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arxiv: 2605.01558 · v2 · submitted 2026-05-02 · 🧮 math.OC · cs.SY· eess.SY

A Measure-Theoretic Formulation of Behavioral Systems

Victor M. Preciado This is my paper

Pith reviewed 2026-05-09 14:10 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords behavioral systemsmeasure-theoretic formulationdynamical systemsnonlinear systemsconvexityprobability measures on trajectoriesWillems theory
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The pith

Representing a dynamical system as the set of all probability measures on its admissible trajectories yields a convex and weakly closed behavior even for nonlinear deterministic dynamics.

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

The paper lifts Willems' behavioral approach from sets of trajectories to sets of probability measures supported on those trajectories. This lift preserves convexity and weak closure for deterministic systems regardless of nonlinearity, since mixtures of trajectory distributions remain admissible while mixtures of the trajectories themselves do not. The classical trajectory-based theory sits inside the new object as its extreme points, namely the Dirac measures on single admissible trajectories. The construction also supplies a route to stochastic behavioral systems via consistent history-conditional kernels.

Core claim

A finite-horizon dynamical system is identified with the set of all Borel probability measures supported on its admissible trajectories. For deterministic systems this set is convex and weakly closed even when the underlying dynamics are nonlinear, because any convex combination of measures supported on admissible trajectories remains supported on admissible trajectories. Its extreme points are precisely the Dirac measures concentrated on individual admissible trajectories, so the original deterministic behavior is recovered as the extremal skeleton of the measure-valued object.

What carries the argument

The behavioral-measure set: the collection of all Borel probability measures whose support is contained in the set of admissible trajectories.

Load-bearing premise

The space of trajectories can be equipped with a Borel sigma-algebra such that the collection of probability measures supported on admissible trajectories is weakly closed and its extreme points are exactly the Dirac measures on single admissible trajectories.

What would settle it

An explicit pair of admissible trajectory distributions whose convex combination is supported only on trajectories that violate the system dynamics would falsify the claimed convexity.

Figures

Figures reproduced from arXiv: 2605.01558 by Victor M. Preciado.

Figure 1
Figure 1. Figure 1: Scalar polynomial experiment for (V.1). Feasible regions of M(r) B projected onto (E[xtut ], E[xt+1]) for r = 1 (dotted), r = 2 (dashed), and r = 3 (solid), together with sam￾pled true trajectories (gray). The exact boundary (black) is indistinguishable from the r = 3 boundary. V-B Nonlinear control via a moment-SOS relaxation The second experiment applies the occupation-measure framework of Section III to… view at source ↗
Figure 2
Figure 2. Figure 2: Data-driven LTI validation for (V.6). Left: histogram of out-of-sample Hankel projection residuals (log10 scale) for 200 validation trajectories. Right: singular values of the length-6 Hankel matrix built from the persistently exciting dataset, showing the expected rank-8 truncation. Trajectory-level validation. We generate 200 independent length-6 trajectories with random initial states and inputs. For ea… view at source ↗
read the original abstract

In Willems' behavioral systems theory, a dynamical system is identified with the set of all trajectories compatible with its laws of motion. In the linear time-invariant setting this trajectory set is a linear subspace, and its algebraic structure underpins the Fundamental Lemma: a single persistently exciting data trajectory generates the entire finite-horizon behavior. For nonlinear or stochastic systems, however, the admissible trajectory set is generally nonconvex, obstructing direct optimization over the behavior. In this paper, we lift the behavioral viewpoint from trajectories to probability measures on trajectories by representing a finite-horizon dynamical system with the set of all Borel probability measures supported on its admissible trajectories. For deterministic systems, this behavioral-measure set is convex and weakly closed even when the dynamics are nonlinear, because convex combinations of trajectory distributions remain dynamically admissible even when convex combinations of trajectories do not. Its extreme points are precisely the Dirac masses on individual admissible trajectories, so the classical deterministic theory is embedded as the extremal skeleton of the richer measure-valued object. On this foundation we establish two core deterministic results and outline a stochastic extension based on history-conditional kernel consistency.

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 / 3 minor

Summary. The paper lifts Willems' behavioral systems theory from trajectory sets to the set of all Borel probability measures supported on admissible trajectories. For deterministic finite-horizon systems the behavioral-measure set is shown to be convex and weakly closed regardless of linearity of the dynamics, because convex combinations and weak limits preserve support inclusion in the admissible set B. The extreme points are precisely the Dirac measures on individual trajectories in B, recovering the classical deterministic theory. Two core deterministic results are established on this foundation and a stochastic extension via history-conditional kernel consistency is outlined.

