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EN FR
DISCO - 2017
Overall Objectives
New Software and Platforms
Bilateral Contracts and Grants with Industry
Bibliography
Overall Objectives
New Software and Platforms
Bilateral Contracts and Grants with Industry
Bibliography


Section: New Results

Stability of time-varying systems with delay and Switched Nonlinear Systems

Participants : Frederic Mazenc, Hitay Ozbay [Blikent University, Turkey] , Saeed Ahmed [Blikent University, Turkey] , Silviu Niculescu, Michael Malisoff [LSU, USA] .

Switched systems is a family of systems which is frequently encountered in practice and can be used to approximate time-varying systems to ease their stability analysis or control. In the two works [20] and [17], we provided results that are useful when it comes to analyze the stability of time-varying or switched systems with delay. In [20] we provided several significant applications of the trajectory approach developed recently by Mazenc and Malisoff. In two results, we used a Lyapunov function for a corresponding undelayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. Our main results used upper bounds on an integral average involving the delay. We also provided 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.

Three of our other works are devoted to switched systems. In [55] and [21], a new technique is proposed to ensure global asymptotic stability for nonlinear switched time-varying systems with time-varying discontinuous delays. It uses an adaptation of Halanay's inequality to switched systems and the trajectory based technique mentioned above. The result is applied to a family of linear time-varying systems with time-varying delays. In [53], we presented an extension of the trajectory based approach mentioned above for state feedback stabilization of switched linear continuous-time systems with a time-varying input delay. In contrast with finding classical common Lyapunov function or multiple Lyapunov functions for establishing the stability of the closed-loop switched system, the new trajectory based approach relies on verifying certain inequalities along the solution of a supplementary system. This study does not make any assumption regarding the stabilizability of all of the constituent modes of the switched system. Moreover, no assumption is needed about the differentiability of the delay and no constraint is imposed on the upper bound of the delay derivative.

In [17], we proved extensions of the celebrate Razumikhin's theorem for a general family of time-varying continuous and discrete-time nonlinear systems. Our results include a novel "strictification" technique for converting a nonstrict Lyapunov function into a strict one. We also provided new constructions of Lyapunov-Krasovskii functionals that can be used to prove robustness to perturbations. Our examples include a key model from identification theory, and they show how our method can sometimes allow broader classes of delays than the results in the literature.