On determining unknown functions in differential systems, with an application to biological reactors

Busvelle, Éric; Gauthier, Jean-Paul

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 509-551.

In this paper, we consider general nonlinear systems with observations, containing a (single) unknown function $\varphi$. 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 $\varphi$). 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 general no hope to do something better than using “approximate differentiators”, as show very elementary examples. However, a practical methodology is proposed in some cases. It shows very reasonable performances. As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, [2, 4], in which a population is fed by some substrate. The model heavily depends on a “growth function”, expressing the way the population grows in presence of the substrate. The problem is to identify this “growth function”. We give several identifiability results, and identification methods, adapted to this problem.

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