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arxiv: 2607.00383 · v1 · pith:V6G4QG7Knew · submitted 2026-07-01 · 🧮 math.OC

Regularized Model Predictive Control via Contractivity and Implicit Lur'e Analysis

Ryotaro Shima , Anand Gokhale , Alexander Davydov , Francesco Bullo This is my paper

Pith reviewed 2026-07-02 08:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords regularized model predictive controlcontraction analysisimplicit Lur'e systemslinear matrix inequalitiesstability guaranteesconvex regularizersclosed-loop dynamics
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The pith

Regularized model predictive control admits contraction certificates via a multiplier framework for implicit Lur'e systems and associated LMI conditions.

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

The paper develops a contraction-based stability analysis for regularized model predictive control, where the feedback law arises from a finite-horizon optimal control problem augmented by a regularizing cost. It interprets the resulting closed-loop dynamics as an implicit Lur'e system in which the regularizer perturbs the optimality conditions. A multiplier-based contraction framework is introduced for such systems and converted into linear matrix inequality conditions that certify contraction for three classes of regularizers: convex smooth stage costs, convex closed proper stage costs, and differentiable regularizers with Lipschitz gradients. Numerical examples with soft penalties, hard constraints, and sparsity penalties show that regularization can shape performance while the LMI conditions retain the contraction guarantee.

Core claim

Regularized MPC feedback can be represented as an implicit Lur'e system whose contraction is certified by a multiplier-based framework that yields explicit LMI conditions for the three listed classes of regularizers, thereby providing formal stability guarantees that hold uniformly across those regularizers.

What carries the argument

Multiplier-based contraction framework for implicit Lur'e systems, which converts the perturbed optimality conditions of regularized MPC into verifiable linear matrix inequalities.

If this is right

  • Regularized MPC retains contraction-based stability for convex smooth, convex closed proper, and Lipschitz-gradient regularizers.
  • Input and state soft penalties, hard input constraints, and sparsity-promoting penalties all fall under the certified classes.
  • Regularization parameters can be tuned to shape transient performance without invalidating the stability certificate.
  • The same LMI conditions apply to any finite-horizon optimal control problem whose regularizer belongs to one of the three classes.

Where Pith is reading between the lines

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

  • The LMI conditions could be solved offline to select regularizer weights that achieve desired performance margins.
  • The implicit Lur'e representation may allow similar analysis for other optimization-based controllers beyond MPC.
  • The multiplier framework might connect to incremental passivity or robust control certificates for nonlinear feedback laws.

Load-bearing premise

The regularizing cost must perturb the optimality conditions so that the closed-loop map becomes an implicit Lur'e system that admits a multiplier-based contraction certificate expressible as LMIs.

What would settle it

A concrete closed-loop trajectory or eigenvalue computation for one of the three regularizer classes where the derived LMIs are feasible yet the system state does not contract under the regularized feedback.

Figures

Figures reproduced from arXiv: 2607.00383 by Alexander Davydov, Anand Gokhale, Francesco Bullo, Ryotaro Shima.

