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arxiv: 2604.07225 · v3 · submitted 2026-04-08 · 🧮 math.OC · cs.SY· eess.SY

A Trajectory-Based Approach to Controlled Invariance and Recursively Feasible MPC

Emmanuel Junior Wafo Wembe , Adnane Saoud This is my paper

Pith reviewed 2026-05-10 17:11 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords controlled invariant setsmodel predictive controlrecursive feasibilitytrajectory-based approachlinear discrete-time systemsconvex feasible pointsmaximal invariant set
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The pith

Convex feasible points along finite trajectories characterize controlled invariance and enable recursively feasible MPC without terminal sets.

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

This paper re-examines controlled invariant sets for linear discrete-time systems by viewing them through finite state trajectories. It defines convex feasible points to capture the property that a state can be kept inside the set indefinitely using allowed controls. Merging this definition with the standard backward reachability algorithm produces the largest possible controlled invariant set. The same idea directly yields a model predictive controller whose feasibility is preserved at every step without needing an auxiliary terminal constraint set. The method is made practical by recasting the point search as a convex optimization problem.

Core claim

For linear discrete-time systems, a state belongs to a controlled invariant set if and only if it is the start of a finite trajectory that satisfies the dynamics and state-input constraints in a manner that admits convex combinations of feasible points. This characterization, when inserted into the classical fixed-point iteration, computes the maximal controlled invariant set exactly. The resulting sets support an MPC formulation in which recursive feasibility follows immediately from the trajectory property.

What carries the argument

Convex feasible points: initial states of finitely long trajectories for which convex combinations of the points remain feasible under the linear dynamics and constraints.

Load-bearing premise

The finite-trajectory convex points fully describe every controlled invariant set without leaving out states that could be kept safe by infinite or non-convex sequences.

What would settle it

Finding a linear system and a controlled invariant set larger than the one obtained by the convex feasible point iteration, or an MPC closed-loop trajectory that becomes infeasible at some finite time despite starting from a computed set.

Figures

Figures reproduced from arXiv: 2604.07225 by Adnane Saoud, Emmanuel Junior Wafo Wembe.

Figure 1
Figure 1. Figure 1: Convex feasible Trajectory initiated at x0. The points x1 = Φ(1, x0, u) at the top right, x2 = Φ(2, x0, u) and x3 = Φ(3, x0, u) at the bottom left and right respectively. Finally x4 = Φ(4, x0, u) ∈ ch  S 3 t=0 {Φ(t, x0, u)}  . established in this paper extend to this setting, with the exception of Theorem 4. In particular, the condition ϵ > 0 is required in Theorem 4 to ensure convergence towards the max… view at source ↗
Figure 2
Figure 2. Figure 2: Invariant set computation. (a) Comparison between [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop trajectories under the proposed MPC [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

In this paper, we revisit the computation of controlled invariant sets for linear discrete-time systems through a trajectory-based viewpoint. We begin by introducing the notion of convex feasible points, which provides a new characterization of controlled invariance using finitely long state trajectories. We further show that combining this notion with the classical backward fixed-point algorithm allows for the computation of the maximal controlled invariant set. Building on these results, we propose a model predictive control (MPC) scheme that guarantees recursive feasibility without relying on precomputed terminal sets. Finally, we formulate the search for convex feasible points as an optimization problem, yielding a practical computational method for constructing controlled invariant sets. The effectiveness of the approach is illustrated through numerical examples.

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

3 major / 3 minor

Summary. The paper introduces the notion of convex feasible points to characterize controlled invariance for linear discrete-time systems via finite-length state trajectories. It claims that combining this characterization with the classical backward fixed-point algorithm computes the maximal controlled invariant set. Building on this, the authors propose an MPC scheme that guarantees recursive feasibility by online optimization for convex feasible points without precomputed terminal sets, formulate the search as an optimization problem, and illustrate the approach with numerical examples.

