What are hybrid dynamical systems and how are they controlled?

Hybrid systems combine continuous dynamics with discrete events (switching, logic, resets). Examples include bouncing ball, thermostat, gear shifting, and supervisory control. Modeling uses hybrid automata, switched systems, or piecewise affine systems. Stability analysis uses multiple Lyapunov functions, dwell-time conditions, or common Lyapunov functions. Control design includes switched controller synthesis, optimal switching, and MPC for hybrid systems. Challenges include Zeno behavior (infinite switches in finite time), stability under arbitrary switching, and computational complexity of hybrid MPC.