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Deductive Stability Proofs for Ordinary Differential Equations

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12652)


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.


  • differential equations
  • stability
  • differential dynamic logic

This research was sponsored by the AFOSR under grant number FA9550-16-1-0288.

The first author was supported by A*STAR, Singapore.


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Tan, Y.K., Platzer, A. (2021). Deductive Stability Proofs for Ordinary Differential Equations. In: Groote, J.F., Larsen, K.G. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2021. Lecture Notes in Computer Science(), vol 12652. Springer, Cham.

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