Differential Equation Inductive Robustness Axiomatization

2026-06-17

AI Summary

This work addresses the problem of deciding robust safety for dynamical systems governed by polynomial differential equations over bounded time horizons. By reducing δ-robust safety to the sound axiomatization of polynomial invariants, the authors construct the first complete logical proof system and integrate it with a computable algorithm to decide safety for arbitrary perturbation parameters δ. Innovatively leveraging subanalytic geometry, the approach enables inductive safety proofs and approximate decidability for general hybrid dynamical systems without requiring positive separation between initial and postconditions. This paper establishes, for the first time, a complete axiomatization framework for robust safety and provides an effective symbolic verification algorithm.

Abstract

This article establishes the completeness of an axiomatization for the robust safety of dynamical systems with polynomial differential equations on bounded time horizons. Safety properties of robust systems are uniformly reduced to a sound axiomatization of polynomial invariants, resulting in reliable logical proofs of correctness. Approximate decidability results are also established: there is a computable algorithm such that, given any perturbation parameter $δ$, it either produces a symbolic proof of robust safety (hence correctly decides the dynamical system to be robustly safe), or correctly decides that the system is not robustly safe under a perturbation of level $δ$. In contrast to earlier works, this article crucially leverages results from subanalytic geometry to retain a level of exactness, thereby establishing positive results of provability/decidability allowing for arbitrary bounded (semialgebraic) initial/post conditions even without positive separation at their (topological) boundaries. This enables the generation of proofs of inductive safety beyond finite time horizons for general hybrid dynamical systems.

Problem

• robust safety
• polynomial differential equations
• axiomatization
• hybrid dynamical systems
• inductive invariants

Innovation

• robust safety
• polynomial differential equations
• axiomatization
• subanalytic geometry
• hybrid dynamical systems