Implications of Sharp Trace Regularity Results on Boundary Stabilization of the System of Linear Elasticity

Abstract

Uniform exponential decay of solution is established for the elastodynamic system of elasticity using boundary feedback control. Energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces. Two primary goals are achieved: First, these results are proven without the imposition of strong geometric assumptions on the controlled portion of the boundary, thus extending earlier work which required that the domain be “star shaped.” Second, the feedback is only a function of velocity, as opposed to also containing the tangential derivative of the displacement, resulting in a physically viable feedback. Proof is based on the multiplier method and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system.