A Vibrating String with Shuttle-Like Point Dampers and Related Observability Properties

Alexander Y. Khapalov
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Alexander Y. Khapalov: Washington State University, Department of Mathematics and Statistics

Chapter Chapter 12 in Mobile Point Sensors and Actuators in the Controllability Theory of Partial Differential Equations, 2017, pp 203-222 from Springer

Abstract: Abstract In this chapter we consider a vibrating string equipped with a single mobile vertical tie-down viscous point damper. We show that when such a damper bounces between the endpoints on the string with a speed exceeding the speed of propagation, it stabilizes the string at hand. On the contrary, as we pointed it out in Chap. 11 , a static point feedback is principally ill-posed with respect to its position on the string. We link our discussion to the issue of observability with similarly moving point sensors measuring the displacement of the string from its state of rest. We also show that, if the aforementioned point damper bounces between any two given points on the string, the same stabilization result holds if it is complemented by a certain (conventional) locally distributed damper with arbitrarily small support.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-60414-5_12

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DOI: 10.1007/978-3-319-60414-5_12

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