Forward completeness and bounded reachability sets for time-delay systems: the role of the state space

Abstract: A dynamical system is forward complete (FC) if its solutions are well-defined for all future times. If moreover solutions initialized in bounded sets and corresponding to bounded inputs remain uniformly bounded over bounded time-intervals, then it is said to have the bounded reachability sets (BRS) property, also called robust forward completeness (RFC). BRS is a bridge between the pure well-posedness theory (that studies existence and uniqueness) and the stability theory (which is interested in establishing certain bounds for solutions). For example, BRS is a crucial property to establish ISS superposition theorems. It is known that under standard assumptions, FC implies BRS for finite-dimensional systems, while this fails for infinite-dimensional ones. It was recently shown by J.L. Mancilla-Aguilar and H. Haimovich that the implication also fails in the case of time-delay systems with a finite number of delays. In this talk, we aim to show that the implication FC => BRS for time-delay systems actually heavily depends on the choice of the state space. In particular, we propose to consider the state space of essentially bounded functions (instead of the usual one of continuous functions). On this new state space, we prove that FC implies BRS, which raises the question of revisiting stability theory of time-delay systems in this state space.

Biography. Lucas Brivadis is a CNRS researcher at the Laboratory of Signals and Systems (L2S, CentraleSupélec, CNRS, Université Paris-Saclay). He defended his PhD in 2021 at LAGEPP, Université Lyon 1. From 2021 to 2022, he was a postdoctoral researcher at L2S. His research interests include stability properties of nonlinear delay systems, observer design for nonlinear or infinite dimensional systems, and output feedback stabilization.

systems and controlanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

( slides | video )

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Input-to-State Stability and its Applications

Series comments: This is a seminar for the exchange of ideas in input-to-state stability (ISS) theory and related fields.

The scope of the Seminar includes but is not limited to

- ISS for finite-dimensional systems (ODEs, hybrid, impulsive, switched, discrete-time systems),

- Infinite-dimensional ISS theory (PDEs, evolution equations in Banach spaces, time-delay systems, infinite networks)

- Applications to robust control and observation, nonlinear control, network analysis, etc.

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