{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T02:55:20Z","timestamp":1787712920777,"version":"build-2784847793"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>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 $L^2$-regularity for the initial position and $H^{-1}$-regularity for the initial velocity, in this paper we allow for initial positions with $L^1$-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 ($L^1$, $W^{-1,1}$) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined $L^1$-energy. For more regular initial states with $(L^2, \\,H^{-1})$ 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 $H^1$-regularity of the initial position and $L^2$-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy.<\/jats:p>","DOI":"10.1137\/140977023","type":"journal-article","created":{"date-parts":[[2015,2,24]],"date-time":"2015-02-24T12:47:24Z","timestamp":1424782044000},"page":"526-546","source":"Crossref","is-referenced-by-count":6,"title":["Exponential Stabilization of the Wave Equation by Dirichlet Integral Feedback"],"prefix":"10.1137","volume":"53","author":[{"given":"Martin","family":"Gugat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,2,24]]},"reference":[{"issue":"1747","key":"atypb1","first-page":"214","volume":"3","author":"Alembert J.-B.","journal-title":"Mem. Acad. Sci. 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