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Indirect Optimization of Multi-Phase Trajectories Involving Arbitrary Discrete Logic

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that arbitrary boolean switching logic in multi-phase trajectory optimization can be smoothed into continuous dynamics, reducing the necessary conditions from a multi-point boundary value problem to a two-point boundary…

desk verdict GRASHS is a clean, useful extension of RASHS for OR logic, but the paper leaves the convergence question open and its validation is partly circular, so the optimality claim is not yet established. read the letter →

arxiv 2412.07960 v1 pith:K433I5DN submitted 2024-12-10 math.OC

Harish Saranathan This is my paper · ORCID
classification math.OC MSC 49K1565L10
keywords trajectoryoptimizationindirectmethodsentrydescentandlandinghybridsystemsmixedintegerprogrammingGRASHSsigmoidsmoothingdisjunctivenormalform
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Multi-phase trajectories, such as a Mars entry, descent, and landing profile, switch between flight segments when conditions on velocity, altitude, or time are met. When those conditions involve OR logic, the standard indirect optimization approach forces the planner to guess which trigger fires first. This paper claims that arbitrary boolean logic can be converted to disjunctive normal form and smoothed, using products of sigmoids for AND and a hyperbolic tangent of the sum for OR, so that the piecewise equations of motion and Lagrangian become continuous and differentiable. The payoff is that the necessary conditions of optimality reduce from a multi-point boundary value problem, whose interior-point conditions are hard to guess and enforce, to a simpler two-point boundary value problem. In two Mars EDL example profiles, the same smoothed formulation recovers both a velocity-triggered and an altitude-triggered parachute deployment, matching the separately solved multi-point solution.

What carries the argument

The central object is Eq. (13), the smoothed switching-function embedding. It converts any boolean segment-activation expression to disjunctive normal form, represents each AND minterm as a product of sigmoids $1/(1+e^{s\,g_{i,j,k}(X,t)})$, and represents the OR of minterms as $\tanh\!\big(\zeta\sum_i \prod_j [\cdot]\big)$. The sigmoid slope $s$ and the hyperbolic-tangent slope $\zeta$ act as homotopy parameters pushed to large values, so the smooth two-point boundary value problem is asserted to approximate the original multi-point problem arbitrarily closely while the interior-point jump conditions are automatically encoded in the dynamics and cost.

What would settle it

Solve a minimal two-mode, two-trigger example, such as switching when $x<0$ or $y<0$, exactly as a multi-point boundary value problem and compare its states and co-state jumps with the GRASHS limit as $s$ and $\zeta$ are increased; if the limits do not match for some initial and boundary data, the central convergence claim fails.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the discrete switching function for a flight segment can be represented exactly as the signum of a sum of products of horizontally flipped unit step functions, one product per minterm of the segment's boolean condition in disjunctive normal form, and then approximated smoothly. Replacing each unit-step factor by a sigmoid $1/(1+e^{s g})$ and the signum by $\tanh(\zeta \cdot)$ yields Eq. (13), a smooth version of $\dot{X}=\sum_k \xi_k f_k$ and $L=\sum_k \xi_k L_k$. Because the smoothed dynamics embed the switching conditions, the interior-point boundary conditions at segment transitions are implicitly satisfied, and the necessary conditions for optimality become a two-point boundary value problem. The paper demonstrates that, in a Mars EDL problem with parachute descent active when $(v<v_P \text{ or } h<h_P)$ and $h\ge h_{PDI}$, the same GRASHS equations reproduce the velocity-triggered solution when $h_P=3.5$ km and the altitude-triggered solution when $h_P=6.5$ km, without any prior knowledge of which trigger fires first.

Load-bearing premise

The load-bearing premise is that letting the smoothing slopes $s$ and $\zeta$ grow without bound makes the smoothed two-point problem's solution converge to the solution of the original piecewise multi-point problem; the paper asserts this via homotopy but supplies no theorem or error bound.

