Input-to-state dynamical stability and its Lyapunov function characterization

L. Grüne: Input-to-state dynamical stability and its Lyapunov function characterization
IEEE Transactions on Automatic Control 47, 1499 - 1504, 2002

DOI: 10.1109/TAC.2002.802761
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Abstract:

We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class KL function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems.

The preprint that was available until January 25, 2002 (carrying the date May 16, 2001) had a systematic typo in several formulas:
each inequality of the type
DV(x)f(x,u) <= -g(x) (like, e.g., in formula (3.4))
has to be replaced by
DV(x)f(x,u) <= -g(V(x))
This error was due to multiple copying and pasting an erroneous formula and has no impact on the proofs and results in this paper, except for some minor changes in the statement and proof of Theorem 3.3.

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