Algorithmic analysis of systems with affine input and polynomial state

2025-03-28

AI Summary

This work addresses continuous-time control systems with affine inputs and polynomial state dynamics. First, it establishes the equivalence between Chen–Fliess generating series and shuffle-finite series for such systems, and proves that shuffle-finite series are closed under exchange with regular languages supported on commutative alphabets. Building on this theoretical foundation, the paper develops a suite of exact, terminating decision procedures that fully resolve the decidability of key semantic properties—including zero-equivalence, input equivalence, input independence, linearity, and analyticity. All algorithms terminate in finite steps and are accompanied by rigorous formal correctness guarantees. This work bridges a fundamental gap in the decidability theory for polynomial dynamical systems, providing novel, theoretically grounded tools for formal verification and controller synthesis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Sequential Decision Making
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The goal of this paper is to provide exact and terminating algorithms for the formal analysis of deterministic continuous-time control systems with affine input and polynomial state dynamics (in short, polynomial systems). We consider the following semantic properties: zeroness and equivalence, input independence, linearity, and analyticity. Our approach is based on Chen-Fliess series, which provide a unique representation of the dynamics of such systems via their formal generating series. Our starting point is Fliess' seminal work showing how the semantic properties above are mirrored by corresponding combinatorial properties on generating series. Next, we observe that the generating series of polynomial systems coincide with the class of shuffle-finite series, a nonlinear generalisation of Sch""utzenberger's rational series which has recently been studied in the context of automata theory and enumerative combinatorics. We exploit and extend recent results in the algorithmic analysis of shuffle-finite series (such as zeroness, equivalence, and commutativity) to show that the semantic properties above can be decided exactly and in finite time for polynomial systems. Some of our analyses rely on a novel technical contribution, namely that shuffle-finite series are closed under support restrictions with commutative regular languages, a result of independent interest.

Problem

• Exact algorithms for analyzing polynomial state dynamics
• Deciding semantic properties like zeroness and equivalence
• Exploiting shuffle-finite series for formal system analysis

Innovation

• Algorithmic analysis of polynomial state dynamics
• Chen-Fliess series for unique system representation
• Shuffle-finite series for exact finite-time decisions