Generic Observability of Differentiable Systems

A dynamical system consists of a smooth vectorfield defined on a differentiable manifold, and a smooth mapping from the manifold to the real numbers. The vectorfield represents the dynamics of a physical system. The mapping stands for a measuring device by which experimental information on the dynamics is made available. The information itself is modeled as a sampled version of the image of the state trajectory under the smooth mapping. In this paper the observability of this set-up is discussed from the viewpoint of genericity. First the observability property is expressed in terms of transversality conditions. Then the theory of transversal intersection is called upon to yield the desired results. It is shown that almost any measuring device will combine with a given physical system to form an observable dynamical system, if (2n + 1) samples are taken and not fewer, where n is the dimension of the manifold. Dually, it is shown that almost any physical system will combine with a given measuring device to form an observable dynamical system, if (2n + 1) samples are taken and not fewer. The analysis leads to the corollary that for nonlinear systems observability is a generic property, a fact well known for linear systems. The relation of the theory to the study of turbulence and to control theory is explained.