A Measure-Theoretic Formulation of Behavioral Systems

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 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.