Set-Induced Stability Results for Delay Difference Equations

Rob H. Gielen 1 Mircea Lazar 1 Sorin Olaru 2, 3
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : This chapter focuses on the relation between stability of delay difference equations (DDEs) and the existence of D -contractive sets. Such sets are of importance as they provide a region of attraction, which is difficult to obtain for delay systems. Firstly, it is established that a DDE admits a D -contractive set if and only if it admits a Lyapunov-Razumikhin function. However, it is also shown that there exist stable DDEs that do not admit a D -contractive set. Therefore, secondly, further necessary conditions for the existence of a D -contractive set are established. These necessary conditions provide a first step towards the derivation of a notion of asymptotic stability for DDEs which is equivalent to the existence of a D -contractive set.
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Submitted on : Monday, November 12, 2012 - 4:23:57 PM
Last modification on : Thursday, April 5, 2018 - 12:30:14 PM

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Rob H. Gielen, Mircea Lazar, Sorin Olaru. Set-Induced Stability Results for Delay Difference Equations. Rifat Sipahi ; Tomáš Vyhlída ; Silviu-Iulian Niculescu ; Pierdomenico Pepe. Time Delay Systems: Methods, Applications and New Trends, 423, Springer, pp.73-84, 2012, Lecture Notes in Control and Information Sciences, ⟨10.1007/978-3-642-25221-1_6⟩. ⟨hal-00750901⟩

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