Section: New Results

Stabilization of various systems with pointwise delays

Participants : Frederic Mazenc, Michael Malisoff [LSU] , Delphine Bresch-Pietri [Mines Paris Tech.] , Nicolas Petit [Mines Paris Tech.] , Robledo Gonzalo [Univ. de Chile, Chile] , Maruthi Akella [University of Texas, USA] , Xi-Ming Sun [Dalian University of Technology, China] , Xue-Fan Wang [Dalian University of Technology, China] .

The presence of delays to big for being neglected is an obstacle to the design of stabilizing controllers in many cases. We have made efforts to overcome this challenge by developping several techniques.

In the paper [14], we investigated the design of a prediction-based controller for a linear system subject to a problematic time-varying input delay: the delay we considered is not necessarily "First-In/First-Out". The feedback law we proposed uses the current delay value in the prediction. It does not exactly compensate the delay in the closed-loop dynamics but does not require to predict future delay values, contrary to classical prediction techniques. Modeling the input delay as a transport Partial Differential Equation, we proved asymptotic stabilization of the system state, provided that the average $L_2$-norm of the first derivative of the delay over some time-window is sufficiently small and that the average time between two discontinuities (average dwell time) is sufficiently large.

In the paper [51], we adopted another type of strategy: we used a new sequential predictors approach to build uniformly globally exponentially stabilizing feedback controls for a large class of linear time-varying systems that contain an arbitrary number of different delays. This allows different delays in different components of the input. We illustrated our work in an example from identification theory, and in an Euler-Lagrange system arising from two-link manipulator systems.

The paper [31] continues our works on the chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allowed delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derived bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. In the presence of delays and uncertainties, we provided bounds on the delays and on the uncertainties that ensure, with respect to uncertainties, the robustness property called "input-to-state stability".