Please don't forget about today's talk of the
3rd Online Seminar on Input-to-State Stability and its Applications
Speaker: Fabian Wirth (University of Passau, Germany)
Title: The Characterization of ISS for time-delay systems: Results and Counterexamples
Date: Wednesday, 31 January 2024, at 17:00 (Time zone of Amsterdam, Berlin, Paris, Rome, Vienna)
Zoom Link: 自己搜
Abstract: We consider a general class of time-delay systems with inputs. While this defines a class of infinite-dimensional systems, it has been observed for a long time that the analytic properties of such systems are more benign than in the general infinite-dimensional case. To give an example, it is a classic result that for systems without input asymptotic stability of a fixed point implies this property with a rate that is uniform in the state space. As the state space is not locally compact this is by no means as self-evident as for ODEs.
On the other hand, it is also known that the infinite-dimensional character of time-delay systems results in properties that are not possible for systems of ordinary differential equations. For instance, in a recent paper, Mancilla-Aguilar and Haimovich have shown that forward completeness does not imply boundedness of finite-time reachability sets (the BRS property, for short). Even more strikingly, the example provided is even globally asymptotically stable and uniformly globally attractive in the fixed point zero.
All this touches upon the characterization of input-to-state stability for time-delay systems in terms of other dynamic properties of the system. The properties that have turned out to be important are
stability properties: (uniform) local stability (LS/ULS), the (uniform) limit property (LIM/ULIM), describing the long term behaviour of trajectories, (uniform) asymptotic gains (AG/UAG), describing asymptotic bounds on trajectories.
For finite-dimensional control systems, it has been shown by Sontag and Wang that we have the equivalence
ISS <=> LIM & LS <=> AG & LS
In contrast for a broad class of nonlinear infinite-dimensional systems, it has been shown that stronger requirements are needed, namely,
ISS <=> ULIM & ULS & BRS <=> UAG & CEP & BRS
And in the context of the general infinite-dimensional class this characterization cannot be relaxed. However, delay systems are more benign so that it possible to relax the characterization to
ISS <=> LIM & 0-ULS & BRS <=> AG & 0-ULS & BRS
On the other hand the example of Mancilla-Aguilar and Haimovich may be extended to show that, in the absence of BRS, LIM does not imply ULIM. Also global asymptotic stability does not imply uniform global attractivity. Thus the requirements on bounded reachability sets and uniformity properties of the stability requirements cannot be removed in the time-delay case.
joint work with
Lucas Brivadis, Université Paris-Saclay, CNRS, CentraleSupélec, LSS, France,
Antoine Chaillet, Université Paris-Saclay, CNRS, CentraleSupélec, LSS, France,
Andrii Mironchenko, University of Klagenfurt, Austria and University of Passau, Germany.
Biography: Fabian Wirth received his PhD from the Institute of Dynamical Systems at the University of Bremen in 1995. He has since held positions in Bremen, at the Centre Automatique et Systèmes of Ecole des Mines, the Hamilton Institute at NUI Maynooth, Ireland, the University of Würzburg and IBM Research Ireland. He now holds the chair for Dynamical Systems at the University of Passau in Germany. His current interests include stability theory, switched systems, the joint spectral radius of matrix sets and large scale networks with applications to networked systems and distributed control.
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