WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 15, 2016
Exponential Stabilization of 1-d Wave Network with One Circuit
Authors: ,
Abstract: In this paper, we consider a complex network of strings. Suppose that the network is comprised of eight strings with a fixed vertex, and other exterior vertices that are imposed velocity feedback controller. The displacement is not continuous at one interior node, while at the other interior nodes continuity holds and the force is not balanced at all interior nodes. We design controllers for the nodes with discontinuous displacement and with unbalanced force. We show that the operator determined by the closed loop system has a compact resolvent and generates C0 semigroup in an appropriate Hilbert space. Under certain condition, we prove that the closed loop system is asymptotically stable. We also show that there is a sequence of generalized eigenvectors of the system operator, which forms a Riesz basis. Hence the spectrum determined growth condition holds. If the imaginary axis is not an asymptote of the spectrum, then the closed loop system is exponentially stable.
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Pages: 13-22
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #2