Gugat M (2015)
Publication Language: English
Publication Type: Journal article, Original article
Publication year: 2015
Publisher: Society for Industrial and Applied Mathematics
Book Volume: 53
Pages Range: 526-546
Journal Issue: 1
DOI: 10.1137/140977023
We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required L2-regularity for the initial position and H-1-regularity for the initial velocity, in this paper we allow for initial positions with L1-regularity and initial velocities in W-1, 1 on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the (L1, W-1,1) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined L1-energy. For more regular initial states with (L2, H-1) regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with H1-regularity of the initial position and L2-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy.
APA:
Gugat, M. (2015). EXPONENTIAL STABILIZATION OF THE WAVE EQUATION BY DIRICHLET INTEGRAL FEEDBACK. SIAM Journal on Control and Optimization, 53(1), 526-546. https://dx.doi.org/10.1137/140977023
MLA:
Gugat, Martin. "EXPONENTIAL STABILIZATION OF THE WAVE EQUATION BY DIRICHLET INTEGRAL FEEDBACK." SIAM Journal on Control and Optimization 53.1 (2015): 526-546.
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