Controllability for a string with attached masses and Riesz bases for asymmetric spaces

Sergei Avdonin; Julian Edward

doi:10.3934/mcrf.2019021

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2019, Volume 9, Issue 3: 453-494. Doi: 10.3934/mcrf.2019021

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Controllability for a string with attached masses and Riesz bases for asymmetric spaces

Sergei Avdonin^1, and
Julian Edward^2, ,

1.
Department of Mathematics and Statistics, University of Alaska at Fairbanks, Fairbanks, AK 99775, USA
2.
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

^* Corresponding author
^* Corresponding author

Received: March 31, 2017

Revised: August 31, 2018

Published: September 2019

The research of Sergei Avdonin was supported in part by the National Science Foundation, grant DMS 1411564 and by the Ministry of Education and Science of Republic of Kazakhstan, grant no. AP05136197.

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Abstract

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control of Dirichlet, or Neumann, or mixed type is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for an $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results. The results are sharp with respect to both the regularity of the solution and with respect to time. The eigenfunctions of the associated Sturm--Liouville problem are used to construct Riesz bases for a family of asymmetric spaces that include the sets of reachable positions and velocities.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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