  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 (dL). The key insight is to specify ODE stability by
suitably nesting the dynamic modalities of dL 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 dL's axioms with dL'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 dL.
