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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (5), P. 835–841

Control and inverse problems for networks of vibrating strings with attached masses

F. Al-Musallam – Department of Mathematics, Kuwait University, P.O Box 5969 Safat, Kuwait; musallam@sci.kuniv.edu.kw
S. Avdonin – Department of Mathematics and Statistics, University of Alaska, Fairbanks, AK 99775, USA; s.avdonin@alaska.edu
N. Avdonina – Department of Mathematics and Statistics, University of Alaska, Fairbanks, AK 99775, USA; navdonina@alaska.edu
J. Edward – Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA; edwardj@fiu.edu

We consider the control and inverse problems for serially connected and tree-like networks of strings with point masses loaded at the internal vertices. We prove boundary controllability of the systems and the identifiability of varying coefficients of the string equations along with the complete information on the graph, i.e. the loaded masses, the lengths of the edges and the topology (connectivity) of the graph. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincaré operator for the dynamic wave equation on the tree. The general result is obtained by the leaf peeling method which reduces the inverse problem layer-by-layer from the leaves to the fixed root of the tree.

Keywords: wave equation on graphs, inverse problem, boundary control.

DOI 10.17586/2220-8054-2016-7-5-835-841

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