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Towards a Generic Interval Solver for Differential-Algebraic CSP

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Principles and Practice of Constraint Programming (CP 2020)

Abstract

In this paper, we propose an interval constraint programming approach that can handle the differential-algebraic CSP (DACSP), where an instance is composed of real and functional variables (also called dynamic variables or trajectories) together, and differential and/or “static” numerical constraints among those variables. Differential-Algebraic CSP systems can model numerous real-life problems occurring in physics, biology or robotics. We introduce a solver, built upon the Tubex and Ibex interval libraries, that can rigorously approximate the set of solutions of a DACSP system. The solver achieves temporal slicing and a tree search by splitting trajectories domains. Our approach provides a significant step towards a generic interval CP solver for DACSP that has the potential to handle a large variety of constraints. First experiments highlight that this solver can tackle interval Initial Value Problems (IVP), Boundary Value Problems (BVP) and integro-differential equations.

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Notes

  1. 1.

    An Initial Value Problem is composed of an ODE and an initial condition. Numerical integration propagates the initial value through the whole trajectory by integrating the evolution function of the ODE.

  2. 2.

    Note that a high-order problem can be transformed automatically into a first-order ODE shown in Definition 2 by introducing auxiliary variables. Also note that non autonomous ODEs of the form \(\dot{\textit{\textbf{x}}}(t)=\textit{\textbf{f}}\big (\textit{\textbf{x}}(t),t\big )\) can also be transformed into autonomous ODEs \(\dot{\textit{\textbf{x}}}(t)=\textit{\textbf{f}}\big (\textit{\textbf{x}}(t)\big )\) whose derivative depends only on the state.

  3. 3.

    The actual code is a little bit more complicated. An instant is skipped if it is handled by the previous integration step.

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Acknowledgements

This work was supported by the French Agence Nationale de la Recherche (ANR) [grant number ANR-16-CE33-0024].

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Rohou, S. et al. (2020). Towards a Generic Interval Solver for Differential-Algebraic CSP. In: Simonis, H. (eds) Principles and Practice of Constraint Programming. CP 2020. Lecture Notes in Computer Science(), vol 12333. Springer, Cham. https://doi.org/10.1007/978-3-030-58475-7_32

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  • DOI: https://doi.org/10.1007/978-3-030-58475-7_32

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