On the periodicity of symbolic observations of piecewise smooth discrete-time systems

The behavior of discrete-time systems composed of a set of smooth transition maps coupled by a `quantized´ feedback function is studied. The feedback function partitions the state space into disjoint regions, and assigns a smooth transition function to reach region. The main result is that under a constraint on the norm of the derivative of the transition maps, and an assumption on the limit points of the state trajectory, every bounded state trajectory leads to a region index sequence that is eventually periodic