Analytical and numerical estimates of reachable sets in a subdivision scheme

Titelangaben

Riedl, Wolfgang ; Baier, Robert ; Gerdts, Matthias:
Analytical and numerical estimates of reachable sets in a subdivision scheme.
Mathematisches Institut, Universität Bayreuth; Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr in Neubiberg/München
Bayreuth, Neubiberg/München , 2017 . - 15 S.

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Abstract

Reachable sets for (discrete) nonlinear control problems can be described by feasible sets of nonlinear optimization problems. The objective function for this problem is set to minimize the distance from an arbitrary grid point of a bounding box to the reachable set. To avoid the high computational costs of starting the optimizer for all points in an equidistant grid, an adaptive version based on the subdivision framework known in the computation of attractors and invariant measures is studied. The generated box collections provide over-approximations which shrink to the reachable set for a decreasing maximal diameter of the boxes in the collection and, if the bounding box is too pessimistic, do not lead to an exploding number of boxes as examples show. Analytical approaches for the bounding box of a 3d funnel are gained via the Gronwall-Filippov-Wazewski theorem for differential inclusions or by choosing good reference solutions. An alternative self-finding algorithm for the bounding box is applied to a higher-dimensional kinematic car model.