Deductive Stability Proofs for Ordinary Differential Equations

Abstract

Stability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from ’s axioms with ’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on .

Differential Elimination and Algebraic Invariants of Polynomial Dynamical Systems

“…If an invariant does not intersect the unsafe set, then every state in the invariant set is a safe starting point. Beyond safety [4], invariant reasoning is an important part of other significant properties like stability [5] and liveness [6]. Due to its significance, the problem of invariant generation for ODEs has received substantial interest [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and even has a dedicated tool [23], but most methods have nontrivial heuristic search parts.…”

Implicit and Explicit Proof Management in KeYmaera X

“…Correctness is easier to achieve with an LCF-style approach that strips the soundness-critical core to the bare minimum, but, as a consequence, proof convenience has to be regained outside the soundness-critical core with proof management techniques. The KeYmaera X [5] theorem prover for hybrid systems takes an LCF-style approach; previous techniques expanded the capabilities of KeYmaera X primarily by providing tactics [4], e.g., for certifying solutions of differential equations [16], for certifying safety and liveness properties of differential equations [19,21], for stability proofs [22], for code synthesis [3], for component-based modeling and verification [12], and for monitor synthesis [10].…”