A Measure-Theoretic Formulation of Behavioral Systems
Victor M. Preciado

TL;DR
This paper introduces a measure-theoretic approach to behavioral systems, representing systems with probability measures on trajectories to handle nonlinear and stochastic cases more effectively.
Contribution
It extends Willems' behavioral systems theory by lifting the framework to probability measures, enabling convex analysis of nonlinear and stochastic systems.
Findings
Convex set of probability measures supports nonlinear/deterministic systems analysis.
Extreme points are Dirac measures on individual trajectories.
Framework embeds classical deterministic theory as extremal case.
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…
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