Editorial extensions

If this is right

  • The same GRASHS equations apply to any boolean combination of predicates across flight segments, including NOT operations, which are handled by replacing the predicate with its sign-flipped form.
  • Interior-point co-state jumps, which in the multi-point formulation must be explicitly computed through Lagrange multipliers, are produced automatically by the smoothed equations of motion and Lagrangian.
  • For EDL design, a planner no longer needs trial-and-error analysis of trigger thresholds to guarantee that every flight segment activates.
  • The GRASHS solution can serve as a high-quality initial guess for the exact multi-point boundary value problem, making the hard solve tractable.
  • The approach avoids the exponential growth in the number of trajectory optimizations that results from enumerating which OR-trigger fires first.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the convergence of the smoothed problem to the discontinuous problem as $s,\zeta\to\infty$ were proven with an error bound, the same embedding could likely be extended to non-autonomous switched systems where the discrete mode is itself a control decision.
  • A natural testable extension is to study simple two-mode examples with overlapping OR triggers and compare the smoothed limit with the exact multi-point solution for all initial and boundary data, checking whether the limit always selects a physically valid mode sequence.
  • The paper's reliance on manual homotopy suggests that adaptive or stabilized continuation schemes could replace the hand-tuned homotopy steps and broaden the method's reliability.
  • Because the smoothing slopes act as continuation parameters, the method could also be used as a mechanism for homotopy between different trigger logics within one optimization framework.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a Generalized Relaxed Autonomously Switched Hybrid System (GRASHS) approach for indirect multi-phase trajectory optimization when the active-segment logic is an arbitrary Boolean expression over predicates on states and time. The method converts each segment's Boolean condition to disjunctive normal form, smooths each predicate with a sigmoid, represents the OR of minterms as the hyperbolic tangent of their sum, and embeds the smoothed switching weights into the equations of motion and Lagrangian. This reduces the necessary conditions of optimality from a multi-point boundary value problem (MPBVP) to a two-point boundary value problem (TPBVP). The approach is demonstrated on a Mars entry, descent, and landing problem with a parachute-deployment condition that combines AND and OR logic, for two mission profiles in which the trigger is either velocity-first or altitude-first. The GRASHS solutions are compared with RASHS and MPBVP solutions that are built using the trigger information obtained from the GRASHS solution.

Significance. If the central claim is correct, GRASHS would fill a genuine gap in indirect optimization of autonomously switched systems, extending RASHS from AND-only to arbitrary Boolean logic without requiring a priori knowledge of the active trigger. The construction in Section 2 is mathematically clean: DNF conversion, sigmoid smoothing of predicates, and tanh saturation of the OR are all well-defined and the resulting equations are smooth. The paper also provides a demanding aerospace test case with two distinct trigger modes and reproduces the expected co-state jump pattern in the numerical results. However, the work's significance is strongly tempered by two gaps: there is no proof or quantitative error bound for convergence of the smoothed optimal solution to the original OR-logic optimum, and the numerical validation is circular because the comparison baselines are derived from the GRASHS solution itself. As a result, the paper establishes that the smoothed problem is solvable and self-consistent, but it does not yet establish that the GRASHS solution is optimal for the original arbitrary-logic problem.

major comments (3)
  1. [Section 2, after Eq. (13)] The central assertion that increasing s and zeta "arbitrarily large values" makes the TPBVP "an arbitrarily close approximation of the MPBVP" is stated without proof. No theorem or error bound is given for the convergence of the smoothed optimal controls, trajectories, or co-states to the solution of the original OR-logic problem. In particular, the paper does not show that the interior-point jump conditions of Eq. (19) are recovered in the limit, nor that the sequence of minimizers of the smoothed problems converges to a minimizer of the original problem. This is load-bearing because the claimed reduction from MPBVP to TPBVP for arbitrary discrete logic rests entirely on this convergence. The authors should provide a rigorous convergence analysis, an epi-convergence argument, or a carefully constructed numerical convergence study that explicitly demonstrates the approach of co-state jumps and mode sequences to the original problem as s and zeta grow.
  2. [Sections 5.2 and 5.3] The validation is circular. The RASHS and MPBVP comparison solutions are constructed by removing the OR-logic trigger that was identified from the GRASHS solution; the paper states this explicitly when it says the interior-point boundary conditions "will be obtained from the solution of the GRASHS approach" (Section 3.1) and when it removes hP or vP based on that knowledge (Sections 5.2 and 5.3). Consequently, agreement between GRASHS and these baselines only demonstrates consistency with a mode sequence that GRASHS itself selected; it cannot detect whether GRASHS picked the wrong trigger or whether the solution satisfies the true OR-logic optimality conditions. The original MPBVP with the full OR-logic interior-point conditions is never formulated or solved. The examples therefore do not independently confirm the central claim. A non-circular validation would compare against a direct method (e.g., SCP with state-triggered constraints), a combinatorial search over all feasible trigger orders, or an MPBVP that includes the genuinely disjunctive switching conditions.
  3. [Section 2, Eq. (13)] The smoothed dynamics in Eq. (13) do not form a partition of unity over the m flight segments. For finite s and zeta, the sum of the smoothed segment weights can be greater than or less than 1 in transition regions, because the weights are independent sigmoid products and tanh-saturated sums rather than normalized convex weights. This means the smoothed equations of motion are not a convex combination of the segment dynamics, and the implied co-state dynamics may not correspond to any true switching system even in the limit of large slopes. The paper does not address this non-unit partition or quantify its effect on the claimed implicit interior-point conditions. The authors should either prove that the error vanishes as s and zeta tend to infinity in the appropriate sense or reformulate the smoothing so that the weights sum to one identically.
minor comments (5)
  1. [Eq. (13)] The product index in the smoothed equations uses "gi,k" instead of "gi,j,k"; the subscript j is missing.
  2. [General] The paper contains several typographical inconsistencies, including "GRASH" in Section 5.2 where "GRASHS" is meant, and reference entries such as "665, 683" with a comma instead of a range dash.
  3. [Section 5.1] The notation tP DI in Eq. (27) is used without a clear definition of the subscript spacing; it should be typeset consistently as tPDI or t_PDI throughout.
  4. [References] Several reference entries have formatting issues, including missing spaces after commas and inconsistent page ranges, which would need to be corrected for journal production.
  5. [Section 3.2] The homotopy description would benefit from a figure or pseudocode that clarifies the five steps and the exact scheduling of s, zeta, and the boundary-condition continuation, since this is the practical means by which the claimed convergence is achieved.

