Local input-to-state stability: Characterizations and counterexamples |
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Affiliation: | 1. College of Information and Control Engineering, Weifang University, Weifang 261061, China;2. Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China;3. Department of Engineering, Faculty of Technology and Science, University of Agder, N-4898 Grimstad, Norway |
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Abstract: | 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). |
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Keywords: | Nonlinear control systems Infinite-dimensional systems Input-to-state stability Lyapunov methods |
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