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IMA Journal of Mathematical Control and Information Advance Access originally published online on May 11, 2007
IMA Journal of Mathematical Control and Information 2008 25(1):111-121; doi:10.1093/imamci/dnm014
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Optimal boundary feedback stabilization of a string with moving boundary

Martin Gugat{dagger}

Lehrstuhl 2 für Angewandte Mathematik, Martensstrasse 3, 91058 Erlangen, Germany

{dagger} Email: gugat{at}am.uni-erlangen.de

Received on April 30, 2006; We consider a finite string that is fixed at one end and subject to a feedback control at the other end which is allowed to move. We show that the behaviour is similar to the situation where both ends are fixed: As long as the movement is not too fast, the energy decays exponentially and for a certain parameter in the feedback law it vanishes in finite time. We consider movements of the boundary that are continuously differentiable with a derivative whose absolute value is smaller than the wave speed. We solve a problem of worst-case optimal feedback control, where the parameter in the feedback law is chosen such that the worst-case Lp-norm of the space derivative at the fixed end of the string is minimized (p isin [1, {infty})). We consider the worst case both with respect to the initial conditions and with respect to the boundary movement. It turns out that the parameter for which the energy vanishes in finite time is optimal in this sense for all p.

Keywords: PDE constrained optimization; optimal control of PDEs; optimal boundary control; wave equation; feedback; moving boundary.


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