Section: New Results

Reduction model approach: new advances

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 delays in the input.

1) In [46] , solutions to the problem of globally exponentially stabilizing linear systems with an arbitrarily long pointwise delay with sampled feedbacks are presented. The main result of a contribution by F. Mazenc and D. Normand-Cyrot is recalled and compared with other results available in the literature.

2) We considered in [41] a stabilization problem for continuous-time linear systems with discrete-time measurements and a sampled input with a pointwise constant delay. In a first step, we constructed a continuous-discrete observer which converges when the maximum time interval between two consecutive measurements is sufficiently small. We also constructed a dynamic output feedback through a technique which is strongly reminiscent of the reduction model approach. It stabilizes the system when the maximal time between two consecutive sampling instants is sufficiently small. No limitation on the size of the delay was imposed.

3) In [43] , we studied a general class of nonlinear systems with input delays of arbitrary size. We adapted the reduction model approach to prove local asymptotic stability of the closed loop input delayed systems, using feedbacks that may be nonlinear. We determined estimates of the basins of attraction for the closed loop systems using Lyapunov-Krasovskii functionals.

4) The contribution [21] is devoted to stabilization problems for time-varying linear systems with constant input delays. The reduction model approach we proposed ensures a robustness property (input-to-state stability) with respect to additive uncertainties, under arbitrarily long delays. It applies to rapidly time-varying systems, and gives a lower bound on the admissible rapidness parameters. We also covered slowly time-varying systems, including upper bounds on the allowable slowness parameters. We illustrated our work using a pendulum model.