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Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems
Author(s) -
Kobayashi Toshihiro
Publication year - 2004
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.543
Subject(s) - mathematics , control theory (sociology) , exponential stability , multiplier (economics) , exponential function , closed loop , function (biology) , stability (learning theory) , energy (signal processing) , mathematical analysis , nonlinear system , computer science , physics , control (management) , quantum mechanics , artificial intelligence , control engineering , evolutionary biology , biology , machine learning , engineering , economics , macroeconomics , statistics
In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.