Significance. If the central claims hold, the work offers a trajectory-based alternative for computing maximal controlled invariant sets and designing recursively feasible MPC without terminal sets for constrained linear systems. The optimization-based formulation provides a practical computational method, and the combination of a new finite-trajectory notion with an existing algorithm is a clear strength when properly substantiated.

major comments (3)
  1. [§3] §3 (definition of convex feasible points): The finite-trajectory characterization must be shown to be equivalent to the standard infinite-horizon definition of controlled invariance. Specifically, every state admitting an infinite admissible input sequence must admit a finite trajectory of the chosen length that extends indefinitely; without an explicit proof of closure under infinite extensions or a counter-example check, the claim that the backward algorithm recovers the maximal set is not yet load-bearing.
  2. [§5] §5 (MPC recursive feasibility): The proof that the online optimization always succeeds precisely when the current state lies in the maximal controlled invariant set relies on the equivalence in §3. If convexity or the finite length introduces conservatism or hidden feasibility cuts, recursive feasibility fails; the manuscript must provide a derivation showing that the optimization recovers all required points without such restrictions.
  3. [§4] §4 (backward fixed-point algorithm combination): The paper asserts that the new notion plus the classical iteration computes the maximal set, but supplies no explicit derivation or verification that the fixed-point iteration on convex feasible points converges to the same set as the standard algorithm on the infinite-horizon definition.
minor comments (3)
  1. [§6] The choice of trajectory length N is introduced without guidance on how it is selected in practice or its effect on computational complexity.
  2. [§3] Notation for the convex feasible point set and the optimization variables could be clarified to avoid confusion with standard reachable-set notation.
  3. [§7] The numerical examples would benefit from a table comparing computation times or set sizes against a standard terminal-set-based MPC baseline.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our results. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (definition of convex feasible points): The finite-trajectory characterization must be shown to be equivalent to the standard infinite-horizon definition of controlled invariance. Specifically, every state admitting an infinite admissible input sequence must admit a finite trajectory of the chosen length that extends indefinitely; without an explicit proof of closure under infinite extensions or a counter-example check, the claim that the backward algorithm recovers the maximal set is not yet load-bearing.

    Authors: We agree that the equivalence requires a more explicit treatment. The current manuscript introduces convex feasible points via finite trajectories and states their relation to controlled invariance, but the proof of equivalence to the infinite-horizon definition (including closure under infinite extensions) is only sketched. In the revision we will add a dedicated lemma with a complete proof: for linear discrete-time systems, a state admits an infinite admissible input sequence if and only if it is a convex feasible point for a trajectory of length at least equal to the system dimension plus the number of active constraints; the extension is constructed explicitly by concatenating the finite trajectory with a tail obtained from the invariance property, using linearity to preserve convexity and feasibility. revision: yes

  2. Referee: [§5] §5 (MPC recursive feasibility): The proof that the online optimization always succeeds precisely when the current state lies in the maximal controlled invariant set relies on the equivalence in §3. If convexity or the finite length introduces conservatism or hidden feasibility cuts, recursive feasibility fails; the manuscript must provide a derivation showing that the optimization recovers all required points without such restrictions.

    Authors: The recursive-feasibility argument in §5 is indeed predicated on the equivalence in §3. We will strengthen the derivation by adding a theorem that directly links the feasibility of the online convex-feasible-point optimization to membership in the maximal controlled invariant set. The proof will show that any infinite admissible sequence can be represented by a finite convex feasible point of sufficient length without loss of feasibility; because the dynamics are linear and the constraint sets are convex, truncating the sequence and optimizing over the initial segment recovers exactly the same set, with no hidden cuts introduced by the finite horizon. revision: yes

  3. Referee: [§4] §4 (backward fixed-point algorithm combination): The paper asserts that the new notion plus the classical iteration computes the maximal set, but supplies no explicit derivation or verification that the fixed-point iteration on convex feasible points converges to the same set as the standard algorithm on the infinite-horizon definition.

    Authors: We accept that an explicit convergence argument is missing. In the revised manuscript we will insert a proposition establishing that the backward fixed-point operator defined on convex feasible points is monotone and that its least fixed point coincides with the maximal controlled invariant set obtained from the classical infinite-horizon operator. The argument relies on the equivalence lemma added to §3 and on the fact that the finite-trajectory representation preserves the inclusion properties required by the standard iteration. revision: yes

Circularity Check

0 steps flagged

No circularity: new finite-trajectory characterization combined with standard backward iteration yields independent computational method

full rationale

The paper defines convex feasible points as a novel finite-trajectory characterization of controlled invariance for linear discrete-time systems, then combines this definition with the classical backward fixed-point algorithm to compute the maximal controlled invariant set. It further casts the search as an optimization problem and derives an MPC scheme guaranteeing recursive feasibility without terminal sets. None of these steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the central claims rest on the explicit new definition and its algebraic properties rather than on any self-referential loop. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, background axioms, or invented entities are stated beyond the new definition of convex feasible points.

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