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.

[1]  Irena Lasiecka,et al.  Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations , 1993 .

[2]  Irena Lasiecka,et al.  Asymptotic Behavior with Respect to Thickness of Boundary Stabilizing Feedback for the Kirchoff Plate , 1994 .

[3]  Jacques-Louis Lions,et al.  Modelling Analysis and Control of Thin Plates , 1988 .

[4]  M. Horn.  Sharp trace regularity for the solutions of the equations of dynamic elasticity , 1996 .

[5]  R. Triggiani,et al.  Uniform stabilization of the wave equation with dirichlet-feedback control without geometrical conditions , 1992 .

[6]  J. Lagnese,et al.  Boundary Stabilization of Linear Elastodynamic Systems , 1983 .

[7]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[8]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[9]  E. Zuazua,et al.  A direct method for boundary stabilization of the wave equation , 1990 .

[10]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .

[11]  Fatiha Alabau,et al.  Boundary observability, controllability and stabilization of linear elastodynamic systems , 1999 .

[12]  L. Robbiano,et al.  La propriété du prolongement unique pour un système elliptique. Le système de Lamé , 1992 .

[13]  J. Lagnese.  Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary , 1990 .

[14]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[15]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .