Input-to-state stability of infinite-dimensional control systems

Titelangaben

Dashkovskiy, Sergey ; Mironchenko, Andrii:
Input-to-state stability of infinite-dimensional control systems.
In: Mathematics of Control, Signals, and Systems. Bd. 25 (2013) Heft 1 . - S. 1-35.
ISSN 0932-4194
DOI: https://doi.org/10.1007/s00498-012-0090-2

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Abstract

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.