Optimal Control of ODE Car-Following Models: Applications to Mixed-Autonomy Platoon Control via Coupled Autonomous Vehicles

Alexander Keimer; Alexandre M. Bayen; Anish Gollakota; Arwa Alanqary; Ashish Pandian; Xiaoqian Gong

REVIEW 4 major objections 5 minor 54 references

Optimal Control of ODE Car-Following Models: Applications to Mixed-Autonomy Platoon Control via Coupled Autonomous Vehicles

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A rigorous optimal control formulation for a mixed-autonomy platoon with the Bando-FtL model is shown to have a minimizer and is solved by adjoint gradient descent, with simulations showing large reductions in acceleration and fuel use for one to five autonomous vehicles.

desk verdict Solid existence theory fixes a gap in prior work; the numerical claims are not yet supported by the experiments as reported. read the letter →

arxiv 2508.19417 v1 pith:YDEIEUGO submitted 2025-08-26 math.OC cs.SYeess.SY

Arwa Alanqary , Alexandre M. Bayen , Xiaoqian Gong , Anish Gollakota , Alexander Keimer , Ashish Pandian This is my paper · ORCID

classification math.OCcs.SYeess.SY

keywords controlplatoonoptimalvehiclesproblemautonomouscar-followingconstraints

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

Traffic jams on highways often come from small speed fluctuations that grow as human drivers react to each other. This paper asks what a few autonomous vehicles (AVs) can do to stop those waves if they can brake and accelerate freely, while human-driven cars follow a standard car-following rule, the Bando-Follow-the-Leader model. The authors set up an optimal control problem: choose accelerations for the AVs to minimize the total squared acceleration of every car in a 20-car platoon over a fixed time, subject to keeping safe gaps and nonnegative speeds.

The theoretical part shows that this optimization problem is well posed. They prove that, with admissible controls, the vehicles' trajectories exist and depend continuously on the control, and that a minimizer exists under a convexity assumption on the cost. The proof uses the fact that the Bando-FtL model guarantees a minimum headway no matter what the leader does, a property taken from an earlier paper by two of the same authors.

The numerical part uses a gradient descent method with adjoint-based gradients, plus penalties for the safety constraints. In simulations on a measured highway trajectory from I-24, one AV placed just behind the leader cuts the platoon's total squared acceleration by about 70% and estimated fuel use by about 16% compared to an all-human platoon. Five AVs cut acceleration by about 85% and fuel by 26%, with diminishing returns. But the human model is not calibrated to the data, and the results are not benchmarked against existing controllers, so the absolute percentages should be read as an upper-bound-style illustration, not a field measurement.

Extended reading notes

Core claim

Theorem 4.3: given Assumptions 2.1-2.3, convex running cost L and final cost S, and sufficiently small amin, the optimal control problem in Definition 2.10 admits a minimizer u* in U. Numerically, Algorithm 1 on the I-24 trajectory produces a 70.4% reduction in total L2 acceleration with one AV and 84.6% with five AVs (Table 1).

Load-bearing premise

The paper relies on Theorem 3.3, whose proof is not in this paper but is cited from [17], for the global well-posedness and the uniform lower bounds on headway and velocity (eqs. 12-14) of the Bando-FtL model. Lemma 4.1 and Theorem 4.3 depend on these bounds to ensure the control-to-state map is continuous and the admissible set U is nonempty; if the bounds fail for any admissible leader trajectory, the existence proof collapses. The numerical experiments additionally assume that Bando-FtL with alpha=0.1, beta=525 and the unstated vmax, ds, l reproduces human driving on I-24.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The existence theorem is the real contribution. Theorem 4.3 gives a minimizer for the Bando-FtL optimal control problem via weak-star compactness plus a continuity lemma that uses the uniform headway bounds from the model's well-posedness. That fixes a genuine gap in [52], whose compactness claim is wrong for box-constrained controls in L^p. The fix is clean: keep weak-star convergence and add convexity in the control. This part is solid and publishable.

The numerical section is not yet support for the Table 1 claims (70-85% acceleration, 16-26% fuel). The experiment is under-specified: vmax, ds, and l are never stated, initial conditions are omitted, and the Bando-FtL parameters are chosen, not calibrated, to the I-24 trajectory. The baseline is the model's own unstable behavior, so the percentages are relative to an artificial human surrogate, not to real driving. No benchmark against existing controllers is given; the conclusion says that's future work. The abstract promises distribution experiments that are not in the paper. This is a real weakness in the applied framing, not a nitpick. The fuel metric is independent of the optimized objective, which earns some credit, but it inherits the same unvalidated model.

