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Decoupling Abstractions of Non-linear Ordinary Differential Equations

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FM 2016: Formal Methods (FM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9995))

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Abstract

We investigate decoupling abstractions, by which we seek to simulate (i.e. abstract) a given system of ordinary differential equations (ODEs) by another system that features completely independent (i.e. uncoupled) sub-systems, which can be considered as separate systems in their own right. Beyond a purely mathematical interest as a tool for the qualitative analysis of ODEs, decoupling can be applied to verification problems arising in the fields of control and hybrid systems. Existing verification technology often scales poorly with dimension. Thus, reducing a verification problem to a number of independent verification problems for systems of smaller dimension may enable one to prove properties that are otherwise seen as too difficult. We show an interesting correspondence between Darboux polynomials and decoupling simulating abstractions of systems of polynomial ODEs and give a constructive procedure for automatically computing the latter.

This work was supported by the Air Force Research Laboratory (AFRL) through contract number FA8750-15-1-0105 and the Air Force Office of Scientific Research (AFOSR) under contract numbers FA9550-15-1-0258 and FA9550-16-1-0246.

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Notes

  1. 1.

    By this we understand a finite expression in terms of polynomials and elementary functions such as \(\sin ,\cos ,\exp ,\ln \), etc.

  2. 2.

    i.e. p is a first integral if \(\mathfrak {L}_f(p) = \lambda p\) where \(\lambda =0\).

  3. 3.

    When q is a constant, the Darboux polynomial is trivial [11, Definition 2.14]. In this paper we will generally be interested in the non-trivial case.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading and judicious critique and extend their thanks to Dr. André Platzer at Carnegie Mellon University for his technical questions and helpful insights into differential ghosts.

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Correspondence to Andrew Sogokon or Khalil Ghorbal .

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Sogokon, A., Ghorbal, K., Johnson, T.T. (2016). Decoupling Abstractions of Non-linear Ordinary Differential Equations. In: Fitzgerald, J., Heitmeyer, C., Gnesi, S., Philippou, A. (eds) FM 2016: Formal Methods. FM 2016. Lecture Notes in Computer Science(), vol 9995. Springer, Cham. https://doi.org/10.1007/978-3-319-48989-6_38

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