Control via decoupling of a class of second order linear hybrid systems

David L. Russell

doi:10.3934/dcdss.2014.7.1321

Article Contents

2014, Volume 7, Issue 6: 1321-1334. Doi: 10.3934/dcdss.2014.7.1321

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Control via decoupling of a class of second order linear hybrid systems

David L. Russell^1,

1.
Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24060

Received: March 31, 2013

Revised: October 31, 2013

Published: November 2014

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Abstract

We study a terminal state control (reachability) problem for certain elastic systems of ``hybrid" type consisting of a single space dimension distributed parameter part coupled, at one endpoint of the relevant spatial, $x$, interval, to a lumped mass component. Two such systems are studied in detail. The first is a vibrating string system fixed at $x = 0$ and attached to a point mass at the right hand endpoint $x = L$. The second example concerns an Euler - Bernoulli beam system ``clamped" at $x = 0$ and attached, at $x = L$, to a mass with both translational and rotational inertia. In both cases the controls act on the mass, which is modeled by a finite dimensional system of differential equations. Analysis of the reachability problem is facilitated by a preliminary ``feedback type" transformation of the control variable which decouples the point mass from the distributed system. In both examples a concluding analysis is required to show that the component of the control generated by feedback lies in the same space as the originally applied control.

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Mathematics Subject Classification: Primary: 74C05, 74K10, 74K20, 90C25, 93D15.

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