The small stuff: eqs. (16)-(17) are missing squares on v0; likely a typo, but dimensionally off as printed. The reliance on [17] is fine, since it's a published, parameter-free well-posedness result; not circular. The whole theory sits on the assumption that the headway bounds hold for any admissible leader; that's explicit, and if it fails the proof collapses. Acceptable as an assumption, but worth stating as a limitation in the final version.

Who this is for: specialists in ODE optimal control for car-following, and mixed-autonomy researchers who want a theoretical baseline. The theory deserves a serious referee; the numerics need a revision pass: report all constants, benchmark against a simple controller, calibrate or justify parameters, ship code and data, and either add the promised experiments or cut them from the abstract.

My recommendation: send it to review. Conditional acceptance is the right call, with the numerical revisions as conditions. The theoretical core holds up.

Referee Report

4 major / 5 minor

Summary. The paper studies optimal control of a single-lane mixed-autonomy platoon with a prescribed leader trajectory. Human-driven vehicles follow the Bando-Follow-the-Leader (Bando-FtL) model, while autonomous vehicles have directly controlled acceleration. The authors define an admissible control set with acceleration bounds, nonnegative velocity, and a minimum safety gap for the AVs. Their main theoretical result (Theorem 4.3) asserts existence of a minimizer for a convex running cost and C^1 terminal cost, relying on well-posedness and headway/velocity bounds for Bando-FtL from [17]. They propose a direct adjoint-based gradient descent method with a penalty formulation for state constraints, and report numerical experiments on an I-24 trajectory showing large reductions in total L2 acceleration and fuel consumption as the number of AVs increases. The exposition is generally clear, but several load-bearing proof steps and the numerical validation need correction.

Significance. If completed and corrected, the existence theorem is a meaningful contribution: it transfers the known well-posedness of Bando-FtL into an optimal-control guarantee for a convex-in-control problem under explicit safety constraints. The reliance on [17] is legitimate and not circular, and the one-sided coupling structure is exploited cleanly. The numerical algorithm is standard in spirit, but the adjoint derivation currently has sign errors, and the experiments are under-specified; the applied claims in Table 1 are therefore not yet supported. The framework is potentially useful for benchmarking AV smoothing controllers, provided the numerical section is made reproducible and compared against existing controllers.

major comments (4)

[§3, Corollary 3.4, Eq. (16)-(17)] The constructed braking control has a dimensional error: amin := -v° / (2d) has units 1/s, not m/s^2. Solving (15) gives amin = -(v°)^2 / (2(x_l(0)-x_1(0)-l-dsafe)) and T* = 2(...)/v°. As written, with v°=10 m/s and d=20 m, the claimed amin=-0.25 m/s^2 stops far too late. The zero-velocity case v°=0 is also not covered. Since U ≠ ∅ is needed for Theorem 4.3, the proof must be corrected; this is fixable but not merely cosmetic.
[§4.1, Theorem 4.3 proof] The final chain J(u*) = lim_n J(y_n,u*) ≤ lim_n J(y_n,u_n) is not justified as stated. The pointwise inequality J(y_n,u*) ≤ J(y_n,u_n) need not hold for a weak-star limit. The correct argument is to use convexity and the established uniform convergence of states to show J(u*) ≤ liminf_n J(u_n). Please rewrite the limiting step explicitly. This is load-bearing because the existence conclusion depends on it.
[§5.1, Eq. (30)-(32), (40)] The variational derivative of the constraint term in H should be ζ^T(δ\dot y - f_y δy - f_u δu), not ζ^T(δ\dot y + f_y δy + f_u δu). With the printed signs, the adjoint equation and the gradient G_i = L_{u_i} - ζ_{i+|I|} are incorrect: for L(u)=u^2, S=0, one obtains G=2u, which is the direction of steepest ascent, not descent. This sign error affects the core of Algorithm 1 and the numerical results. Please verify and correct the adjoint derivation.
[§6.2, Table 1] The numerical validation is not reproducible and does not support the headline reductions. The optimal velocity function (8) depends on vmax and ds, and the dynamics depend on l, but none are reported; initial conditions are also omitted. The Bando-FtL parameters α=0.1, β=525 are not calibrated to the I-24 data, and no baseline controller is compared (the conclusion acknowledges this). Since the objective being optimized is the same L2-acceleration metric reported in Table 1, the acceleration reductions are not independent evidence; the fuel metric is independent but inherits the uncalibrated dynamics. The abstract also promises experiments on 'distributions of controlled vehicles', but only one fixed distribution is tested.

