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In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram's lemma and semigroup theory. The validity of cutting-edge method is proved by spectral analysis approach. In particular, we give a detailed procedure of cutting-edge for the bush-type wave networks. The results show that if we impose feedback controllers, consisting of velocity and position terms, at all the boundary vertices and at most three velocity feedback controllers on the cycle, the system is asymptotically stabilized. Finally, some examples are given.
@article{COCV_2012__18_1_208_0,
author = {Zhang, Yaxuan and Xu, Genqi},
title = {Controller design for bush-type 1-d wave networks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {208--228},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010050},
mrnumber = {2887933},
zbl = {1243.35167},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010050/}
}
TY - JOUR AU - Zhang, Yaxuan AU - Xu, Genqi TI - Controller design for bush-type 1-d wave networks JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 208 EP - 228 VL - 18 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010050/ DO - 10.1051/cocv/2010050 LA - en ID - COCV_2012__18_1_208_0 ER -
%0 Journal Article %A Zhang, Yaxuan %A Xu, Genqi %T Controller design for bush-type 1-d wave networks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 208-228 %V 18 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010050/ %R 10.1051/cocv/2010050 %G en %F COCV_2012__18_1_208_0
Zhang, Yaxuan; Xu, Genqi. Controller design for bush-type 1-d wave networks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 208-228. doi: 10.1051/cocv/2010050
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