Section: New Results

Trajectory based approach

Participants : Frederic Mazenc [Disco] , Silviu Niculescu [Disco] , Michael Malisoff [LSU] .

In the work [22] , we provided a new stability analysis technique, which is based on the study of the behaviors of the solutions over any interval $[t-T,T]$, where $t$ represents the time and $T$ is an appropriately chosen constant. Thus trajectory-based approach is completely new in the sense that it neither reply on Lyapunov functions nor on the small gain theorem. One of its most striking feature is that it applies to a broad number of systems (systems with delay, continuous/discrete systems, ODE coupled with difference equations).

In [30] and [55] we provided several significant applications of the main result of [22] . In [30] in two results, we use a Lyapunov function for a corresponding undelayed system to provide a new method to prove stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provided in [55] a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provided an alternative to the reduction model method, based on a different dynamic extension.