## Section: New Results

### Reduction model approach: new advances

Participants : Frédéric Mazenc [correspondent] , Michael Malisoff [Louisiana State University] , Silviu Iulian Niculescu, Dorothé Normand-Cyrot [L2S, CNRS] .

We solved several distinct problems entailing to the celebrated reduction model approach. Let us recall that this technique makes it possible to stabilize systems with arbitrarily large pointwise or distributed delay in the input.

1) We proposed in [25] a new construction of exponentially stabilizing sampled feedbacks for continuous-time linear time-invariant systems with an arbitrarily large constant pointwise delay in the inputs. Stability is guaranteed under an assumption on the size of the largest sampling interval. The proposed design is based on an adaptation of the reduction model approach. The stability of the closed loop systems is proved through a Lyapunov-Krasovskii functional of a new type, from which is derived a robustness result

2) The paper [59] presents several results pertaining to the stabilization with feedbacks given by an explicit formula of linear time varying systems in the case where there is a constant delay in the input. In addition, it establishes input-to-state stability with respect to additive uncertainties. As a particular case, we considered a large class of rapidly time varying systems and provided a lower bound on the admissible rapidness parameters. We illustrated our results using a pendulum model.

3) The paper [24] , which is devoted to the original problem of stabilizing nonlinear systems with input with distributed delay, is actually not an extension of the reduction model approach, but it complements it and uses operators which have been inspired by those used in the classical context of the reduction model theory.