TY - JOUR A1 - Gugat, Martin ED - Bloch, Anthony M. T1 - Exponential Stabilization of the Wave Equation by Dirichlet Integral Feedback JF - SIAM Journal on Control and Optimization (SICON) N2 - 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. KW - exponential stability KW - Dirichlet boundary control KW - energy decay KW - exact control Y1 - 2016 U6 - https://doi.org/DOI: 10.1137/140977023 VL - 53 IS - 1 SP - 526 EP - 546 ER -