minor comments (5)

[Definition 2.7] Typo: 'saftey' should be 'safety'.
[§5.3, Eq. (47)-(53)] The grid indexing is inconsistent: p is called a positive real in (47), but it is a count; in (53), t_p=T should presumably be t_r, and the quantifier uses j while the set index is i/r.
[§5.1, Eq. (28)] The first-variation formula should include \tilde u(t) in the L_u term and evaluate the terminal term at \tilde u(T); as written the expression is dimensionally inconsistent.
[Definition 3.1 and references] The Bando-FtL model is attributed to [12], but [12] is the Delle Monache et al. reference; the original Bando model is [4] and the Bando-FtL well-posedness is [17]. Please check the citation.
[§6.1] The text says box constraints on the control are not imposed in the experiments. This means Theorem 4.3, which assumes controls in U with finite amin, amax, does not directly cover the numerical problem; a remark should clarify the theoretical status of the unconstrained instance.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: existence proof is self-contained modulo an independent prior well-posedness theorem; numerical L2 metric is self-referential but not a derivation-step.

full rationale

The main theoretical result (Theorem 4.3) is obtained by the standard direct method: take a minimizing sequence, use weak-* compactness from the L∞ box constraints on the controls, pass to the limit in the safety constraints using Lemma 4.1 (control-to-state continuity), and rely on convexity of the running cost for weak lower semicontinuity. No step defines the minimizer in terms of the objective or fits a parameter to the result. The only imported mathematical fact is Theorem 3.3, whose proof is omitted and cited to [17] (Gong and Keimer, 2023). This is a self-citation and is load-bearing for Lemma 4.1 and Corollary 3.4, but [17] is a published, parameter-free well-posedness theorem about the Bando-FtL model; its assumptions do not include the optimal control problem or its minimizer. Under the review rules, that citation is independent support and does not by itself constitute circularity. In the numerical section, the paper optimizes exactly the L2-acceleration objective in eq. (56) and then reports the reduction in total L2 acceleration in Table 1; the paper explicitly acknowledges that this metric is also the objective function. Thus the 70.42% acceleration figure is a self-referential measure of the optimizer's own loss rather than an independent validation. However, the paper also reports an independent fuel-consumption metric (eq. (57), Table 1), and no fitted constants feed the theoretical derivation. The remaining issues — unreported values for vmax, ds, and l, uncalibrated Bando-FtL parameters, and the absence of a benchmark controller (explicitly left to future work in the conclusion) — are reproducibility and validation concerns, not circularity. Therefore no circular derivation step is present; the score reflects the minor self-referential numerical metric and the self-cited but independent Theorem 3.3.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central derivation requires no fitted constants; the only external theorem is [17] (self-cited but published). The numerical claims, however, rest on several hand-chosen model parameters and unspecified values, which are the main source of uncertainty.

free parameters (8)

alpha (Bando-FtL sensitivity) = 0.1
Hand-chosen in Section 6.2; no calibration to I-24 data. Central to the simulated human behavior and hence all percentage reductions.
beta (Bando-FtL velocity difference sensitivity) = 525
Hand-chosen in Section 6.2; same rationale as alpha.
vmax (optimal velocity function parameter)
Appears in eq. (8) but its value in the experiments is never stated.
ds (optimal velocity safety distance)
Appears in eq. (8); value not stated.
l (vehicle length)
Used in dynamics and safety constraints; numerical value not stated.
dsafe (safety distance constraint) = 5 m
Problem parameter, chosen by hand in Section 6.2.
dmax (maximum headway constraint) = 120 m
Added in eq. (58) to prevent the AV from falling behind; chosen by hand.
mu (penalty parameter)
Iteratively increased without a published schedule (Section 5.2, Algorithm 1).

