Perturbed Turing machines and hybrid systems

Investigates the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider models for both discrete- and continuous-time dynamical systems: Turing machines, piecewise affine maps, linear hybrid automata and piecewise-constant derivative systems (a simple model of hybrid systems). We associate with each of these models a notion of perturbed dynamics by a small ε (w.r.t. to a suitable metric), and define the perturbed reachability relation as the intersection of all reachability relations obtained by ε-perturbations, for all possible values of ε. We show that, for the four kinds of models we consider, the perturbed reachability relation is co-recursively enumerable (co-r.e.), and that any co-r.e. relation can be defined as the perturbed reachability relation of such models. A corollary of this result is that systems that are robust (i.e. whose reachability relation is stable under infinitesimal perturbation) are decidable