The probabilistic convolution regularization of Zeno hybrid systems

Zeno hybrid systems exhibit an infinite number of switches between different configurations in finite time. This is the result of the abstraction of modeling and the assumption that the switching is instantaneous. Simulation of such systems will generally stop or give false results after a finite time, called the Zeno time. In reality, physical systems do not exhibit such behavior, because the switching is not in reality instantaneous, but also often because they are subject to a small amount of noise. This noise makes possible the continuous evolution of the real systems even if extremely fast switching arises. Thus, the problem is how to predict the behavior of the system after the Zeno time. In this paper a new technique is proposed to extend simulations beyond the Zeno time. It consists of a probabilistic regularization by a convolution method that includes noise in the system. For example, low-amplitude Gaussian noise can be used to transform discontinuous vector fields to continuous vector fields. This convolution approach makes the value observed for the continuous evolution over time slightly different from what would be predicted without the noise, but in a way that should correspond to what happens in reality. An example of a Zeno hybrid system is employed to illustrate the result. The evolution beyond the Zeno time varies depending on the noise that exists in physical systems and on how the noise is included in the regularized model. However, a useful idealization is obtained in the limit of low noise amplitude.