Research interests

My master of science deals with the construction of observers for dynamical systems modeling a bioreactor in two dimension when a function of the state variables is bad known that is often the case in biology. A new tool linking classical observer as the high gain and the asymptotic one is built.

First, a one dimensional bounded error observer is built: the error between observer and initial system dis assumed not tend to zero. The main problem of this bounded observer is that we don't have any way to improve the convergence rate. Thus, a two dimensional bounded error observer, using the classical methods of Gauthier Kupka is constructed. The observed error become  as small as we want with an adjustable convergence rate at the beginning when the observed error is large and with a fixed convergence rate when this error is less than a fixed constant. A generalization in higher dimension is done for particular dynamical system.

<>Another research interest is cell growth modeling in a chemostat. For particular cells (phytoplankton), oscillatory behavior is observed (Nisbet and Gurney). The main mathematical models in the literature that reproduce this phenomenon are always complex. Indeed, their mathematical analysis is not very simple since their dimension is too high or the mathematical approach (PDE, Pascual and Caswell) does not allow some analytical studies.
Then, we built two three dimensional models in ODE based on the main biological phenomena. This mechanist approach called sometimes "compartmental" approach gives simple ODE model which can be studied analytically. In my thesis for example, the growth and the division are taken into account to describe cell cycle. Different units are chosen to describe more precisely some phenomenon. Two models are built considering either the cell biomass or the number of the cell. With the number of the cell,  cell division is more specified: one mother cell gives two daughter cells. We prove existence of limit cycle for the model in number and existence of a globally asymptotic steady state for the model in biomass. Then a mechanist approach some complex behavior can be reproduced.
To describe with more accuracy the cell cycle, we added a constant mortality due to the inner cell mechanisms oldness. The same behavior mathematical behavior is obtained. This models are generalized in higher dimension using the slow fast method. The study of this the n-dimensional models is easy since reduced model in 3-dimensional are obtained and this systems are the same than the above one.

With our mechanist approach, another growth cell model espacilly for phytoplankton is built considering stored nutrient in a cell. Indeed, Droop described a model which take into account the store average of the cell. This model based on biological experiences reproduce with accuracy the phytoplankton growth but the main biological phenomena can not be put into relief. Then, a biochemically based model is constructed and all the variables have a biological meaning. The mathematical analysis prove the global asymptotic stability of the non trivial steady state. A comparison with the classical model (Droop) is done and we prove that both are equivalent. Then with different approaches (compartmental and based on biological data), the same behavior is put into relief. Nevertheless, our model gives more dynamical informations and justifies a model based on biological data.