Figure 1
Figure 1. Figure 1: Series mass-spring-damper system used in the mass-spring￾damper case studies. The state is x = [q1; v1; q2; v2], and the control inputs u1 and u2 are external forces applied to the two masses. Ac =     0 1 0 0 a21 a22 a23 a24 0 0 0 1 k2 m2 c2 m2 − k2 m2 − c2 m2     , Bc =     0 0 1 m1 0 0 0 0 1 m2     where a21 = − k1+k2 m1 , a22 = − c1+c2 m1 , a23 = k2 m1 , and a24 = c2 m1 . The parameters… view at source ↗
Figure 2
Figure 2. Figure 2: Soft input-penalty example. Top: distance between two closed￾loop trajectories, measured in the certified norm ∥ · ∥P , with the gray line indicating the prescribed decay rate η. Middle and bottom: input trajectories u1 and u2. The input-penalized MPC preserves the certified contraction behavior while keeping the inputs closer to the desired interval [−1, 0.5] than the nominal MPC. B. Tracking MPC for Seri… view at source ↗
Figure 4
Figure 4. Figure 4: Communication graph of the consensus network used in the sparse-control example. Nodes represent scalar agents, and edge labels indicate the eight control inputs acting through the incidence matrix BG. LG =         2 −1 0 0 −1 0 −1 3 −1 0 −1 0 0 −1 3 −1 0 −1 0 0 −1 2 0 −1 −1 −1 0 0 3 −1 0 0 −1 −1 −1 3         , BG =         0 0 0 0 0 0 1 1 1 0 1 0 0 0 −1 0 −1 1 0 1 0 0 0 0 0 −1 0 0 … view at source ↗
Figure 5
Figure 5. Figure 5: Sparse consensus-control example with λ = 1.5. Top: distance between two reduced-state trajectories, measured in the certified norm ∥ · ∥P , with the gray line indicating the prescribed decay rate η. Middle and bottom: representative edge inputs u2 and u4. The sparse MPC retains contraction while driving selected inputs exactly to zero, illustrating the sparsity-promoting effect of the ℓ1 regularizer. 0 5 … view at source ↗
Figure 6
Figure 6. Figure 6: Sparsity statistics for the consensus example over 50 randomly generated initial conditions. The curves show the number of zero￾valued edge inputs over time for different values of the regularization coefficient λ. Larger λ values produce sparser control actions, whereas the nominal MPC case (λ = 0) rarely sets inputs exactly to zero. To illustrate the certified contraction, we compute two trajectories {x … view at source ↗
read the original abstract

This paper develops a contraction-based stability analysis for regularized model predictive control (MPC), whose feedback law is defined implicitly by a finite-horizon optimal control problem with an additional regularizing cost. The proposed approach interprets regularized MPC as an implicit Lur'e system, in which the regularizing cost perturbs the optimality conditions. We develop a multiplier-based contraction framework for implicit Lur'e systems and derive linear matrix inequality conditions for regularized MPC with three broad classes of regularizers: convex smooth stage costs, convex closed proper stage costs, and differentiable regularizers with Lipschitz gradients. Numerical studies on input and state soft penalties, hard input constraints, and sparsity-promoting penalties illustrate that regularization shapes closed-loop performance while retaining formal contraction-based stability guarantees.

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 / 1 minor

Summary. The paper develops a contraction-based stability analysis for regularized model predictive control by representing the implicitly defined feedback law as an implicit Lur'e system in which the regularizing cost perturbs the optimality conditions. It introduces a multiplier-based contraction framework for such systems and derives LMI conditions for three classes of regularizers (convex smooth stage costs, convex closed proper stage costs, and differentiable regularizers with Lipschitz gradients), with numerical examples on soft penalties, hard constraints, and sparsity-promoting penalties.

Significance. If the LMI derivations and contraction certificates hold, the work supplies a systematic, multiplier-based route to contraction guarantees for regularized MPC. This is significant because it connects contraction theory and implicit Lur'e analysis to a practically relevant class of optimization-based controllers, enabling regularization to shape performance while retaining formal stability certificates. The explicit treatment of three broad regularizer classes and the accompanying numerical validation are strengths.

minor comments (1)
  1. [Abstract] The abstract asserts that LMI conditions are derived but does not exhibit any explicit LMIs, multiplier selections, or key proof steps; the full manuscript should make these derivations self-contained and verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recognizing the significance of the multiplier-based contraction framework for implicit Lur'e representations of regularized MPC. The recommendation is listed as uncertain, yet no specific major comments or concerns are provided in the report. We therefore have no points to address point-by-point and maintain that the LMI derivations, contraction certificates, and numerical validation stand as presented.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core derivation represents regularized MPC optimality conditions as an implicit Lur'e interconnection and obtains contraction certificates via multiplier theory cast as LMIs for three standard regularizer classes. This proceeds from established contraction analysis and Lur'e system theory without any step that reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The modeling choice itself is the explicit contribution being justified, not an input assumed without derivation. No self-definitional, fitted-prediction, or uniqueness-imported patterns appear in the stated claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5659 in / 1014 out tokens · 31750 ms · 2026-07-02T08:49:14.446179+00:00 · methodology

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Reference graph

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