Multi-dimensional multi-mode systems: Structure and optimal control

The dynamics of multi-mode multi-dimensional (M
³
D) hybrid systems is described. Such systems have modes of different dimensions and the notion of a state space needs to be carefully defined. Also the behavior at the transitions between modes is particular, and may be actively controlled (exo-M
³
D), or passively switched via a fixed switching surface (auto-M
³
D). The optimal timing (switching) control for such multi-mode multi-dimensional (M
³
D) systems is solved. Necessary conditions for a stationary solution are derived and shown to extend those of the equal dimension case. We illustrate the problem with the optimal control for a spring assisted high jump (aka man on a trampoline).