Circularity Check

1 steps flagged · score 6.0 of 10

Core GRASHS derivation is not circular, but the numerical validation is: the RASHS and MPBVP baselines are built from the GRASHS solution's own trigger decision, so agreement with them cannot independently confirm the OR-logic optimum.

  1. fitted input called prediction [Sec. 5.2 (Mission Profile 1; repeated in Sec. 5.3), with the comparison construction specified in Sec. 3.1]
    "With the knowledge provided by the GRASHS solution that the parachute deployment was triggered by v < vP , the trajectory is solved again using the RASHS approach by removing hP from the conditions listed in Table 4, thereby eliminating the OR logic in Segment 2. ... For comparison, the MPBVP from Eq. (19) is also solved by implementing vP and ignoring hP in the interior-point boundary conditions using the apriori knowledge gained from the GRASHS solution that the vehicle first decelerates to vP before descending to hP ."

    The comparison solutions are constructed only after reading the active OR minterm out of the GRASHS solution: the OR logic is eliminated by hard-coding whichever trigger GRASHS selected. Agreement between GRASHS and these baselines is therefore agreement with a problem that was built from GRASHS's own mode sequence. It cannot independently establish that GRASHS found the optimum of the original OR-logic problem, because the true OR-logic MPBVP is explicitly declared beyond scope in Sec. 3.1 and is never solved. The consistency shown is essentially consistency of the smoothed solution with the simplified problem it itself selected, so the claimed validation of the OR-logic prediction is circular.

full rationale

The central derivation in Sec. 2 is self-contained: Eq. (13) is constructed by converting arbitrary boolean logic to DNF, representing minterms as products of sigmoid-smoothed predicates, and OR as a tanh-smoothed signum of the sum. This is a smoothing ansatz, not a derivation that assumes the conclusion, and the reduction from MPBVP to TPBVP follows from the smoothness of the construction rather than from the comparison examples. The author's prior RASHS work is cited for the sigmoid approximation in Eq. (11), but that approximation is a standard mathematical fact and is not the load-bearing source of the GRASHS claim. The serious circularity is in the validation: the RASHS and MPBVP benchmark trajectories are not independent solutions of the original OR-logic problem; they are generated after eliminating the OR logic using the trigger information obtained from the GRASHS solution. Consequently, the observed agreement demonstrates only that GRASHS is consistent with the particular simplified problem it selected, not that it converges to the true optimum of the arbitrary-logic MPBVP. The paper also asserts without proof that increasing s and zeta makes the TPBVP arbitrarily close to the MPBVP; that is a correctness gap rather than a circularity, but it compounds the validation problem because no independent error bound or OR-logic MPBVP solution is provided. Overall, the derivation has independent content, but the numerical demonstration of the central claim is partially circular, giving a score of 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces smoothing parameters and homotopy choices rather than physical free parameters, and relies on unproved convergence and partition-of-unity assumptions. No new physical entities are postulated.

