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

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...

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Bibliographic Details
Published in:Mathematics of control, signals, and systems Vol. 25; no. 1; pp. 1 - 35
Main Authors: Dashkovskiy, Sergey, Mironchenko, Andrii
Format: Journal Article
Language:English
Published: London Springer-Verlag 01.03.2013
Springer Nature B.V
Subjects:
Approximation
Banach space
Banach spaces
Communications Engineering
Construction
Control
Control systems
Interconnections
International Space Station
Investigations
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Mechatronics
Networks
Nonlinear equations
Original Article
Partial differential equations
Robotics
Stability
Systems Theory
Nonlinear control systems
Infinite-dimensional systems
Input-to-state stability
Lyapunov methods
Linearization
ISSN:0932-4194, 1435-568X
Online Access:Get full text
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Summary: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.
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ISSN:0932-4194
1435-568X
DOI:10.1007/s00498-012-0090-2