Control Closure Certificates

2025-08-05

AI Summary

This paper addresses controller synthesis for discrete-time control systems under ω-regular specifications—including Linear Temporal Logic (LTL)—by introducing the *Control Closure Certificate* (CCC), a novel certificate framework. The CCC generalizes traditional transition invariants to functional forms and integrates disjunctive well-foundedness arguments, enabling unified characterization of both infinite- and finite-visit constraints over the product space of the system and a parity automaton. Synthesis of CCCs is automated via a combination of Lyapunov-like functions, barrier certificates, ranking functions, and sum-of-squares optimization. Experimental results demonstrate that the proposed framework efficiently synthesizes controllers satisfying complex temporal specifications, significantly improving scalability and automation in controller synthesis for ω-regular objectives.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

This paper introduces the notion of control closure certificates to synthesize controllers for discrete-time control systems against $ω$-regular specifications. Typical functional approaches to synthesize controllers against $ω$-regular specifications rely on combining inductive invariants (for example, via barrier certificates) with proofs of well-foundedness (for example, via ranking functions). Transition invariants, provide an alternative where instead of standard well-foundedness arguments one may instead search for disjunctive well-foundedness arguments that together ensure a well-foundedness argument. Closure certificates, functional analogs of transition invariants, provide an effective, automated approach to verify discrete-time dynamical systems against linear temporal logic and $ω$-regular specifications. We build on this notion to synthesize controllers to ensure the satisfaction of $ω$-regular specifications. To do so, we first illustrate how one may construct control closure certificates to visit a region infinitely often (or only finitely often) via disjunctive well-founded arguments. We then combine these arguments to provide an argument for parity specifications. Thus, finding an appropriate control closure certificate over the product of the system and a parity automaton specifying a desired $ω$-regular specification ensures that there exists a controller $κ$ to enforce the $ω$-regular specification. We propose a sum-of-squares optimization approach to synthesize such certificates and demonstrate their efficacy in designing controllers over some case studies.

Problem

• Synthesize controllers for discrete-time systems with ω-regular specifications
• Use control closure certificates for disjunctive well-foundedness arguments
• Automate verification via sum-of-squares optimization for parity specifications

Innovation

• Control closure certificates for ω-regular specifications
• Disjunctive well-founded arguments for control synthesis
• Sum-of-squares optimization for certificate synthesis