Significance. If the derivations hold, the construction supplies a convex set in measure space for optimization over behaviors even when the underlying trajectory set is nonconvex, which is a direct response to a known obstruction in nonlinear behavioral control. The embedding of the deterministic case as the extremal skeleton is a clean application of standard facts on the weak topology and support. Credit is due for the parameter-free character of the lift and for relying only on classical measure-theoretic properties rather than additional dynamical structure.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 (weak closure): the argument that weak limits preserve support inclusion in B requires the trajectory space to be Polish (or at least completely regular) so that the support map is upper semicontinuous; the manuscript should state the precise topological assumptions on the trajectory space to make this rigorous for general nonlinear finite-horizon dynamics.
  2. [§4, Proposition 4.2] §4, Proposition 4.2 (extreme points): while the claim that extreme points are exactly the Diracs follows from the definition of extremality in the simplex, the proof should explicitly invoke the relevant fact from convex analysis (e.g., that any non-Dirac measure is a nontrivial convex combination) rather than leaving it implicit.
minor comments (3)
  1. [Definition 2.3] Definition 2.3: the notation for the behavioral-measure set could be introduced with an explicit symbol (e.g., M(B)) to improve readability when it is used repeatedly in later sections.
  2. [Introduction] Introduction, paragraph 3: the discussion of the Fundamental Lemma would benefit from a direct citation to Willems' original work rather than only secondary references.
  3. [§6] The stochastic extension in §6 is described as an outline; if the authors intend it as a contribution, a brief statement of the precise consistency conditions would help readers assess its scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comments. We address each major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (weak closure): the argument that weak limits preserve support inclusion in B requires the trajectory space to be Polish (or at least completely regular) so that the support map is upper semicontinuous; the manuscript should state the precise topological assumptions on the trajectory space to make this rigorous for general nonlinear finite-horizon dynamics.

    Authors: We agree that the topological assumptions should be stated explicitly. In the revised manuscript we will add, at the beginning of Section 3, the standing assumption that the trajectory space is a Polish space (complete separable metric space). This is the natural setting for finite-horizon behaviors whose trajectories take values in Euclidean space. Under this hypothesis the support map is upper semicontinuous with respect to the weak topology, so any weak limit of measures supported on the closed admissible set B remains supported on B. The proof of Theorem 3.1 is therefore rigorous once the assumption is recorded; no change to the argument itself is required. revision: yes

  2. Referee: [§4, Proposition 4.2] §4, Proposition 4.2 (extreme points): while the claim that extreme points are exactly the Diracs follows from the definition of extremality in the simplex, the proof should explicitly invoke the relevant fact from convex analysis (e.g., that any non-Dirac measure is a nontrivial convex combination) rather than leaving it implicit.

    Authors: We thank the referee for this suggestion. In the revised manuscript we will make the invocation explicit in the proof of Proposition 4.2 by adding the standard fact that the extreme points of the simplex of probability measures are precisely the Dirac measures. Equivalently, any non-Dirac measure μ admits a nontrivial convex decomposition μ = λν₁ + (1-λ)ν₂ with 0 < λ < 1 and ν₁ ≠ ν₂ (obtained, for instance, by splitting the support into two sets of positive μ-measure). This renders the identification of the extreme points with the Dirac measures on admissible trajectories fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard measure theory

full rationale

The paper defines the behavioral-measure set as the collection of all Borel probability measures whose support lies inside the admissible trajectory set B. Convexity and weak closure then follow immediately from the definition of support (any convex combination of measures supported on B remains supported on B) and the fact that weak limits preserve support inclusion when B is closed in a Polish space. Extreme points being the Dirac measures is likewise a direct consequence of the geometry of the probability simplex. No equations reduce a claimed result to a fitted parameter or prior self-citation; the argument invokes only textbook facts about weak topology and extremal points. The construction is therefore self-contained against external mathematical benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard properties of Borel measures and the weak topology; the new object is the behavioral-measure set itself.

axioms (1)
  • standard math The trajectory space is equipped with a Borel sigma-algebra and the weak topology on probability measures is well-defined.
    Invoked to guarantee that the set of supported measures is weakly closed and that extreme points are Dirac measures.
invented entities (1)
  • Behavioral-measure set no independent evidence
    purpose: Convex set of probability measures representing the admissible behavior of a dynamical system.
    New object introduced to restore convexity for nonlinear dynamics.

pith-pipeline@v0.9.0 · 5488 in / 1297 out tokens · 66098 ms · 2026-05-09T14:10:28.724152+00:00 · methodology

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