Reference : Tracking Control in Billiards Using Mirrors without Smoke, Part I: Lyapunov-Based Local ...
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Tracking Control in Billiards Using Mirrors without Smoke, Part I: Lyapunov-Based Local Tracking in Polyhedral Regions
Forni, Fulvio[Université de Liège - ULg > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation >]
Teel, Andrew[UC Santa Barbara, CA > Electrical and Computer Engineering (ECE) > > >]
Zaccarian, Luca[University of Rome, Tor Vergata > Dipartimento di Informatica, Sistemi e Produzione > > >]
50th IEEE Conference on Decision and Control (cdc 2011)
50th IEEE Conference on Decision and Control
[en] tracking ; impacts ; hybrid systems
[en] In this paper we formulate the tracking problem of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous dynamics arising from nonsmooth impacts, the tracking problem is formulated within a hybrid systems framework and a Lyapunov function is given, which decreases during flow (continuous motion) and remains constant across jumps (impacts of the masses). To guarantee non-increase of the Lyapunov function at jumps, we introduce a novel concept of mirrored images of the target mass and prove that, with this concept, local tracking is achieved. Several simulations illustrate the effectiveness of the proposed approach as compared to alternative solutions. In a companion paper  we address global results and generalize the local approach to curved billiards.