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

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

2012-02-15

arxiv.org/abs/1202.3325v1

Mathematics

Optimization and Control

29 pages

Scientific paper

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 input-to-state stability of a system. Then for the case of the 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 which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide the way to construct an ISS-Lyapunov function for an entire interconnection, if a ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate theory on examples of linear and semilinear reaction-diffusion equations.

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