A Lyapunov-based small-gain approach to ISS of infinite nonlinear networks

Abstract: In this talk, I present an approach to the verification of input-to-state stability for networked control systems composed of a countably infinite number of nonlinear subsystems. The essential requirements on these subsystems are that they are finite-dimensional, continuous in time and each of them is influenced only by finitely many other subsystems. Assuming that each subsystem admits an ISS Lyapunov function with respect to both internal inputs (influences from other subsystems) and external inputs (control inputs), our result provides sufficient conditions for the existence of an ISS Lyapunov function for the whole network. This Lyapunov function is built from the Lyapunov functions of the subsystems and it is important to note that the ISS estimates for the later are given in the max-type formulation. This formulation allows for the definition of an associated max-type gain operator Gamma, encoding the influence of the subsystems on each other via nonlinear gain functions. The operator Gamma acts as a monotone operator on the positive cone of \ell_{\infty}. The essential requirement on Gamma is that it admits a so-called path of strict decay, a condition which is known to be equivalent to the classical small-gain condition in the case of finite networks. For infinite networks, however, this equivalence does not hold. Still, as in finite dimensions, the existence of a path of strict decay is linked to the stability properties of the discrete-time system generated by the gain operator. In my talk, I will try to explain the difficulties involved with the stability analysis of this system.

Joint work with Andrii Mironchenko and Majid Zamani

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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