STUDY OF STABILIZATION AND ENERGY DISSIPATION FOR SECOND ORDER VIBRATING SYSTEMS

In this dissertation, we are concerned with the stabilization and energy dissipation of second order vibrating systems. The main tolls in our work are linear and nonlinear functional analysis and the method of characteristics.;Next, we propose a model for nonlinear passive damping devices. We study the effects of nonlinear boundary stabilization and analyze the asymptotic behavior of solutions of such systems. We are able to determine the ;Finally, we prove the exponential stability of two-dimension vibrating membranes of annular shape with radial symmetry with dissipative boundary conditions.;We first study the placement of point stabilizers in a structure which is made up of two coupled strings. We point out an error in a recent paper by T. H. Qin and then use the corrected version along with Routh's stability criterion to derive a sufficient condition on feedback equations. We also apply a theorem of F. L. Huang to prove exponential stability for coupled linear equation with positive feedback gains. Our results indicate that installing a stabilizer on the boundary will provide robustness for exponential stability in both nonlinear and linear cases.