## Section: New Results

### Biological systems

We solved several problems of control and stability analysis for different types of biological models.

#### Control of continuous bioreactors

Participants : Frédéric Mazenc, Jiang Zhong Ping, Michael Malisoff.

The paper [29] is devoted to the persistence issue for chemostat models with an arbitrary number of species competing for a single limiting substrate. In a first part, we have shown that there exist fundamental limitations for the existence of nonlinear feedback control which ensure persistence of several species in some chemostats. More precisely, we have exhibited models for which there are some admissible bounded periodic trajectories for which there is no feedback control law guaranteeing their local asymptotic stability. In a second part, for models with an arbitrary number of species associated to growth rates of Monod type, we have shown that a dilution rate and input substrate time-varying nonlinear controllers can be designed so that a positive trajectory of the chemostat model becomes globally asymptotically stable. In this case, the designed control laws ensure persistence of all the species.

The paper [28] extends the results of [29] and [96] by designing a dilution rate and input substrate feedback controllers when only the substrate concentration is measured. More precisely, we achieved the coexistence by designing a novel output-feedback controller that globally asymptotically stabilizes a periodic reference state trajectory of the system. It is worth mentioning that, in practice, measuring the values of the concentration of each species is not feasible but measuring the substrate concentration is. Therefore, considering the substrate concentration as the output is a reasonable choice which is frequently made in the literature. The dynamic output feedback we proposed relies on an observer.

The stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem, and is of significant ongoing interest in bioengineering and population dynamics. In [32] (see also [58] ), we solved an output feedback stabilization problem for chemostat models having two species, one limiting substrate, and either Haldane or Monod growth functions. Our feedback stabilizers depend on a given linear combination of the species concentrations, which are both measured with a constant time delay. Our work is based on a Lyapunov-Krasovskii argument.

In [31] (see also [57] ), we studied feedback stabilization problems for chemostats with two species and one limiting substrate, which play an important role in systems biology and population dynamics. We constructed new dilution rate output feedbacks that stabilize a componentwise positive equilibrium, and only depend on the sum of the species levels. We proved that the feedbacks are robust to model uncertainty. The novelty and importance of our new contribution is in our dropping the usual assumption on the relative sizes of the growth yield constants, and our allowing uncertain uptake functions that are not necessarily concave. The proofs are based on classical results of ordinary differential equations (as for instance the Poincaré-Bendixson theorem and Dulac' criterion).

#### Study of a model of anaerobic digestion process

Participants : Frédéric Mazenc, Miled El Hajji, Jérôme Harmand.

In [24] , a mathematical model involving the syntrophic relationship of two major populations of bacteria (acetogens and methanogens), each responsible for a stage of the methane fermentation process is proposed. We carried out a detailed qualitative analysis. We performed the local and global stability analyses of the equilibria. Under general assumptions of monotonicity, we demonstrated relevant from an applied point of view, the global asymptotic stability of a positive equilibrium point which corresponds to the coexistence of acetogenic and methanogenic bacteria. The proofs are based on classical results of ordinary differential equations.

#### Control of a model of human heart

Participants : Frédéric Mazenc, Michael Malisoff, Marcio De Queiroz.

The control of human heart rate during exercise is an important problem that has implications for the development of protocols for athletics, assessing physical fitness, weight management, and the prevention of heart failure. In [55] , we provided a new stabilization technique for a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The control input is the treadmill speed, and the control objective is to make the heart rate track a prescribed reference trajectory. We used a strict Lyapunov function analysis to design new state and output feedback tracking controllers that render the error dynamics globally exponentially stable. This allowed us to prove robustness stability properties for our feedback stabilized systems relative to actuator errors.

#### Modelling of hematopoiesis with application to acute myeloid leukemia

Participants : José Luis Avila Alonso, Houda Benjelloun, Catherine Bonnet, Jean Clairambault [BANG Project-Team, INRIA Paris-Rocquencourt] , Jean-Pierre Marie [INSERM Paris (team 18 of UMR 872), Cordeliers Research Center ans St Antoine Hospital, Paris] , Faten Merhi [BANG Project-Team, INRIA Paris-Rocquencourt] , Hitay Özbay, Ruopping Tang [INSERM Paris (team 18 of UMR 872), Cordeliers Research Center ans St Antoine Hospital, Paris] .

We have worked on a nonlinear PDE model of hematopoeisis designed by Adimy and Crauste [76] , more precisely on its approximation by a nonlinear system with multiple distributed delays.

A complete stability analysis with therapeutic implications has been performed in [61] , [60] and [98] . Moreover, a simulation program of this model is already available.

Through the DIGITEO project ALMA, parameters of this model will be identified through experiments and the model will be changed in order to take more precisely into account some biological phenomena in hematopoiesis.