Tracking Control in Billiards Using Mirrors without Smoke, Part II: Additional Lyapunov-Based Local and Global Results
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) > > >]
50th IEEE Conference on Decision and Control (cdc 2011)
50th IEEE Conference on Decision and Control
[en] tracking ; impacts ; hybrid systems
[en] Two control results are described: 1) local tracking control for convex billiards with piecewise locally Lipschitz boundary, and 2) global tracking control for special polyhedral billiards, including rectangles and equilateral triangles. The controllers are based on Lyapunov functions and a mirroring concept introduced in a companion paper. The local results require the impacts to satisfy an average dwell-time condition with parameters that depend on the Lipschitz constant of the function that characterizes the boundary. For piecewise constant boundary, and for the global results, the average dwell-time parameters are arbitrary. Tools from stability analysis for hybrid systems are used to establish the results.