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Linearization, Model Reduction and Reachability in Nonlinear odes

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Reachability Problems (RP 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13608))

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Abstract

In the analysis of nonlinear ordinary differential equations (odes), linear and Taylor approximations are fundamental tools. Such approximations are generally accurate only in a local sense, that is near a given expansion point in space or time. We study conditions and methods to compute linear approximations of nonlinear odes that are accurate also non locally. Relying on Carleman linearization and Krylov projection, our method yields a small, hence tractable linear system that is shown to produce accurate approximate solutions, under suitable stability conditions. In the general, possibly non stable case, we provide an algorithm that, given an initial set and a finite time horizon, builds a tight overapproximation of the reachable states at specified times. Experiments conducted with a proof-of-concept implementation have given encouraging results.

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Notes

  1. 1.

    This can be weakened to analyticity in some open set containing all the trajectories \(x(t;x_0)\) for \(x_0\in \varOmega \).

  2. 2.

    For an introduction to Krylov spaces, see e.g. [27].

  3. 3.

    An explicit expression for \(y_1\) is:

    \(y_1(t;y_0)= -1/3\,{\textrm{e}^{t/2}} ( \sqrt{3} ( { x_1}-2\,{ x_2} ) \sin ( \frac{1}{2}\,t\sqrt{3} ) -3\,{ x_1}\,\cos ( \frac{1}{2}\,t\sqrt{3} ) ) \).

  4. 4.

    That is, all eigenvalues of \(H_m\) have a nonnegative real part and every imaginary eigenvalue, if any, has geometric multiplicity equal to the algebraic one. See e.g. [16, pp. 135–136].

  5. 5.

    The method can be extended without much difficulty to more sophisticated and scalable types of sets, like zonotopes.

  6. 6.

    These are useful in case one wants a flowpipe encapsulating the flow \(x(t;x_0)\) for all t’s in a given interval, not only at specified time points \(t_k\)’s.

  7. 7.

    More precisely: , \( b_1=(-0.11067,-0.10006,0.20011,0.22134)^T\) and \(\eta _1=(-0.11065,-0.10002,0.20015,0.22142)^T\).

  8. 8.

    Code and examples available at https://github.com/Luisa-unifi/CKR.

  9. 9.

    It should be noted, though, that it makes little sense to compare a proof-of-concept implementation with highly optimized tools in this respect. At any rate, all execution times are below 100 seconds.

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Authors and Affiliations

  1. Università di Firenze, Dipartimento di Statistica, Informatica, Applicazioni, V.le Morgagni 65, 50134, Florence, IT, Italy

    Michele Boreale & Luisa Collodi

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  1. Michele Boreale
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Correspondence to Michele Boreale .

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Editors and Affiliations

  1. Department of Computer Science, TU Kaiserslautern, Kaiserslautern, Germany

    Anthony W. Lin

  2. Max Planck Institute for Software Systems, Kaiserslautern, Germany

    Georg Zetzsche

  3. University of Liverpool, Liverpool, UK

    Igor Potapov

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Boreale, M., Collodi, L. (2022). Linearization, Model Reduction and Reachability in Nonlinear odes. In: Lin, A.W., Zetzsche, G., Potapov, I. (eds) Reachability Problems. RP 2022. Lecture Notes in Computer Science, vol 13608. Springer, Cham. https://doi.org/10.1007/978-3-031-19135-0_4

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