free parameters (4)
  • sigmoid slope s = homotoped to 40,000
    Smoothing slope for predicates; chosen by hand and increased via homotopy, with no convergence criterion tying the final value to solution accuracy.
  • hyperbolic tangent slope zeta = homotoped to 40,000
    Smoothing slope for the OR saturation; chosen by hand and increased via homotopy alongside s.
  • cost weights K1, K2, K3 = 1, 1, 1e-5
    Chosen to balance the heat-load terms and the thrust term in the demonstration; standard but hand-picked for this problem.
  • homotopy schedule = five steps, 500 or 1000 iterations per step
    The number of steps, iteration counts, and parameter variations are chosen by trial and error, as acknowledged in Sec. 3.2; these affect convergence but are not part of the method itself.
assumptions (4)
  • standard math Every boolean expression can be converted to disjunctive normal form.
    Used in Sec. 2 to represent arbitrary logic as a sum of minterms; a standard result cited to Pahl and Damrath.
  • ad hoc to paper As s and zeta tend to infinity, the sigmoid and tanh approximations converge pointwise to step and signum functions, and the smoothed optimal control solutions converge to the original optimal solution.
    The paper asserts that the TPBVP becomes an arbitrarily close approximation of the MPBVP via homotopy (Sec. 2), but provides no proof of convergence of the optimal solutions and no error bound.
  • ad hoc to paper The interior-point boundary conditions of the original MPBVP are implicitly satisfied by the smoothed equations of motion and Lagrangian.
    Stated in Secs. 2 and 3.1; the co-state jumps are observed numerically, but not derived from the optimality conditions of the smoothed problem.
  • ad hoc to paper The smoothed dynamics with non-unit partition of unity still give a valid approximation of the piecewise dynamics in the transition regions.
    The sigmoid and tanh coefficients for adjacent segments do not provably sum to one during finite-smoothing transitions, which can introduce spurious blended dynamics; the paper does not analyze this error.

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Cite this review

Pith. "Pith review of Indirect Optimization of Multi-Phase Trajectories Involving Arbitrary Discrete Logic." pith.science (2026). https://pith.science/paper/K433I5DN

@misc{pith2026241207960,
  author       = {Pith},
  title        = {Pith review of: Indirect Optimization of Multi-Phase Trajectories Involving Arbitrary Discrete Logic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K433I5DN}},
  note         = {Machine review of arXiv:2412.07960}
}
read the original abstract

Multi-phase trajectories of aerospace vehicle systems involve multiple flight segments whose transitions may be triggered by boolean logic in continuous state variables, control and time. When the boolean logic is represented using only states and/or time, such systems are termed autonomously switched hybrid systems. The relaxed autonomously switched hybrid system approach (RASHS) was previously introduced to simplify the trajectory optimization process of such systems in the indirect framework when the boolean logic is solely represented using AND operations. This investigation enables cases involving arbitrary discrete logic. The new approach is termed the Generalized Relaxed Autonomously Switched Hybrid System (GRASHS) approach. Similar to the RASHS approach, the outcome of the GRASHS approach is the transformation of the necessary conditions of optimality from a multi-point boundary value problem to a two-point boundary value problem, which is simpler to handle. This is accomplished by converting the arbitrary boolean logic to the disjunctive normal form and applying smoothing using sigmoid and hyperbolic tangent functions. The GRASHS approach is demonstrated by optimizing a Mars entry, descent, and landing trajectory, where the parachute descent segment is active when the velocity is below the parachute deployment velocity or the altitude is below the parachute deployment altitude, and the altitude is above the powered descent initiation altitude. This set of conditions represents a combination of AND and OR logic. The previously introduced RASHS approach is not designed to handle such problems. The proposed GRASHS approach aims to fill this gap.

Figures

Figures reproduced from arXiv: 2412.07960 by the authors.

Figure 1
Figure 1. Approximation of unit step function using sigmoid function. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Approximation of signum function using hyperbolic tangent function. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Relationship between coordinate frames. Adopted from Saranathan and Grant [2018a]. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Illustration of the mission architecture. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Altitude vs. atmospheric-relative velocity for GRASHS solution for mission profile 1, zoomed in. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Comparison of trajectory and control for mission profile 1. [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Propellant consumption vs. time for mission profile 1. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Comparison of co-state history for mission profile 1. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Altitude vs. atmospheric-relative velocity for GRASHS solution for mission profile 2, zoomed in. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Comparison of trajectory and control for mission profile 2. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Propellant consumption vs. time for mission profile 2. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Comparison of co-state history for mission profile 2. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.