# Set-valued Euler's method with interpolated distance functions and optimal control solvers

R. Baier:
**Set-valued Euler's method with interpolated distance functions and optimal control solvers**

in:
*Proceedings on the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), July 9-13, 2012, Melbourne, Australia,*
*
*
Melbourne, Australia,
2012,
4 pages,
CD-ROM, Paper No. 85, extended abstract

ISBN/ISSN/ISMV Nummer: 978-0-646-58062-3

Keywords: set-valued Euler's method; reachable sets; distance functions; differential inclusions; optimal control

Mathematics Subject Classification Code: 34A60 (93B03 49M25 65D18 65L06 49J15)

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## Abstract:

For a given nonlinear control system resp. a differential inclusion, the reachable set at a given end time collects the end points of all admissible solutions. There are various applications, e.g. in collision avoidance in safety systems, climate research, motion planning or population models.

Besides level-set methods and solutions to the Hamilton-Jacobi-Bellman partial differential equation, set-valued Runge-Kutta methods such as Euler's method are studied for approximating reachable sets. To be able to guarantee the overall order of convergence 1 for Euler's method with respect to step-size, each set iterate of Euler's method must be discretized up to an error *O(h ^{2})*. To reduce the computational complexity and high memory consumption, distance functions are used.

The set operations applied in Euler's method can be easily expressed via distance functions. To avoid the state-space discretization of order *O(h ^{2})*, the spatial piecewise linear interpolation of the distance function based on its values on a coarse state-space grid with distance

*O(h)*is evaluated for intermediate grid points on a fine grid with distance

*O(h*. First numerical tests still indicate order of convergence 1 and a considerable speedup.

^{2})A second approach formulates an appropriate optimal control problem in which the distance function from an arbitrary grid point to the reachable set appears in the optimal value. Discretization via Euler's method and varying the grid point leads to discrete reachable sets avoiding completely the state-space discretization of *O(h ^{2})*. Numerical test examples indicate good performance even for higher state-space dimension and lower-dimensional projections of the reachable set.