Approximately bisimilar abstractions of incrementally stable finite or infinite dimensional systems

In this paper, we establish a certain number of results for abstraction of a class of incrementally stable dynamical systems, in the framework of approximate bisimulation. Our approach does not rely on a discretization of the state space, it is therefore applicable indifferently to finite dimensional systems such as those modeled by differential equations, or infinite dimensional systems, such as those modeled by time-delay or partial differential equations. Our first result states that the sampled dynamics of an incrementally stable dynamical system is approximately bisimilar to a family of finite dimensional systems; this is of particular interest for infinite dimensional dynamical systems. The second result shows that these finite dimensional systems admit approximately bisimilar symbolic abstractions. In both cases, any precision can be achieved either by increasing the dimension or the number of states of the abstractions.