ESAIM: Control, Optimisation and Calculus of Variations Will be set by the publisher DOI: (will be inserted later) ON DETERMINING UNKNOWN FUNCTIONS IN DIFFERENTIAL SYSTEMS, WITH AN APPLICATION TO BIOLOGICAL REACTORS.

Abstract. In this paper, we consider general nonlinear systems with observations, containing a (sin-gle) unknown function ϕ. We study the possibility to learn about this unknown function via the observations: if it is possible to determine the [values of the] unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, we say that the system is “identifiable”. For systems without controls, we give a more or less complete picture of what happens for this identifiability property. This picture is very similar to the picture of the “observation theory ” in [7]: 1. if the number of observations is three or more, then, systems are generically identifiable; 2. if the number of observations is 1 or 2, then the situation is reversed. Identifiability is not at all generic. In that case, we add a more tractable infinitesimal condition, to define the “infinitesimal identifiability ” property. This property is so restrictive, that we can almost characterize it (we can characterize it by geometric properties, on an open-dense subset of the product of the state space X by the set of values of ϕ). This, surprisingly, leads to a non trivial classification, and to certain corresponding “identifiability normal forms”. Contrarily to the case of the observability property, in order to identify in practice, there is in genera