Local input-to-state stability : Characterizations and counterexamples

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Mironchenko, Andrii:
Local input-to-state stability : Characterizations and counterexamples.
In: Systems & Control Letters. Vol. 87 (2016) . - pp. 23-28.
ISSN 1872-7956
DOI: https://doi.org/10.1016/j.sysconle.2015.10.014

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Abstract in another language

We show that a nonlinear locally uniformly asymptotically stable infinite-dimensional system is automatically locally input-to-state stable (LISS) provided the nonlinearity possesses some sort of uniform continuity with respect to external inputs. Also we prove that LISS is equivalent to existence of a LISS Lyapunov function. We show by means of a counterexample that if this uniformity is not present, then the equivalence of local asymptotic stability and local ISS does not hold anymore. Using a modification of this counterexample we show that in infinite dimensions a uniformly globally asymptotically stable at zero, globally stable and locally ISS system possessing an asymptotic gain property does not have to be ISS (in contrast to finite dimensional case).