A Measure-Theoretic Formulation of Behavioral Systems

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.

• The paper introduces the behavioral-measure set, a convex, weakly closed set of probability measures capturing all admissible trajectories.
• It extends the Fundamental Lemma to data-driven control, enabling optimal control via occupation measures and semidefinite approximations.
• The framework generalizes deterministic behavior and provides a unified approach for robust analysis in nonlinear and stochastic systems.

Measure-Theoretic Behavioral Systems: A Convex Invariant Framework

Overview and Motivation

This paper presents a measure-theoretic extension of Willems' behavioral theory, providing a rigorous framework for representing dynamical systems—deterministic, stochastic, linear, or nonlinear—via probability measures on trajectory spaces. The classical behavioral framework identifies a system with the set of all admissible trajectories, which for LTI (Linear Time-Invariant) systems forms a linear subspace. However, for nonlinear or stochastic systems, the admissible set is nonconvex, limiting the applicability of convex optimization and data-driven methods. The central contribution here is the formulation and analysis of the behavioral-measure set: the convex, weakly closed set of all Borel probability measures supported on admissible trajectories. This construction embeds the classical deterministic behavior as an extremal subset and extends powerful behavioral results—including the Fundamental Lemma—to the measure-theoretic and stochastic domains.

Behavioral-Measure Set: Formalism and Properties

The behavioral-measure set $\mathcal{M}_{\mathcal{B}}$ is defined as the set of Borel probability measures supported entirely on admissible system trajectories. For a finite horizon $T$, with Polish state, input, and output spaces $(X, U, Y)$, and dynamics governed by continuous functions $f$ and $h$, the product trajectory space is $\Omega_T = X^{T+1} \times U^T \times Y^T$, and admissible paths constitute a closed subset $\mathfrak{B}_T$. The set of interest is $\mathcal{M}_{\mathcal{B}} := P(\mathfrak{B}_T)$, where $P(S)$ denotes all Borel probability measures on $S$.

The principal structural results established for $T$0 are:

• Convexity and Weak Closedness: Both the full set and its slices for fixed initial laws are convex and weakly closed in $T$1.
• Extreme Point Characterization: The extreme points of $T$2 are precisely the Dirac measures concentrated on admissible trajectories, so the classical deterministic behavior is fully recoverable.
• System-Theoretic Invariance: The set is defined solely by the system dynamics, independent of cost, constraints, or optimization problems posed over it.
• Operator Identities: Membership in $T$3 enforces a family of weak operator (moment) identities, which, under polynomial dynamics, admit semidefinite outer approximations (Lasserre hierarchy).

This formalism generalizes previous occupation-measure and relaxed-control frameworks, which operate at the per-step or marginal level, in that it retains full trajectory-level couplings essential for distributions and data-driven results.

Occupation-Measure Optimal Control and Bellman Duality

Optimal control problems are posed as linear programming problems over the behavioral-measure set (or its slice for a prescribed initial law $T$4). For cost functionals linear in the occupation measures (i.e., expected stage and terminal costs), strong duality is achieved, and the dual problem is the well-known Bellman dynamic-programming recursion. Under compactness and continuity conditions, strong duality holds:

• Existence of Solutions: Compactness of underlying spaces or tightness via moment bounds guarantees that optimal occupation measures exist.
• Policy Extraction: Any solution can be disintegrated into stochastic feedback policies, with deterministic selectors attainable via measurable selection results; randomized strategies confer no cost advantage in this context.
• Cost Independence: The feasible set is independent of the objective, so structural results are universally applicable.

This extends classical Markov decision process theory to the measure-theoretic behavioral context.

Data-Driven Control: Measure-Theoretic Fundamental Lemma

A key contribution is the measure-level Fundamental Lemma for controllable LTI systems, which generalizes the classical result due to Willems and underpins data-driven control methodologies:

• Hankel Factorization of Measures: Every probability measure supported on the LTI system's finite-horizon behavior is the pushforward of a coefficient-space measure via the data Hankel matrix.
• Equivalence for Optimization: Any expectation, moment, or distributional constraint on trajectories transfers to the coefficient space; all data-driven and distributional DeePC-like methods are captured as special cases.
• Reduction to Data: No structural knowledge beyond a single persistently exciting trajectory is required—remarkably, distributional optimal control and validation are fully determined by this data.
• Higher-Order Moment Structure: Means, covariances, and higher moments of trajectory-level distributions map exactly to corresponding coefficient-space moments via linear transformations dictated by the Hankel matrix.

This generalization clarifies the limitations of occupation-based approaches and demonstrates the precise scope and power of data-driven control, especially in the presence of uncertainty or ensemble objectives.

Stochastic Extensions

The measure-theoretic framework admits an extension to stochastic systems modeled by Feller transition kernels. The behavioral-measure set is then characterized by conditions on the consistency of regular conditional distributions (not marginal distributions alone), ensuring preservation of full trajectory laws:

• Convexity and Weak Closedness: These properties survive under the appropriate Feller/continuity conditions on kernels and outputs.
• Future Directions: Extension of results such as the extremal structure and measure-level Fundamental Lemma in this stochastic context is identified as an open research avenue.

Prior polynomial-chaos behavioral approaches only partially address these issues, as they do not retain full trajectory coupling at the measure level.

Numerical Validation

The proposed framework is validated via several low-dimensional experiments:

• Polynomial Nonlinear System: Moment-SOS relaxations show that semidefinite approximations converge rapidly to the feasible moment set, validating the utility for polynomial dynamics.
• Nonlinear Control Synthesis: The approach yields certified global optimality in short-horizon scenarios, outperforming local approximations and demonstrating tractable policy extraction.
• LTI Data Validation: The measure-level Fundamental Lemma is confirmed empirically up to second-order moments, using data from SISO systems and validating covariance transfer identities to numerical precision.

Non-polynomial systems can be approached via polynomial approximations, empirical/sampling approaches, or Koopman-theoretic lifting for which the measure-level results remain directly applicable.

Implications and Perspectives

The measure-theoretic behavioral framework fundamentally enhances system representation in several ways:

• Universal Convexification: Nonlinear and stochastic control problems can be reformulated as convex optimizations over measure sets, expanding computational and theoretical tractability.
• General Data-Driven Control: Exact data-driven optimization over distributions, not just trajectories, is made possible without model knowledge—deepening the reach of modern data-driven control.
• Trajectory Law Invariance: By preserving full trajectory-level coupling (beyond marginal and per-step statistics), this approach retains the requisite structure for invariant, compositional, and hierarchical system analysis, supporting interconnections and modularity.

Future research should further extend these results to infinite-horizon, continuous-time, and general stochastic systems, including characterizing the structure of the extremal measures and lifting the measure-level Fundamental Lemma to broader classes of stochastic systems. The computational aspects—especially scaling semidefinite and data-driven algorithms—form another critical direction.

Conclusion

This paper provides a rigorous measure-theoretic foundation for behavioral systems, unifying deterministic, stochastic, linear, and nonlinear systems under a convex, invariant framework. The resulting theory bridges classical behavioral analysis, occupation-measure optimal control, and modern data-driven methods, supporting both theoretical insight and practical computational methods for contemporary systems and control challenges.

Reference: "A Measure-Theoretic Formulation of Behavioral Systems" (2605.01558)