assumptions (6)

domain assumption Well-posedness and bounds for Bando-FtL (Theorem 3.3), proof in [17]
The paper states Theorem 3.3 and cites [17]; Lemma 4.1 and Theorem 4.3 depend on the uniform lower headway bound (12) and velocity bounds (13)-(14).
domain assumption Leader trajectory regularity (Assumption 2.1): x_l in W^{2,infty}, v_l >= 0
Ensures the leader has bounded acceleration and nonnegative speed, needed for Theorem 3.3 and for the integral estimates.
domain assumption Initial spacing (Assumption 2.2): gaps exceed d_o > 0
Guarantees initial feasibility and defines d_o for the safety constraint.
domain assumption Admissible control set (Assumption 2.3): bounded controls and nonnegative AV velocities
Defines U^{a_max}_{a_min}; weak-star compactness of this set is used in Theorem 4.3.
domain assumption Convexity and C^1 of the running and final costs (Definition 2.10)
Used to pass to the limit in the objective (Theorem 4.3) and to derive the adjoint system.
domain assumption Feasibility of the control set U for sufficiently small amin (Corollary 3.4)
The construction in eqs. (16)-(17) is sketched; as written it has a dimensional typo, and nonemptiness of U is required for the existence theorem.

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Pith. "Pith review of Optimal Control of ODE Car-Following Models: Applications to Mixed-Autonomy Platoon Control via Coupled Autonomous Vehicles." pith.science (2026). https://pith.science/paper/YDEIEUGO

@misc{pith2026250819417,
  author       = {Pith},
  title        = {Pith review of: Optimal Control of ODE Car-Following Models: Applications to Mixed-Autonomy Platoon Control via Coupled Autonomous Vehicles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDEIEUGO}},
  note         = {Machine review of arXiv:2508.19417}
}

read the original abstract

In this paper, we study the optimal control of a mixed-autonomy platoon driving on a single lane to smooth traffic flow. The platoon consists of autonomous vehicles, whose acceleration is controlled, and human-driven vehicles, whose behavior is described using a microscopic car-following model. We formulate the optimal control problem where the dynamics of the platoon are describing through a system of non-linear ODEs, with explicit constraints on both the state and the control variables. Theoretically, we analyze the well-posedness of the system dynamics under a reasonable set of admissible controls and establish the existence of minimizers for the optimal control problem. To solve the problem numerically, we propose a gradient descent-based algorithm that leverages the adjoint method, along with a penalty approach to handle state constraints. We demonstrate the effectiveness of the proposed numerical scheme through several experiments, exploring various scenarios with different penetration rates and distributions of controlled vehicles within the platoon.

Figures

Figures reproduced from arXiv: 2508.19417 by the authors.

**Figure 1.** The leader xℓ(t) ≡ x0(t) with its dynamics determined by the acceleration ulead(t) and the following cars x1(t), x2(t), x3(t), x4(t) with its dynamics governed system eq. (4). Here, cars x1(t) and x4(t) are autonomous and cars x2(t) and x3(t) are human-driven. indicating the functional dependency of the solution (x, v) with regard to the control u. The fact that there is such a functional dependency is not obvious a… view at source ↗

**Figure 2.** (a) Velocity and acceleration profiles of the considered lead vehicle. (b) Segments showing velocity and acceleration of a platoon of 20 human-driven vehicles following the leader’s trajectory in (a). These segments highlight the instability in the behavior of the Bando-FtL car following model used in this study. Human-driven Vehicles In all experiments, the human-drive vehicles’ (HVs) acceleration is modeled by the… view at source ↗

**Figure 3.** Illustration of a the positions of the first 2 AVs (red and orange) in a platoon of 20 vehicles. As mentioned earlier, the proposed numerical approach is only guaranteed to converge locally and it is essential to find a good initialization for the optimization algorithm. To that end we initialize the control function of the first AV using the solution of the problem instance in eq. (59) with a single AV following th… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Acceleration (left) and velocity (middle) profiles of the 20 vehicles in the platoon with 1AV. The right tile shows the space headway of the AVs in the platoon. One thing we note from fig. 4 is that the effect of the AV diminishes for further followers. In fig. 5 we sh…

**Figure 5.** Figure 5: Acceleration (left) and velocity (middle) profiles of the 20 vehicles in the platoon with 2AV. The right tile shows the space headway of the AVs in the platoon. As we increase the penetration rate of the AVs, both the total acceleration and fuel consumption reduce, how…

**Figure 6.** Figure 6: Acceleration (left) and velocity (middle) profiles of the 20 vehicles in the platoon with 5AV. The right tile shows the space headway of the AVs in the platoon. Lastly, we comment on the general qualitative behavior of the AVs. We note from figs. 4, 5 and 6 that the AV…

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