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Abstract
The theory of beams and plates has been long established due to works spanning many fields, and has been explored through many investigations of beam and plate mechanics, controls, stability, and the well-posedness of systems of equations governing the motions of plates and beams. Additionally, recent investigations of flutter phenomena by Dowell, Webster et al. have reignited interest into the mechanics and stability of nonlinear beams. In this thesis, we wish to revisit the seminal well-posedness results of Lagnese and Leugering for the one dimensional, nonlinear beam from their 1991 paper, “Uniform stabilization of a nonlinear beam by nonlinear boundary feedback.” To date, these remarkable results of Lagnese and Leugering are the only such well-posedness treatments for this particular model. We will provide a modified version of the proofs of well-posedness which will also elucidate details omitted in the original proofs, and we will place the treatment of the nonlinear coupling in the now-standard context of locally Lipschitz perturbations of m-accretive generators. Additionally, the second part of this thesis is dedicated to numerical experiments that illustrate stability and controllability properties of this nonlinear problem.
