• DOI:10.1093/imamci/dnq022
• Corpus ID: 11974785

Generalized invariance principles for switched delay systems

@article{Liu2011GeneralizedIP,
  title={Generalized invariance principles for switched delay systems},
  author={Jun Liu and Xinzhi Liu and Wei-Chau Xie},
  journal={IMA J. Math. Control. Inf.},
  year={2011},
  volume={28},
  pages={19-39}
}

• Jun Liu, Xinzhi Liu, W. Xie
• Published 1 March 2011
• Mathematics
• IMA J. Math. Control. Inf.

In many applications, the physical processes are governed by more than one dynamics in which the dynamic changes among a family of choices depending on the time t or the statex. Such processes are often described by switched systems or more generally hybrid systems, which have been studied extensively in recent years (see Goebelet al., 2009;Liberzon,2003;Shortenet al., 2007;van der Schaft & Schumacher,2000 and references therein). A good many of current studies on switched systems have been focused on finding conditions such that the systems are stable under arbitrary switching rules or under some kind of constrained switching rules (see Agrachev & Liberzon, 2001; Boscain, 2002;Chatterjee & Liberzon, 2006;Geromel & Colaneri,2006;Liberzon,2003;Masonet al., 2006). Besides switching properties, real processes always involve time delay, i.e., the future state of the system depends not only on the present state but also on the past states. In many instances, time-delay systems (or functional differential equations) are more appropriate models to describe practical processes (see Hale & Lunel, 1993). However, the fact that the future state of a time-delay system depends not only on its current state but also on its past trajectories brings more difficulty to its stability analysis (see Chapter 5 ofHale & Lunel, 1993and references therein). Since the classical LaSalle’s invariance principle (see LaSalle,1960,1968) has been proved to be a powerful tool for the stability analysis of autonomous ordinary differential equations, numerous extensions of the original invariance principle have been derived for various differential systems. In the 1960s, Hale (1965) extended the invariance principle to functional differential equations in which Lyapunov functionals were employed to discuss the extensions of LaSalle’s invariance principles. In the 1980s,