On the limits of stabilizability for networks of strings
Gugat M, Gerster S (2019)
Publication Type: Journal article
Publication year: 2019
Journal
Book Volume: 131
Article Number: 104494
DOI: 10.1016/j.sysconle.2019.104494
Abstract
An example by Bastin and Coron illustrates that the boundary stabilization of 1-d hyperbolic systems with certain source terms is only possible if the length of the space interval is sufficiently small. We show that related phenomena also occur for networks of vibrating strings that are governed by the wave equation with a certain source term. It turns out that for a tree of strings with Neumann velocity feedback control at one boundary node and a homogeneous Dirichlet boundary condition at at least one boundary node and homogeneous Dirichlet or Neumann conditions at the other boundary nodes, boundary feedback stabilization is not possible if one of the strings is sufficiently long. However, if the number of strings in the tree is sufficiently large, also for arbitrarily short strings for certain parameters in the source term stabilization is not possible. The wave equation with source term that we consider is equivalent to a certain 2 ×2 system. For the examples that illustrate the limits of stabilizability, the matrix of the source term is not positive definite. However if the system parameters are chosen in such a way that the matrix is positive semi-definite, the tree of strings can be stabilized exponentially fast by the boundary feedback control for arbitrary long space intervals.
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Germany (DE)How to cite
APA:
Gugat, M., & Gerster, S. (2019). On the limits of stabilizability for networks of strings. Systems & Control Letters, 131. https://dx.doi.org/10.1016/j.sysconle.2019.104494
MLA:
Gugat, Martin, and Stephan Gerster. "On the limits of stabilizability for networks of strings." Systems & Control Letters 131 (2019).
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