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.
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.
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.
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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