Point-wise dissipation in time-delay systems: recent results and open questions

Abstract: In the existing characterizations of input-to-state stability (ISS) for time-delay systems, the Lyapunov-Krasovskii functional (LKF) has a $\mathcal K_\infty$ dissipation rate that involves the whole LKF itself (LKF-wise dissipation) or even the supremum norm of the state history (history-wise dissipation). A similar characterization holds for integral input-to-state stability (iISS), in which the dissipation rate is just a positive definite function. These characterizations have allowed to extend several results on ISS and iISS from finite dimension to time-delay systems.

Nevertheless, in practice, obtaining a LKF-wise or history-wise dissipation is not always an easy task and often resorts to rather artificial tricks. More crucially, in the absence of inputs, it is known from the work of N. Krasovskii that a dissipation involving merely the current value of the state norm (point-wise dissipation) is enough to guarantee global asymptotic stability.

In this talk, we investigate whether a point-wise dissipation suffices to conclude ISS or iISS for time-delay systems. We give a positive answer to this question for iISS. More precisely, we show that point-wise, LKF-wise and history-wise dissipations through a positive definite function all ensure iISS.

For ISS, despite strong efforts, this question remains open: it has not yet be proved or disproved that ISS is equivalent to the existence of a point-wise dissipation. We identify two classes of systems for which this is the case, by imposing a growth restriction either on the upper bound of the LKF or on the vector field. We also provide some insights on what can be said about a system having a point-wise dissipation to hopefully foster some creative discussion.

Finally, while asymptotic stability is known for long to be equivalent to a point-wise dissipation for input-free systems, this question remains open for exponential stability. We show that, at least for systems ruled by a globally Lipschitz vector field, global exponential stability is guaranteed under a point-wise dissipation.

systems and controlanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

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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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