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Section: New Results

Systems with pointwise delays

Participants : Frédéric Mazenc, Michael Malisoff [LSU] , Robledo Gonzalo [Univ. de Chile, Chile] , Silviu Niculescu.

Frequently, the presence of delays is an obstacle to the stability analysis or the control of systems. Two of our works originate in the will to overcome this obstacle, in two distinct contexts.

The contribution [26] is devoted to the study of a model of a chain of two bio-reactors called 'chemostats'. One contains two microbial species in competition for a single limiting nutrient and receives an external input of the less advantaged competitor, which is cultivated in the other one. Pointwise delays are present. Under a condition on their size, we obtained sufficient conditions ensuring coexistence of all the species in competition. To prove the result, we adopted a Lyapunov based technique.

The contribution of [37] is twofold. In a first part, we exhibited a fundamental feature of the systems with a pointwise periodic time-varying delay: we have shown by a counterexample that the asymptotic stability of such a system cannot be deduced from the average value of the delay (even when the delay is 'rapidly' varying). In a second part, motivated by this counterexample, we proposed a new representation of systems with time-varying delays, which is helpful to carry out stability analyses and to develop a new state feedback stabilization method.