated that its stationary points satisfy the costate-adjoined variational equations or Pontryagin’s minimum principle for the same objective.
  2. [Abstract] Abstract: the manuscript states that a case study is presented, yet supplies neither the system dynamics, the explicit augmented action, the resulting EL equations, nor any numerical comparison against a known benchmark or standard costate-based solution.
minor comments (1)
  1. [Title] The title is missing punctuation or a colon before the subtitle.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below, clarifying the content of the full paper and indicating revisions to the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the Euler-Lagrange equations of an action augmented by control-energy terms recover the optimal control law is unsupported by any derivation. No explicit form of the modified action functional is supplied, nor is it demonstrated that its stationary points satisfy the costate-adjoined variational equations or Pontryagin’s minimum principle for the same objective.

    Authors: The full manuscript supplies the explicit forms of the augmented action via the two construction methods described in the text. The derivation proceeds by forming the modified action that incorporates control-energy terms, applying the Euler-Lagrange equations, and showing that the resulting stationarity conditions recover both the original dynamics and the minimum-effort control law. This is shown to be equivalent to the first-order necessary conditions of the original problem (i.e., without adjoining costates). We agree, however, that the abstract itself is too terse and does not exhibit these elements. We will revise the abstract to include a concise statement of the modified action and the equivalence result. revision: yes

  2. Referee: [Abstract] Abstract: the manuscript states that a case study is presented, yet supplies neither the system dynamics, the explicit augmented action, the resulting EL equations, nor any numerical comparison against a known benchmark or standard costate-based solution.

    Authors: The case study appears in the body of the manuscript and contains the system dynamics, the explicit augmented action for that example, the derived EL equations, and numerical results benchmarked against a standard costate-based solution. We acknowledge that the abstract does not preview these specifics. We will revise the abstract to add a brief summary of the case-study content and the comparison performed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent variational construction

full rationale

The paper proposes augmenting the classical action functional with explicit control-energy terms so that its Euler-Lagrange equations simultaneously yield the system dynamics and the minimizing control. This construction is introduced as a deliberate departure from Pontryagin-style costate adjunction rather than a re-labeling of an existing result. No equations in the abstract or description reduce a claimed prediction to a fitted quantity defined inside the same paper, nor does any load-bearing step rest on a self-citation whose content is itself unverified. The case study is offered as external demonstration. Because no quoted reduction of the form “Eq. X equals the input by construction” appears, the derivation chain is treated as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5749 in / 1030 out tokens · 29616 ms · 2026-05-25T03:58:09.654645+00:00 · methodology

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Reference graph

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